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- Question : 1E - If f sxd ? x 1 s2 2 x and tsud ? u 1 s2 2 u , is it true that f ? t?
- Question : 2E - If f sxd ? x 2 2 x x 2 1 and tsxd ? x is it true that f ? t?
- Question : 3E - The graph of a function f is given. (a) State the value of f s1d. (b) Estimate the value of f s21d. (c) For what values of x is f sxd ? 1? (d) Estimate the value of x such that f sxd ? 0. (e) State the domain and range of f. (f) On what interval is f increasing? y 0 x
- Question : 4E - The graphs of f and t are given. (a) State the values of f s24d and ts3d. (b) For what values of x is f sxd ? tsxd?(c) Estimate the solution of the equation f sxd ? 21. (d) On what interval is f decreasing? (e) State the domain and range of f. (f) State the domain and range of t.
- Question : 5E - Figure 1 was recorded by an instrument operated by the California Department of Mines and Geology at the University Hospital of the University of Southern California in Los Angeles. Use it to estimate the range of the vertical ground acceleration function at USC during the Northridge earthquake.
- Question : 6E - In this section we discussed examples of ordinary, everyday functions: Population is a function of time, postage cost is a function of weight, water temperature is a function of time. Give three other examples of functions from everyday life that are described verbally. What can you say about the domain and range of each of your functions? If possible, sketch a rough graph of each function.
- Question : 7E - Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. In fig.
- Question : 8E - Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. In fig.
- Question : 9E - Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. In fig.
- Question : 10E - Determine whether the curve is the graph of a function of x. If it is, state the domain and range of the function. In fig.
- Question : 11E - Shown is a graph of the global average temperature T during the 20th century. Estimate the following. (a) The global average temperature in 1950 (b) The year when the average temperature was 14.2
- Question : 12E - Trees grow faster and form wider rings in warm years and grow more slowly and form narrower rings in cooler years. The fgure shows ring widths of a Siberian pine from 1500 to 2000. (a) What is the range of the ring width function? (b) What does the graph tend to say about the temperature of the earth? Does the graph re?ect the volcanic eruptions of the mid-19th century?
- Question : 13E - You put some ice cubes in a glass, fll the glass with cold water, and then let the glass sit on a table. Describe how the temperature of the water changes as time passes. Then sketch a rough graph of the temperature of the water as a function of the elapsed time
- Question : 14E - Three runners compete in a 100-meter race. The graph depicts the distance run as a function of time for each runner. Describe in words what the graph tells you about this race. Who won the race? Did each runner fnish the race?
- Question : 15E - The graph shows the power consumption for a day in September in San Francisco. (P is measured in megawatts; t is measured in hours starting at midnight.) (a) What was the power consumption at 6 am? At 6 pm? (b) When was the power consumption the lowest? When was it the highest? Do these times seem reasonable?
- Question : 16E - Sketch a rough graph of the number of hours of daylight as a function of the time of year.
- Question : 17E - Sketch a rough graph of the outdoor temperature as a function of time during a typical spring day.
- Question : 18E - Sketch a rough graph of the market value of a new car as a function of time for a period of 20 years. Assume the car is well maintained.
- Question : 19E - Sketch the graph of the amount of a particular brand of coffee sold by a store as a function of the price of the coffee.
- Question : 20E - You place a frozen pie in an oven and bake it for an hour. Then you take it out and let it cool before eating it. Describe how the temperature of the pie changes as time passes. Then sketch a rough graph of the temperature of the pie as a function of time.
- Question : 21E - A homeowner mows the lawn every Wednesday afternoon. Sketch a rough graph of the height of the grass as a function of time over the course of a four-week period.
- Question : 22E - An airplane takes off from an airport and lands an hour later at another airport, 400 miles away. If t represents the time in minutes since the plane has left the terminal building, let xstd be the horizontal distance traveled and ystd be the altitude of the plane. (a) Sketch a possible graph of xstd. (b) Sketch a possible graph of ystd (c) Sketch a possible graph of the ground speed. (d) Sketch a possible graph of the vertical velocity
- Question : 23E - Temperature readings T (in
- Question : 24E - Researchers measured the blood alcohol concentration (BAC) of eight adult male subjects after rapid consumption of 30 mL of ethanol (corresponding to two standard alcoholic drinks). The table shows the data they obtained by averaging the BAC (in gydL) of the eight men. (a) Use the readings to sketch the graph of the BAC as a function of t. (b) Use your graph to describe how the effect of alcohol varies with time. t (hours) BAC t (hours) BAC 0 0 1.75 0.022 0.2 0.025 2.0 0.018 0.5 0.041 2.25 0.015 0.75 0.040 2.5 0.012 1.0 0.033 3.0 0.007 1.25 0.029 3.5 0.003 1.5 0.024 4.0 0.001
- Question : 25E - If f sxd ? 3x 2 2 x 1 2, fnd f s2d, f s22d, f sad, f s2ad, f sa 1 1d, 2f sad, f s2ad, f sa2d, [ f sad]2, and f sa 1 hd.
- Question : 26E - A spherical balloon with radius r inches has volume Vsrd ? 4 3 r 3. Find a function that represents the amount of air required to in?ate the balloon from a radius of r inches to a radius of r 1 1 inches
- Question : 27E - Evaluate the difference quotient for the given function. Simplify your answer. f sxd ? 4 1 3x 2 x 2, f s3 1 hd 2 f s3d
- Question : 28E - f sxd ? x 3, f sa 1 hd 2 f sad
- Question : 29E - f sxd ? 1 x , f sxd 2 f sad/x 2 a
- Question : 30E - f sxd ? x 1 3 x 1 1 , f sxd 2 f s1d x 2 1
- Question : 31E - Find the domain of the function. f sxd ? x 1 4 x 2 2 9
- Question : 32E - f sxd ? x 221 x 3 x 22 5 6
- Question : 33E - f std ? s 3 2t 2 1
- Question : 34E - tstd ? s3 2 t 2 s2 1 t
- Question : 35E - hsxd ? 1 s 4 x 2 2 5x
- Question : 36E - f sud ? u 1 1 1 1 1 u 1 1
- Question : 37E - Fspd ? s2 2 sp
- Question : 38E - Find the domain and range and sketch the graph of the function hsxd ? s4 2 x 2 .
- Question : 39E - Find the domain and sketch the graph of the function.f sxd ? 1.6x 2 2.4
- Question : 40E - tstd ? t 2 2 1 t 1 1
- Question : 41E - Evaluate f s23d, f s0d, and f s2d for the piecewise defned function. Then sketch the graph of the function. 41. f sxd ?Hx 1 1 2 2 x if if x x , > 0 0
- Question : 42E - f sxd ?H3 2x2 2 1 2 5 x if if x x , > 2
- Question : 43E - f sxd ?Hx x 21 1 if if x x < 2 . 21 1
- Question : 44E - f sxd ?H2 7 2 1 2x if if x x < . 1 1
- Question : 45E - Sketch the graph of the function. f sxd ? x 1 | x |
- Question : 46E - f sxd ? | x 1 2 |
- Question : 47E - tstd ? |1 2 3t |
- Question : 48E - hstd ? | t | 1 | t 1 1|
- Question : 49E - f sxd ?H| 1x | if if || x x || < . 1
- Question : 50E - tsxd ? || x | 2 1|
- Question : 51E - Find an expression for the function whose graph is the given curve. The line segment joining the points s1, 23d and s5, 7d
- Question : 52E - The line segment joining the points s25, 10d and s7, 210
- Question : 53E - The bottom half of the parabola x 1 sy 2 1d2 ? 0
- Question : 54E - The top half of the circle x 2 1 sy 2 2d2 ? 4
- Question : 55E - FIG.
- Question : 56E - FIG.
- Question : 57E - Find a formula for the described function and state its domain. A rectangle has perimeter 20 m. Express the area of the rectangle as a function of the length of one of its sides.
- Question : 58E - A rectangle has area 16 m2. Express the perimeter of the rectangle as a function of the length of one of its sides
- Question : 59E - Express the area of an equilateral triangle as a function of the length of a side.
- Question : 60E - A closed rectangular box with volume 8 ft3 has length twice the width. Express the height of the box as a function of the width.
- Question : 61E - An open rectangular box with volume 2 m3 has a square base. Express the surface area of the box as a function of the length of a side of the base
- Question : 62E - A Norman window has the shape of a rectangle surmounted by a semicircle. If the perimeter of the window is 30 ft, express the area A of the window as a function of the width x of the window
- Question : 63E - A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 20 in. by cutting out equal squares of side x at each corner and then folding up the sides as in the fgure. Express the volume V of the box as a function of x.
- Question : 64E - A cell phone plan has a basic charge of $35 a month. The plan includes 400 free minutes and charges 10 cents for each additional minute of usage. Write the monthly cost C as a function of the number x of minutes used and graph C as a function of x for 0 < x < 600.
- Question : 65E - In a certain state the maximum speed permitted on freeways is 65 miyh and the minimum speed is 40 miyh. The fne for violating these limits is $15 for every mile per hour above the maximum speed or below the minimum speed. Express the amount of the fne F as a function of the driving speed x and graph Fsxd for 0 < x < 100.
- Question : 66E - An electricity company charges its customers a base rate of $10 a month, plus 6 cents per kilowatt-hour (kWh) for the frst 1200 kWh and 7 cents per kWh for all usage over 1200 kWh. Express the monthly cost E as a function of the amount x of electricity used. Then graph the function E for 0 < x < 2000.
- Question : 67E - In a certain country, income tax is assessed as follows. There is no tax on income up to $10,000. Any income over $10,000 is taxed at a rate of 10%, up to an income of $20,000. Any income over $20,000 is taxed at 15%. (a) Sketch the graph of the tax rate R as a function of the income I. (b) How much tax is assessed on an income of $14,000? On $26,000? (c) Sketch the graph of the total assessed tax T as a function of the income I.
- Question : 68E - The functions in Example 10 and Exercise 67 are called step functions because their graphs look like stairs. Give two other examples of step functions that arise in everyday life.
- Question : 69E - 69
- Question : 70E - FIG.
- Question : 71E - a) If the point s5, 3d is on the graph of an even function, what other point must also be on the graph? (b) If the point s5, 3d is on the graph of an odd function, what other point must also be on the graph?
- Question : 72E - A function f has domain f25, 5g and a portion of its graph is shown. (a) Complete the graph of f if it is known that f is even. (b) Complete the graph of f if it is known that f is odd.
- Question : 73E - 73
- Question : 74E - f sxd ? x 4x 1 2 1
- Question : 75E - f sxd ? x x 1 1
- Question : 76E - f sxd ? x | x |
- Question : 77E - f sxd ? 1 1 3x 2 2 x
- Question : 78E - f sxd ? 1 1 3x 3 2 x 5
- Question : 79E - If f and t are both even functions, is f 1 t even? If f and t are both odd functions, is f 1 t odd? What if f is even and t is odd? Justify your answers.
- Question : 80E - If f and t are both even functions, is the product ft even? If f and t are both odd functions, is ft odd? What if f is even and t is odd? Justify your answers.
- Question : 1E - 1
- Question : 2E - (a) y ? x (b) y ? x (c) y ? x 2s2 2 x 3d (d) y ? tan t 2 cos t (e) y ? s 1 1 s (f) y ? sx 3 2 1 1 1 s 3 x
- Question : 3E - 3
- Question : 4E - (a) y ? 3x (b) y ? 3x (c) y ? x 3 (d) y ? s 3 x
- Question : 5E - 5
- Question : 6E - tsxd ? 1 1 2 tan x
- Question : 7E - (a) Find an equation for the family of linear functions with slope 2 and sketch several members of the family. (b) Find an equation for the family of linear functions such that f s2d ? 1 and sketch several members of the family. (c) Which function belongs to both families?
- Question : 8E - What do all members of the family of linear functions f sxd ? 1 1 msx 1 3d have in common? Sketch several members of the family.
- Question : 9E - What do all members of the family of linear functions f sxd ? c 2 x have in common? Sketch several members of the family
- Question : 10E - Find expressions for the quadratic functions whose graphs are shown.
- Question : 11E - Find an expression for a cubic function f if f s1d ? 6 and f s21d ? f s0d ? f s2d ? 0.
- Question : 12E - Recent studies indicate that the average surface temperature of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T ? 0.02t 1 8.50, where T is temperature in
- Question : 13E - If the recommended adult dosage for a drug is D (in mg), then to determine the appropriate dosage c for a child of age a, pharmacists use the equation c ? 0.0417Dsa 1 1d. Suppose the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent? (b) What is the dosage for a newborn?
- Question : 14E - The manager of a weekend ?ea market knows from past experience that if he charges x dollars for a rental space at the market, then the number y of spaces he can rent is given by the equation y ? 200 2 4x. (a) Sketch a graph of this linear function. (Remember that the rental charge per space and the number of spaces rented can
- Question : 15E - The relationship between the Fahrenheit sFd and Celsius sCd temperature scales is given by the linear function F ? 9 5 C 1 32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent? What is the F-intercept and what does it represent?
- Question : 16E - Jason leaves Detroit at 2:00 pm and drives at a constant speed west along I-94. He passes Ann Arbor, 40 mi from Detroit, at 2:50 pm. (a) Express the distance traveled in terms of the time elapsed. (b) Draw the graph of the equation in part (a). (c) What is the slope of this line? What does it represent?
- Question : 17E - Biologists have noticed that the chirping rate of crickets of a certain species is related to temperature, and the relationship appears to be very nearly linear. A cricket produces 113 chirps per minute at 70
- Question : 18E - The manager of a furniture factory fnds that it costs $2200 to manufacture 100 chairs in one day and $4800 to produce 300 chairs in one day. (a) Express the cost as a function of the number of chairs produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent?
- Question : 19E - At the surface of the ocean, the water pressure is the same as the air pressure above the water, 15 lbyin2. Below the surface, the water pressure increases by 4.34 lbyin2 for every 10 ft of descent. (a) Express the water pressure as a function of the depth below the ocean surface. (b) At what depth is the pressure 100 lbyin2?
- Question : 20E - The monthly cost of driving a car depends on the number of miles driven. Lynn found that in May it cost her $380 to drive 480 mi and in June it cost her $460 to drive 800 mi. (a) Express the monthly cost C as a function of the distance driven d, assuming that a linear relationship gives a suitable model. (b) Use part (a) to predict the cost of driving 1500 miles per month. (c) Draw the graph of the linear function. What does the slope represent? (d) What does the C-intercept represent? (e) Why does a linear function give a suitable model in this situation?
- Question : 21E - 21
- Question : 22E - 0 x (a) y 0 (b) y
- Question : 23E - The table shows (lifetime) peptic ulcer rates (per 100 population) for various family incomes as reported by the National Health Interview Survey. Income Ulcer rate (per 100 population) $4,000 14.1 $6,000 13.0 $8,000 13.4 $12,000 12.5 $16,000 12.0 $20,000 12.4 $30,000 10.5 $45,000 9.4 $60,000 8.2 (a) Make a scatter plot of these data and decide whether a linear model is appropriate. (b) Find and graph a linear model using the frst and last data points. (c) Find and graph the least squares regression line. (d) Use the linear model in part (c) to estimate the ulcer rate for an income of $25,000. (e) According to the model, how likely is someone with an income of $80,000 to suffer from peptic ulcers? (f) Do you think it would be reasonable to apply the model to someone with an income of $200,000?
- Question : 24E - Biologists have observed that the chirping rate of crickets of a certain species appears to be related to temperature. The table shows the chirping rates for various temperatures. (a) Make a scatter plot of the data. (b) Find and graph the regression line. (c) Use the linear model in part (b) to estimate the chirping rate at 100
- Question : 25E - Anthropologists use a linear model that relates human femur (thighbone) length to height. The model allows an anthropologist to determine the height of an individual when only a partial skeleton (including the femur) is found. Here we fnd the model by analyzing the data on femur length and height for the eight males given in the following table. (a) Make a scatter plot of the data. (b) Find and graph the regression line that models the data. (c) An anthropologist fnds a human femur of length 53 cm. How tall was the person? Femur length (cm) Height (cm) Femur length (cm) Height (cm) 50.1 178.5 44.5 168.3 48.3 173.6 42.7 165.0 45.2 164.8 39.5 155.4 44.7 163.7 38.0 155.8
- Question : 26E - When laboratory rats are exposed to asbestos fbers, some of them develop lung tumors. The table lists the results of several experiments by different scientists. (a) Find the regression line for the data. (b) Make a scatter plot and graph the regression line. Does the regression line appear to be a suitable model for the data? (c) What does the y-intercept of the regression line represent? Asbestos exposure (fbersymL) Percent of mice that develop lung tumors Asbestos exposure (fbersymL) Percent of mice that develop lung tumors 50 2 1600 42 400 6 1800 37 500 5 2000 38 900 10 3000 50 1100 26
- Question : 27E - The table shows world average daily oil consumption from 1985 to 2010 measured in thousands of barrels per day. (a) Make a scatter plot and decide whether a linear model is appropriate. (b) Find and graph the regression line. (c) Use the linear model to estimate the oil consumption in 2002 and 2012. Years since 1985 Thousands of barrels of oil per day 0 60,083 5 66,533 10 70,099 15 76,784 20 84,077 25 87,302
- Question : 28E - The table shows average US retail residential prices of electricity from 2000 to 2012, measured in cents per kilowatt hour. (a) Make a scatter plot. Is a linear model appropriate? (b) Find and graph the regression line. (c) Use your linear model from part (b) to estimate the average retail price of electricity in 2005 and 2013. Years since 2000 CentsykWh 0 8.24 2 8.44 4 8.95 6 10.40 8 11.26 10 11.54 12 11.58
- Question : 29E - Many physical quantities are connected by inverse square laws, that is, by power functions of the form f sxd ? kx22. In particular, the illumination of an object by a light source is inversely proportional to the square of the distance from the source. Suppose that after dark you are in a room with just one lamp and you are trying to read a book. The light is too dim and so you move halfway to the lamp. How much brighter is the light?
- Question : 30E - It makes sense that the larger the area of a region, the larger the number of species that inhabit the region. Many ecologists have modeled the species-area relation with a power function and, in particular, the number of species S of bats living in caves in central Mexico has been related to the surface area A of the caves by the equation S ? 0.7A0.3. (a) The cave called Misi
- Question : 31E - The table shows the number N of species of reptiles and amphibians inhabiting Caribbean islands and the area A of the island in square miles. (a) Use a power function to model N as a function of A. (b) The Caribbean island of Dominica has area 291 mi2. How many species of reptiles and amphibians would you expect to fnd on Dominica? Island A N Saba 4 5 Monserrat 40 9 Puerto Rico 3,459 40 Jamaica 4,411 39 Hispaniola 29,418 84 Cuba 44,218 76
- Question : 32E - The table shows the mean (average) distances d of the planets from the sun (taking the unit of measurement to be the distance from planet Earth to the sun) and their periods T (time of revolution in years). (a) Fit a power model to the data. (b) Kepler
- Question : 1E - Suppose the graph of f is given. Write equations for the graphs that are obtained from the graph of f as follows. (a) Shift 3 units upward. (b) Shift 3 units downward. (c) Shift 3 units to the right. (d) Shift 3 units to the left. (e) Re?ect about the x-axis. (f) Re?ect about the y-axis. (g) Stretch vertically by a factor of 3. (h) Shrink vertically by a factor of 3
- Question : 2E - Explain how each graph is obtained from the graph of y ? f sxd. (a) y ? f sxd 1 8 (b) y ? f sx 1 8d (c) y ? 8f sxd (d) y ? f s8xd (e) y ? 2f sxd 2 1 (f) y ? 8f s8 1 xd
- Question : 3E - The graph of y ? f sxd is given. Match each equation with its graph and give reasons for your choices. (a) y ? f sx 2 4d (b) y ? f sxd 1 3 (c) y ? 1 3 f sxd (d) y ? 2f sx 1 4d (e) y ? 2f sx 1 6d
- Question : 4E - The graph of f is given. Draw the graphs of the following functions. (a) y ? f sxd 2 3 (b) y ? f sx 1 1d (c) y ? 1 2 f sxd (d) y ? 2f sxd
- Question : 5E - The graph of f is given. Use it to graph the following functions. (a) y ? f s2xd (b) y ? f (1 2 x) (c) y ? f s2xd (d) y ? 2f s2x
- Question : 6E - 6
- Question : 7E - 5 x y 0 2 3 7. _4 _1 _2.5 x y _1 0
- Question : 8E - (a) How is the graph of y ? 2 sin x related to the graph of y ? sin x? Use your answer and Figure 6 to sketch the graph of y ? 2 sin x. (b) How is the graph of y ? 1 1 sx related to the graph of y ? sx ? Use your answer and Figure 4(a) to sketch the graph of y ? 1 1 sx .
- Question : 9E - 9
- Question : 10E - y ? sx 2 3d
- Question : 11E - y ? x 3 1 1
- Question : 12E - y ? 1 2 1
- Question : 13E - y ? 2 cos 3x
- Question : 14E - y ? 2sx 1 1
- Question : 15E - y ? x 2 2 4x 1 5
- Question : 16E - y ? 1 1 sin x
- Question : 17E - y ? 2 2 sx
- Question : 18E - y ? 3 2 2 cos x
- Question : 19E - y ? sin(2 1 x)
- Question : 20E - y ? | x | 2 2
- Question : 21E - y ? | x 2 2 |
- Question : 22E - y ? 1 4 tanSx 2 4D
- Question : 23E - y ? | sx 2 1|
- Question : 24E - y ? | cos x |
- Question : 25E - The city of New Orleans is located at latitude 30
- Question : 26E - A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by 60.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time.
- Question : 27E - Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada. At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period of oscillation is about 12 hours and on June 30, 2009, high tide occurred at 6:45 am. Find a function involving the cosine function that models the water depth Dstd (in meters) as a function of time t (in hours after midnight) on that day
- Question : 28E - In a normal respiratory cycle the volume of air that moves into and out of the lungs is about 500 mL. The reserve and residue volumes of air that remain in the lungs occupy about 2000 mL and a single respiratory cycle for an average human takes about 4 seconds. Find a model for the total volume of air Vstd in the lungs as a function of time.
- Question : 29E - (a) How is the graph of y ? f (| x |) related to the graph of f ? (b) Sketch the graph of y ? sin | x |. (c) Sketch the graph of y ? s| x |.
- Question : 30E - Use the given graph of f to sketch the graph of y ? 1yf sxd. Which features of f are the most important in sketching y ? 1yf sxd? Explain how they are used.
- Question : 31E - 31
- Question : 32E - . f sxd ? s3 2 x , tsxd ? sx 2 2 1
- Question : 33E - 33
- Question : 34E - f sxd ? x 3 2 2, tsxd ? 1 2 4x
- Question : 35E - f sxd ? sx 1 1, tsxd ? 4x 2 3
- Question : 36E - f sxd ? sin x, tsxd ? x 2 1 1
- Question : 37E - f sxd ? x 1 1 x , tsxd ? x 1 1 x 1 2
- Question : 38E - f sxd ? x 1 1 x , tsxd ? sin 2x
- Question : 39E - 39
- Question : 40E - . f sxd ? | x 2 4 |, tsxd ? 2 x, hsxd ? sx
- Question : 41E - f sxd ? sx 2 3 , tsxd ? x 2, hsxd ? x 3 1 2
- Question : 42E - f sxd ? tan x, tsxd ? x x 2 1 , hsxd ? s 3 x
- Question : 43E - 43
- Question : 44E - Fsxd ? cos2x
- Question : 45E - Fsxd ? s 3 x 1 1 s
- Question : 46E - Gsxd ?
- Question : 47E - vstd ? secst 2d tanst 2d
- Question : 48E - ustd ? tan t 1 1 tan t
- Question : 49E - 49
- Question : 50E - Hsxd ? s 8 2 1 | x |
- Question : 51E - Sstd ? sin2scos td
- Question : 52E - Use the table to evaluate each expression. (a) f sts1dd (b) ts f s1dd (c) f s f s1dd (d) tsts1dd (e) st 8 f ds3d (f) s f 8 tds6d x 1 2 3 4 5 6 f sxd 3 1 4 2 2 5 tsxd 6 3 2 1 2 3
- Question : 53E - Use the given graphs of f and t to evaluate each expression, or explain why it is undefned. (a) f sts2dd (b) ts f s0dd (c) s f 8 tds0d (d) st 8 f ds6d (e) st 8 tds22d (f) s f 8 f ds4d
- Question : 54E - Use the given graphs of f and t to estimate the value of f stsxdd for x ? 25, 24, 23, . . . , 5. Use these estimates to sketch a rough graph of f 8 t.
- Question : 55E - A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cmys. (a) Express the radius r of this circle as a function of the time t (in seconds). (b) If A is the area of this circle as a function of the radius, fnd A 8 r and interpret it
- Question : 56E - A spherical balloon is being in?ated and the radius of the balloon is increasing at a rate of 2 cmys. (a) Express the radius r of the balloon as a function of the time t (in seconds). (b) If V is the volume of the balloon as a function of the radius, fnd V 8 r and interpret it.
- Question : 57E - A ship is moving at a speed of 30 kmyh parallel to a straight shoreline. The ship is 6 km from shore and it passes a lighthouse at noon. (a) Express the distance s between the lighthouse and the ship as a function of d, the distance the ship has traveled since noon; that is, fnd f so that s ? f sdd. (b) Express d as a function of t, the time elapsed since noon; that is, fnd t so that d ? tstd. (c) Find f 8 t. What does this function represent?
- Question : 58E - An airplane is ?ying at a speed of 350 miyh at an altitude of one mile and passes directly over a radar station at time t ? 0. (a) Express the horizontal distance d (in miles) that the plane has ?own as a function of t. (b) Express the distance s between the plane and the radar station as a function of d. (c) Use composition to express s as a function of t
- Question : 59E - The Heaviside function H is defned by Hstd ?H0 1 if if t t > , 0 0 It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function. (b) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t ? 0 and 120 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd. (c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t ? 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for Vstd in terms of Hstd. (Note that starting at t ? 5 corresponds to a translation.
- Question : 60E - The Heaviside function defned in Exercise 59 can also be used to defne the ramp function y ? ctHstd, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y ? tHstd. (b) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t ? 0 and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for Vstd in terms of Hstd for t < 60 (c) Sketch the graph of the voltage Vstd in a circuit if the switch is turned on at time t ? 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for Vstd in terms of Hstd for t < 32
- Question : 61E - . Let f and t be linear functions with equations f sxd ? m1x 1 b1 and tsxd ? m2x 1 b2. Is f 8 t also a linear function? If so, what is the slope of its graph?
- Question : 62E - If you invest x dollars at 4% interest compounded annually, then the amount Asxd of the investment after one year is Asxd ? 1.04x. Find A 8 A, A 8 A 8 A, and A 8 A 8 A 8 A. What do these compositions represent? Find a formula for the composition of n copies of A.
- Question : 63E - (a) If tsxd ? 2x 1 1 and hsxd ? 4x 2 1 4x 1 7, fnd a function f such that f 8 t ? h. (Think about what operations you would have to perform on the formula for t to end up with the formula for h.) (b) If f sxd ? 3x 1 5 and hsxd ? 3x 2 1 3x 1 2, fnd a function t such that f 8 t ? h.
- Question : 64E - If f sxd ? x 1 4 and hsxd ? 4x 2 1, fnd a function t such that t 8 f ? h.
- Question : 65E - Suppose t is an even function and let h ? f 8 t. Is h always an even function?
- Question : 66E - Suppose t is an odd function and let h ? f 8 t. Is h always an odd function? What if f is odd? What if f is even?
- Question : 1E - A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes. t smind 5 10 15 20 25 30 V sgald 694 444 250 111 28 0 (a) If P is the point s15, 250d on the graph of V, fnd the slopes of the secant lines PQ when Q is the point on the graph with t ? 5, 10, 20, 25, and 30. (b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines. (c) Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is ?owing from the tank after 15 minutes.)
- Question : 2E - A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.t smind 36 38 40 42 44 Heartbeats 2530 2661 2806 2948 3080 The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient
- Question : 3E - (a) If Q is the point sx, 1ys1 2 xdd, use your calculator to fnd the slope of the secant line PQ (correct to six decimal places) for the following values of x: (i) 1.5 (ii) 1.9 (iii) 1.99 (iv) 1.999 (v) 2.5 (vi) 2.1 (vii) 2.01 (viii) 2.001 (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at Ps2, 21d. (c) Using the slope from part (b), fnd an equation of the tangent line to the curve at Ps2, 21d
- Question : 4E - The point Ps0.5, 0d lies on the curve y ? cos x. (a) If Q is the point sx, cos xd, use your calculator to fnd the slope of the secant line PQ (correct to six decimal places) for the following values of x: (i) 0 (ii) 0.4 (iii) 0.49 (iv) 0.499 (v) 1 (vi) 0.6 (vii) 0.51 (viii) 0.501 (b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at Ps0.5, 0d. (c) Using the slope from part (b), fnd an equation of the tangent line to the curve at Ps0.5, 0d. (d) Sketch the curve, two of the secant lines, and the tangent line
- Question : 5E - If a ball is thrown into the air with a velocity of 40 ftys, its height in feet t seconds later is given by y ? 40t 2 16t 2. (a) Find the average velocity for the time period beginning when t ? 2 and lasting (i) 0.5 seconds (ii) 0.1 seconds (iii) 0.05 seconds (iv) 0.01 seconds (b) Estimate the instantaneous velocity when t ? 2.
- Question : 6E - If a rock is thrown upward on the planet Mars with a velocity of 10 mys, its height in meters t seconds later is given by y ? 10t 2 1.86t 2. (a) Find the average velocity over the given time intervals: (i) [1, 2] (ii) [1, 1.5] (iii) [1, 1.1] (iv) [1, 1.01] (v) [1, 1.001] (b) Estimate the instantaneous velocity when t ? 1.
- Question : 7E - The table shows the position of a motorcyclist after accelerating from rest. t ssecondsd 0 1 2 3 4 5 6 s (feet) 0 4.9 20.6 46.5 79.2 124.8 176.7 (a) Find the average velocity for each time period: (i) f2, 4g (ii) f3, 4g (iii) f4, 5g (iv) f4, 6g (b) Use the graph of s as a function of t to estimate the instantaneous velocity when t ? 3.
- Question : 8E - The displacement (in centimeters) of a particle moving back and forth along a straight line is given by the equation of motion s ? 2 sin t 1 3 cos t, where t is measured in seconds. (a) Find the average velocity during each time period: (i) [1, 2] (ii) [1, 1.1] (iii) [1, 1.01] (iv) [1, 1.001] (b) Estimate the instantaneous velocity of the particle when t ? 1.
- Question : 9E - The point Ps1, 0d lies on the curve y ? sins10 yxd. (a) If Q is the point sx, sins10 yxdd, fnd the slope of the secant line PQ (correct to four decimal places) for x ? 2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9. Do the slopes appear to be approaching a limit? (b) Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at P. (c) By choosing appropriate secant lines, estimate the slope of the tangent line at P.
- Question : 1E - Explain in your own words what is meant by the equation lim x l 2 f sxd ? 5 Is it possible for this statement to be true and yet f s2d ? 3? Explain.
- Question : 2E - Explain what it means to say that lim x l 12 f sxd ? 3 and lim x l11 f sxd ? 7 In this situation is it possible that limx l 1 f sxd exists? Explain.
- Question : 3E - Explain the meaning of each of the following. (a) lim x l23 f sxd ? ` (b) lim x l 41 f sxd ? 2`
- Question : 4E - Use the given graph of f to state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim x l22 f sxd (b) lim x l 21 f sxd (c) lim x l 2 f sxd (d) f s2d (e) lim x l 4 f sxd (f) f s4d
- Question : 5E - For the function f whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim x l 1 f sxd (b) lim x l 32 f sxd (c) lim x l 31 f sxd (d) lim x l 3 f sxd (e) f s3d
- Question : 6E - For the function h whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim x l 232 hsxd (b) lim x l 231 hsxd (c) lim x l 23 hsxd (d) hs23d (e) lim xl02 hsxd (f) lim x l01 hsxd (g) lim x l 0 hsxd (h) hs0d (i) lim x l 2 hsxd (j) hs2d (k) lim x l51 hsxd (l) lim x l52 hsxd
- Question : 7E - For the function t whose graph is given, state the value of each quantity, if it exists. If it does not exist, explain why. (a) lim t l 02 tstd (b) lim t l 01 tstd (c) lim t l 0 tstd (d) lim t l 22 tstd (e) lim t l 21 tstd (f) lim t l 2 tstd (g) ts2d (h) lim t l 4 tstd
- Question : 8E - For the function A whose graph is shown, state the following. (a) lim x l23 Asxd (b) lim x l22 Asxd (c) lim x l21 Asxd (d) lim x l21 Asxd (e) The equations of the vertical asymptotes
- Question : 9E - For the function f whose graph is shown, state the following. (a) lim x l27 f sxd (b) lim x l23 f sxd (c) lim x l 0 f sxd (d) lim x l 62 f sxd (e) lim x l 61 f sxd (f) The equations of the vertical asymptotes
- Question : 10E - A patient receives a 150-mg injection of a drug every 4 hours. The graph shows the amount f std of the drug in the bloodstream after t hours. Find lim tl 122 f std and lim tl 121 f std and explain the signifcance of these one-sided limits. 4 8 12 16 t f(t) 150 0 300
- Question : 11E - 11
- Question : 12E - f sxd ?H1 cos sin 1x xsin x if if 0 if x x , . < 0 x <
- Question : 13E - 13
- Question : 14E - f sxd ? sx x2 3 1 1 x x 2
- Question : 15E - 15
- Question : 16E - lim x l 0 f sxd ? 1, lim x l 32 f sxd ? 22, lim x l 31 f sxd ? 2, f s0d ? 21, f s3d ? 1
- Question : 17E - lim x l 31 f sxd ? 4, lim x l 32 f sxd ? 2, lim x l 22 f sxd ? 2, f s3d ? 3, f s22d ? 1
- Question : 18E - lim x l 02 f sxd ? 2, lim x l 01 f sxd ? 0, lim x l 42 f sxd ? 3, lim x l 41 f sxd ? 0, f s0d ? 2, f s4d ? 1
- Question : 19E - 19
- Question : 20E - lim x l23 x 2 2 3x x 2 2 9 , x ? 22.5, 22.9, 22.95, 22.99, 22.999, 22.9999, 23.5, 23.1, 23.05, 23.01, 23.001, 23.0001
- Question : 21E - lim x l 0 sin x x 1 tan x , x ? 61, 60.5, 60.2, 60.1, 60.05, 60.01
- Question : 22E - lim hl 0 s2 1 hd5 2 32 h , h ? 60.5, 60.1, 60.01, 60.001, 60.0001
- Question : 23E - 23
- Question : 24E - lim p l 21 1 1 p 9 1 1 p 15
- Question : 25E - lim x l01 x x
- Question : 26E - lim t l 0 5t 2 1 t
- Question : 27E - (a) By graphing the function f sxd ? scos 2x 2 cos xdyx 2 and zooming in toward the point where the graph crosses the y-axis, estimate the value of lim x l 0 f sxd. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0
- Question : 28E - (a) Estimate the value of lim x l 0 sin x sin x by graphing the function f sxd ? ssin xdyssin xd. State your answer correct to two decimal places. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0.
- Question : 29E - 29
- Question : 30E - lim x l52 x 1 1 x 2 5 s
- Question : 31E - lim x l1 2 2 x sx 2 1d2
- Question : 32E - xlim l32 sx s 2x3d5
- Question : 33E - lim x l221 x 2 1 x 2sx 1 2d
- Question : 34E - xlim l 0 x 2 1 x 2sx 1 2d
- Question : 35E - lim xls y2d1 1 x sec x
- Question : 36E - lim x l 2 cot x
- Question : 37E - lim x l2 2 x csc x
- Question : 38E - lim x l 22 x 2 2 2x x 2 2 4x 1 4
- Question : 39E - lim x l21 x 2 2 2x 2 8 x 2 2 5x 1 6
- Question : 40E - (a) Find the vertical asymptotes of the function y ? x 2 1 1 3x 2 2x 2 (b) Confrm your answer to part (a) by graphing the function.
- Question : 41E - Determine lim x l12 1 x 3 2 1 and lim x l11 1 x 3 2 1 (a) by evaluating f sxd ? 1ysx 3 2 1d for values of x that approach 1 from the left and from the right, (b) by reasoning as in Example 9, and (c) from a graph of f
- Question : 42E - a) By graphing the function f sxd ? stan 4xdyx and zooming in toward the point where the graph crosses the y-axis, estimate the value of lim x l 0 f sxd. (b) Check your answer in part (a) by evaluating f sxd for values of x that approach 0.
- Question : 43E - (a) Evaluate the function f sxd ? x 2 2 s2xy1000d for x ? 1, 0.8, 0.6, 0.4, 0.2, 0.1, and 0.05, and guess the value of lim x l 0Sx 2 2 1000 2x D (b) Evaluate f sxd for x ? 0.04, 0.02, 0.01, 0.005, 0.003, and 0.001. Guess again
- Question : 44E - (a) Evaluate hsxd ? stan x 2 xdyx 3 for x ? 1, 0.5, 0.1, 0.05, 0.01, and 0.005. (b) Guess the value of lim x l 0 tan x 2 x x 3 . (c) Evaluate hsxd for successively smaller values of x until you fnally reach a value of 0 for hsxd. Are you still confdent that your guess in part (b) is correct? Explain why you eventually obtained values of 0 for hsxd. (In Section 6.8 a method for evaluating this limit will be explained.) (d) Graph the function h in the viewing rectangle f21, 1g by f0, 1g. Then zoom in toward the point where the graph crosses the y-axis to estimate the limit of hsxd as x approaches 0. Continue to zoom in until you observe distortions in the graph of h. Compare with the results of part (c).
- Question : 45E - Graph the function f sxd ? sins yxd of Example 4 in the viewing rectangle f21, 1g by f21, 1g. Then zoom in toward the origin several times. Comment on the behavior of this function.
- Question : 46E - Consider the function f sxd ? tan 1 x . (a) Show that f sxd ? 0 for x ? 1 , 1 2 , 1 3 , . . . (b) Show that f sxd ? 1 for x ? 4 , 4 5 , 4 9 , . . . (c) What can you conclude about lim x l 01 tan 1 x
- Question : 47E - Use a graph to estimate the equations of all the vertical asymptotes of the curve y ? tans2 sin xd 2 < x < Then fnd the exact equations of these asymptotes
- Question : 48E - In the theory of relativity, the mass of a particle with velocity v is m ? m0 s1 2 v2yc2 where m0 is the mass of the particle at rest and c is the speed of light. What happens as v l c2?
- Question : 49E - a) Use numerical and graphical evidence to guess the value of the limit lim xl1 x3 2 1 sx 2 1 (b) How close to 1 does x have to be to ensure that the function in part (a) is within a distance 0.5 of its limit?
- Question : 1E - Given that lim x l 2 f sxd ? 4 lim x l 2 tsxd ? 22 lim x l 2 hsxd ? 0 fnd the limits that exist. If the limit does not exist, explain why. (a) lim x l 2 f f sxd 1 5tsxdg (b) lim x l 2 ftsxdg3 (c) lim x l 2 sf sxd (d) lim x l 2 3f sxd tsxd (e) lim x l2 tsxd hsxd (f) xlim l 2 tsxdhsxd f sxd
- Question : 2E - The graphs of f and t are given. Use them to evaluate each limit, if it exists. If the limit does not exist, explain why. (a) lim x l2 f f sxd 1 tsxdg (b) lim x l 0 f f sxd 2 tsxdg (c) lim x l21 f f sxdtsxdg (d) lim x l3 f sxd tsxd (e) lim x l2 fx 2f sxdg (f) f s21d 1 lim x l21 tsxd
- Question : 3E - 3
- Question : 4E - lim xl 21 sx 4 2 3xdsx 2 1 5x 1 3d
- Question : 5E - lim t l 22 t 4 2 2 2t 2 2 3t 1 2
- Question : 6E - ulim l22 su 4 1 3u 1 6
- Question : 7E - lim x l 8 s1 1 s 3 x ds2 2 6x 2 1 x 3d
- Question : 8E - lim t l 2S t 3 2 t 2 2 3t 1 2 5D
- Question : 9E - lim x l 2
- Question : 10E - (a) What is wrong with the following equation? x 2 1 x 2 6 x 2 2 ? x 1 3 (b) In view of part (a), explain why the equation lim x l2 x 2 1 x 2 6 x 2 2 ? lim x l2 sx 1 3d is correct.
- Question : 11E - 11
- Question : 12E - lim x l23 x 2 1 3x x 2 2 x 2 12
- Question : 13E - lim x l5 x 2 2 5x 1 6 x 2 5
- Question : 14E - lim x l 4 x 2 1 3x x 2 2 x 2 12
- Question : 15E - lim t l23 t 2 2 9 2t 2 1 7t 1 3
- Question : 16E - lim l21 2x 2 1 3x 1 1 x 2 2 2x 2 3
- Question : 17E - lim h l 0 s25 1 hd2 2 25 h
- Question : 18E - lim h l 0 s2 1 hd3 2 8 h
- Question : 19E - lim x l22 x 1 2 x 3 1 8
- Question : 20E - tlim l 1 t 4 2 1 t 3 2 1
- Question : 21E - lim h l 0 s9 1 h 2 3 h
- Question : 22E - lim ul 2 s4u 1 1 2 3 u 2 2
- Question : 23E - lim x l3 1 x 2 1 3 x 2 3
- Question : 24E - lim h l 0 s3 1 hd21 2 321 h
- Question : 25E - lim t l 0 s1 1 t 2 s1 2 t t
- Question : 26E - lim t l 0S 1 t 2 t 2 1 1 Td
- Question : 27E - lim x l 16 4 2 sx 16x 2 x 2
- Question : 28E - lim l2 x 2 2 4x 1 4 x 4 2 3x 2 2 4
- Question : 29E - lim t l 0S ts111 t 2 1 tD
- Question : 30E - xlim l24 sx 2x1 1942 5
- Question : 31E - lim h l 0 sx 1 hd3 2 x 3 h
- Question : 32E - lim h l 0 1 sx 1 hd2 2 1 2x h
- Question : 33E - (a) Estimate the value of lim x l0 x s1 1 3x 2 1 by graphing the function f sxd ? xyss1 1 3x 2 1d. (b) Make a table of values of f sxd for x close to 0 and guess the value of the limit. (c) Use the Limit Laws to prove that your guess is correct
- Question : 34E - (a) Use a graph of f sxd ? s3 1 x 2 s3 x to estimate the value of limx l 0 f sxd to two decimal places. (b) Use a table of values of f sxd to estimate the limit to four decimal places. (c) Use the Limit Laws to fnd the exact value of the limit.
- Question : 35E - Use the Squeeze Theorem to show that limx l 0 sx 2 cos 20 xd ? 0. Illustrate by graphing the functions f sxd ? 2x 2, tsxd ? x 2 cos 20 x, and hsxd ? x 2 on the same screen.
- Question : 36E - Use the Squeeze Theorem to show that lim x l0 sx 3 1 x 2 sin x ? 0 Illustrate by graphing the functions f, t, and h (in the nota tion of the Squeeze Theorem) on the same screen.
- Question : 37E - If 4x 2 9 < f sxd < x 2 2 4x 1 7 for x > 0, fnd lim x l 4 f sxd.
- Question : 38E - If 2x < tsxd < x 4 2 x 2 1 2 for all x, evaluate lim x l 1 tsxd.
- Question : 39E - Prove that lim x l0 x 4 cos 2 x ? 0.
- Question : 40E - Prove that lim x l01 sx f1 1 sin2s2 yxdg ? 0
- Question : 41E - 41
- Question : 42E - lim x l26 2x 1 12 | x 1 6 |
- Question : 43E - lim x l0.52 2x 2 1 | 2x 3 2 x 2 |
- Question : 44E - xlim l22 2 2 | x | 2 1 x
- Question : 45E - lim x l02S 1 x 2 | 1 x |D
- Question : 46E - xlim l01S 1 x 2 | 1 x |D
- Question : 47E - The signum (or sign) function, denoted by sgn, is defned by sgn x ?H21 0 1 if if if x x x , ? . 0 0 0 (a) Sketch the graph of this function. (b) Find each of the following limits or explain why it does not exist. (i) lim x l01 sgn x (ii) lim x l02 sgn x (iii) lim x l 0 sgn x (iv) lim x l 0 | sgn x |
- Question : 48E - Let tsxd ? sgnssin xd. (a) Find each of the following limits or explain why it does not exist. (i) lim x l01 tsxd (ii) lim x l02 tsxd (iii) lim x l0 tsxd (iv) lim x l 1 tsxd (v) lim x l 2 tsxd (vi) lim x l tsxd (b) For which values of a does lim x la tsxd not exist? (c) Sketch a graph of t.
- Question : 49E - Let tsxd ? x 2 1 x 2 6 | x 2 2 | . (a) Find (i) lim x l21 tsxd (ii) lim x l22 tsxd (b) Does limx l 2 tsxd exist? (c) Sketch the graph of t.
- Question : 50E - Let f sxd ?Hx sx 2 2 1 2 1d2 if if x x , > 1 1 (a) Find lim x l12 f sxd and lim x l11 f sxd. (b) Does lim x l1 f sxd exist? (c) Sketch the graph of f.
- Question : 51E - Let Bstd ?H4 s2 t 1 1 2 c t if if t t , > 2 2 Find the value of c so that lim t l 2 Bstd exists.
- Question : 52E - Let tsxd ? x 3 2 2 x 2 x 2 3 if x , 1 if x ? 1 if 1 , x < 2 if x . 2 (a) Evaluate each of the following, if it exists. (i) lim x l12 tsxd (ii) lim x l 1 tsxd (iii) ts1d (iv) lim x l22 tsxd (v) lim x l 21 tsxd (vi) lim x l 2 tsxd (b) Sketch the graph of t
- Question : 53E - (a) If the symbol v b denotes the greatest integer function defned in Example 10, evaluate (i) lim x l221 v x b (ii) lim x l22 v x b (iii) lim x l22.4 v x b (b) If n is an integer, evaluate (i) lim x ln2 v x b (ii) lim x l n1 v x b (c) For what values of a does limx l a v x b exist?
- Question : 54E - Let f sxd ? vcos x b, 2 < x < . (a) Sketch the graph of f. (b) Evaluate each limit, if it exists. (i) lim x l 0 f sxd (ii) lim x ls y2d2 f sxd (iii) lim x ls y2d1 f sxd (iv) lim x l y2 f sxd (c) For what values of a does limx l a f sxd exist?
- Question : 55E - If f sxd ? v x b 1 v2x b, show that limx l 2 f sxd exists but is not equal to f s2d.
- Question : 56E - In the theory of relativity, the Lorentz contraction formula L ? L0s1 2 v 2yc 2 expresses the length L of an object as a function of its velocity v with respect to an observer, where L0 is the length of the object at rest and c is the speed of light. Find limv lc2L and interpret the result. Why is a left-hand limit necessary?
- Question : 57E - If p is a polynomial, show that lim xl a psxd ? psad.
- Question : 58E - If r is a rational function, use Exercise 57 to show that limx l a rsxd ? rsad for every number a in the domain of r
- Question : 59E - If lim x l 1 f sxd 2 8 x 2 1 ? 10, fnd lim x l 1 f sxd.
- Question : 60E - If lim x l 0 f sxd x 2 ? 5, fnd the following limits. (a) lim x l 0 f sxd (b) lim x l 0 f sxd x
- Question : 61E - If f sxd ?Hx 02 if if x x is irrational is rational prove that lim x l 0 f sxd ? 0.
- Question : 62E - Show by means of an example that limx l a f f sxd 1 tsxdg may exist even though neither lim x l a f sxd nor limx l a tsxd exists
- Question : 63E - Show by means of an example that limx l a f f sxd tsxdg may exist even though neither limx l a f sxd nor limx l a tsxd exists
- Question : 64E - Evaluate lim x l 2 s6 2 x 2 2 s3 2 x 2 1 .
- Question : 65E - Is there a number a such that lim x l22 3x 2 1 ax 1 a 1 3 x 2 1 x 2 2 exists? If so, fnd the value of a and the value of the limit.
- Question : 66E - The fgure shows a fxed circle C1 with equation sx 2 1d2 1 y 2 ? 1 and a shrinking circle C2 with radius r and center the origin. P is the point s0, rd, Q is the upper point of intersection of the two circles, and R is the point of intersection of the line PQ and the x-axis. What happens to R as C2 shrinks, that is, as r l 01?
- Question : 1E - Use the given graph of f to fnd a number such that if | x 2 1 | , then | f sxd 2 1 | , 0.2
- Question : 2E - Use the given graph of f to fnd a number such that if 0 , | x 2 3 | , then | f sxd 2 2 | , 0.5
- Question : 3E - Use the given graph of f sxd ? sx to fnd a number such that if | x 2 4 | , then | sx 2 2 | , 0.4
- Question : 4E - Use the given graph of f sxd 5 x 2 to fnd a number such that if | x 2 1 | , then | x 2 2 1 | , 1 2
- Question : 5E - Use a graph to fnd a number such that if Z x 2 4 Z , then | tan x 2 1| , 0.2
- Question : 6E - Use a graph to fnd a number such that if | x 2 1| , then Z x 22 1 x 4 2 0.4 Z , 0.1
- Question : 7E - For the limit lim x l 2 sx 3 2 3x 1 4d 5 6 illustrate Defnition 2 by fnding values of that correspond to
- Question : 8E - For the limit lim x l2 4x 1 1 3x 2 4 ? 4.5 illustrate Defnition 2 by fnding values of that correspond to
- Question : 9E - (a) Use a graph to fnd a number such that if 4 , x , 4 1 then x 2 1 4 sx 2 4 . 100 (b) What limit does part (a) suggest is true?
- Question : 10E - Given that lim x l csc2 x ? `, illustrate Defnition 6 by fnding values of that correspond to (a) M ? 500 and (b) M ? 1000.
- Question : 11E - A machinist is required to manufacture a circular metal disk with area 1000 cm2. (a) What radius produces such a disk? (b) If the machinist is allowed an error tolerance of 65 cm2 in the area of the disk, how close to the ideal radius in part (a) must the machinist control the radius? (c) In terms of the
- Question : 12E - A crystal growth furnace is used in research to determine how best to manufacture crystals used in electronic components for the space shuttle. For proper growth of the crystal, the temperature must be controlled accurately by adjusting the input power. Suppose the relationship is given by Tswd ? 0.1w 2 1 2.155w 1 20 where T is the temperature in degrees Celsius and w is the power input in watts. (a) How much power is needed to maintain the temperature at 200
- Question : 13E - (a) Find a number such that if | x 2 2| , , then | 4x 2 8| ,
- Question : 14E - Given that limx l 2s5x 2 7d 5 3, illustrate Defnition 2 by fnding values of that correspond to
- Question : 15E - 15
- Question : 16E - lim x l 4 s2x 2 5d 5 3
- Question : 17E - lim xl23 s1 2 4xd ? 13
- Question : 18E - lim xl22 s3x 1 5d ? 21
- Question : 19E - 19
- Question : 20E - lim x l 10 s3 2 4 5 xd ? 25
- Question : 21E - lim xl4 x 2 2 2x 2 8 x 2 4 ? 6
- Question : 22E - lim x l21.5 9 2 4x 2 3 1 2x ? 6
- Question : 23E - lim x l a x ? a
- Question : 24E - lim x l a c ? c
- Question : 25E - lim x l 0 x 2 ? 0
- Question : 26E - lim x l 0 x 3 ? 0
- Question : 27E - lim x l 0 | x | ? 0
- Question : 28E - xl lim 261 s 8 6 1 x ? 0
- Question : 29E - lim x l 2 sx 2 2 4x 1 5d 5 1
- Question : 30E - lim x l 2 sx 2 1 2x 2 7d 5 1
- Question : 31E - lim x l22 sx 2 2 1d 5 3
- Question : 32E - lim x l 2 x 3 5 8
- Question : 33E - Verify that another possible choice of for showing that lim x l3 x 2 5 9 in Example 4 is 5 minh2,
- Question : 34E - Verify, by a geometric argument, that the largest possible choice of for showing that lim x l3 x 2 ? 9 is ? s9 1
- Question : 35E - a) For the limit limx l 1 sx3 1 x 1 1d 5 3, use a graph to fnd a value of that corresponds to
- Question : 36E - Prove that lim x l2 1 x ? 1 2 .
- Question : 37E - Prove that lim x l a sx ? sa if a . 0. FHint: Use | sx 2 sa | ? s|xx 2 1 a s|a . F
- Question : 38E - If H is the Heaviside function defned in Example 1.5.6, prove, using Defnition 2, that lim t l 0 Hstd does not exist. [Hint: Use an indirect proof as follows. Suppose that the limit is L. Take
- Question : 39E - If the function f is defned by f sxd ?H0 1 if if x x is irrational is rational prove that lim x l 0 f sxd does not exist.
- Question : 40E - By comparing Defnitions 2, 3, and 4, prove Theorem 1.6.1
- Question : 41E - How close to 23 do we have to take x so that 1 sx 1 3d4 . 10,000
- Question : 42E - Prove, using Defnition 6, that lim x l23 1 sx 1 3d4 ? `.
- Question : 43E - Prove that lim x l212 5 sx 1 1d3 ? 2`
- Question : 44E - Suppose that lim x l a f sxd 5 ` and lim x l a tsxd 5 c, where c is a real number. Prove each statement. (a) lim x l a f f sxd 1 tsxdg ? ` (b) lim x l a f f sxdtsxdg 5 ` if c . 0 (c) lim xl a f f sxdtsxdg 5 2` if c , 0
- Question : 1E - Write an equation that expresses the fact that a function f is continuous at the number 4.
- Question : 2E - If f is continuous on s2`, `d, what can you say about its graph?
- Question : 3E - (a) From the graph of f , state the numbers at which f is discontinuous and explain why. (b) For each of the numbers stated in part (a), determine whether f is continuous from the right, or from the left, or neither
- Question : 4E - From the graph of t, state the intervals on which t is continuous.
- Question : 5E - 5
- Question : 6E - Discontinuities at 21 and 4, but continuous from the left at 21 and from the right at 4
- Question : 7E - Removable discontinuity at 3, jump discontinuity at 5
- Question : 8E - Neither left nor right continuous at 22, continuous only from the left at 2
- Question : 9E - The toll T charged for driving on a certain stretch of a toll road is $5 except during rush hours (between 7 am and 10 am and between 4 pm and 7 pm) when the toll is $7. (a) Sketch a graph of T as a function of the time t, measured in hours past midnight. (b) Discuss the discontinuities of this function and their signifcance to someone who uses the road.
- Question : 10E - Explain why each function is continuous or discontinuous. (a) The temperature at a specifc location as a function of time (b) The temperature at a specifc time as a function of the distance due west from New York City (c) The altitude above sea level as a function of the distance due west from New York City (d) The cost of a taxi ride as a function of the distance traveled (e) The current in the circuit for the lights in a room as a function of time
- Question : 11E - 11
- Question : 12E - tstd ? t 2 1 5t 2t 1 1 , a ? 2
- Question : 13E - psvd ? 2s3v2 1 1 , a ? 1
- Question : 14E - f sxd ? 3x4 2 5x 1 s 3 x 2 1 4 , a ? 2
- Question : 15E - 15
- Question : 16E - tsxd ? x 2 1 3x 1 6 , s2`, 22d
- Question : 17E - 17
- Question : 18E - f sxd ?H1 x 1 1 2 if if x x
- Question : 19E - f sxd ?H1 1y2 x x 2 if if x x , > 1
- Question : 20E - f sxd ?H1 x x 2 2 2 2 1 x if if x x
- Question : 21E - f sxd ?Hcos 0 1 2xx 2 if if if x x x ? , . 0 0 0
- Question : 22E - f sxd ?H6 2x 2 x 22 5x32 3 if if x x
- Question : 23E - 23
- Question : 24E - f sxd ? x 3 2 8 x 2 2 4
- Question : 25E - 25
- Question : 26E - Gsxd ? 2x 2x2 2 1 x 1 2 1
- Question : 27E - Qsxd ? s 3 x 2 2 x 3 2 2
- Question : 28E - hsxd ? xsin 1x
- Question : 29E - hsxd ? coss1 2 x 2d
- Question : 30E - Bsxd ? tan x s4 2 x 2
- Question : 31E - Msxd ?
- Question : 32E - Fsxd ? sinscosssin xdd
- Question : 33E - 33
- Question : 34E - . y ? tansx
- Question : 35E - lim x l2 x s20 2 x 2
- Question : 36E - lim x l sinsx 1 sin xd
- Question : 37E - lim x l y4 x 2 tan x
- Question : 38E - . lim xl2 x 3ysx 2 1 x 2 2
- Question : 39E - 39
- Question : 40E - f sxd ?Hsin cosx x if if x x > , y y4 4
- Question : 41E - 41
- Question : 42E - . f sxd ?Hx 3 s 2x 2 1x1 if if 1 if x x < . , 1 4 x < 4
- Question : 43E - sxd ?H2 2 xx1 2 2 2 x if if 0 if x x , < . 0 x 1 < 1
- Question : 44E - The gravitational force exerted by the planet Earth on a unit mass at a distance r from the center of the planet is Fsrd ? GMr R 3 if r , R GM r 2 if r > R where M is the mass of Earth, R is its radius, and G is the gravitational constant. Is F a continuous function of r?
- Question : 45E - For what value of the constant c is the function f continuous on s2`, `d? f sxd ?Hcx x 3 22 1cx 2x if if x x , > 2 2
- Question : 46E - Find the values of a and b that make f continuous everywhere. f sxd ? x 2 2 4 x 2 2 ax 2 2 bx 1 3 2x 2 a 1 b if x , 2 if 2 < x , 3 if x > 3
- Question : 47E - Suppose f and t are continuous functions such that ts2d ? 6 and lim x l2 f3f sxd 1 f sxdtsxdg ? 36. Find f s2d.
- Question : 48E - Let fsxd ? 1yx and tsxd ? 1yx 2. (a) Find s f + tdsxd. (b) Is f + t continuous everywhere? Explain.
- Question : 49E - Which of the following functions f has a removable discontinuity at a? If the discontinuity is removable, fnd a function t that agrees with f for x
- Question : 50E - Suppose that a function f is continuous on [0, 1] except at 0.25 and that f s0d ? 1 and f s1d ? 3. Let N ? 2. Sketch two possible graphs of f, one showing that f might not satisfy the conclusion of the Intermediate Value Theorem and one showing that f might still satisfy the conclusion of the Intermediate Value Theorem (even though it doesn
- Question : 51E - If f sxd ? x 2 1 10 sin x, show that there is a number c such that f scd ? 1000
- Question : 52E - Suppose f is continuous on f1, 5g and the only solutions of the equation f sxd ? 6 are x ? 1 and x ? 4. If f s2d ? 8, explain why f s3d . 6.
- Question : 53E - 53
- Question : 54E - 2yx ? x 2 sx , s2, 3d
- Question : 55E - cos x ? x, s0, 1d
- Question : 56E - sin x ? x 2 2 x, s1, 2d
- Question : 57E - 57
- Question : 58E - x 5 2 x 2 1 2x 1 3 ? 0
- Question : 59E - 59
- Question : 60E - sx 2 5 ? 1 x 1 3
- Question : 61E - 61
- Question : 62E - y ? x 2 2 3 1 1yx, s0, 2d
- Question : 63E - Prove that f is continuous at a if and only if lim h l 0 f sa 1 hd ? f sad
- Question : 64E - To prove that sine is continuous, we need to show that lim x l a sin x ? sin a for every real number a. By Exercise 63 an equivalent statement is that lim h l 0 sinsa 1 hd ? sin a Use (6) to show that this is true.
- Question : 65E - Prove that cosine is a continuous function.
- Question : 66E - (a) Prove Theorem 4, part 3. (b) Prove Theorem 4, part 5.
- Question : 67E - For what values of x is f continuous? f sxd ?H0 1 if if x x is irrational
- Question : 68E - For what values of x is t continuous? tsxd ?H0 x if if x x is irrational
- Question : 69E - Is there a number that is exactly 1 more than its cube?
- Question : 70E - If a and b are positive numbers, prove that the equation a x 3 1 2x 2 2 1 1 b x 3 1 x 2 2 ? 0 has at least one solution in the interval s21, 1d.
- Question : 71E - Show that the function f sxd ?Hx 04 sins1yxd if if x x
- Question : 72E - (a) Show that the absolute value function Fsxd ? | x | is continuous everywhere. (b) Prove that if f is a continuous function on an interval, then so is | f |. (c) Is the converse of the statement in part (b) also true? In other words, if | f | is continuous, does it follow that f is continuous? If so, prove it. If not, fnd a counterexample.
- Question : 73E - A Tibetan monk leaves the monastery at 7:00 am and takes his usual path to the top of the mountain, arriving at 7:00 pm. The following morning, he starts at 7:00 am at the top and takes the same path back, arriving at the monastery at 7:00 pm. Use the Intermediate Value Theorem to show that there is a point on the path that the monk will cross at exactly the same time of day on both days.
- Question : 1R - (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?
- Question : 2R - Discuss four ways of representing a function. Illustrate your discussion with examples.
- Question : 3R - (a) What is an even function? How can you tell if a function is even by looking at its graph? Give three examples of an even function. (b) What is an odd function? How can you tell if a function is odd by looking at its graph? Give three examples of an odd function.
- Question : 4R - What is an increasing function?
- Question : 5R - What is a mathematical model?
- Question : 6R - Give an example of each type of function. (a) Linear function (b) Power function (c) Exponential function (d) Quadratic function (e) Polynomial of degree 5 (f) Rational function
- Question : 7R - Sketch by hand, on the same axes, the graphs of the following functions. (a) f sxd ? x (b) tsxd ? x 2 (c) hsxd ? x 3 (d) jsxd ? x 4
- Question : 8R - Draw, by hand, a rough sketch of the graph of each function. (a) y ? sin x (b) y ? cos x (c) y ? tan x (d) y ? 1yx (e) y ? | x | (f) y ? sx
- Question : 9R - Suppose that f has domain A and t has domain B. (a) What is the domain of f 1 t? (b) What is the domain of f t? (c) What is the domain of fyt
- Question : 10R - How is the composite function f 8 t defned? What is its domain?
- Question : 11R - Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Re?ect about the x-axis. (f) Re?ect about the y-axis. (g) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. (i) Stretch horizontally by a factor of 2. (j) Shrink horizontally by a factor of 2.
- Question : 12R - Explain what each of the following means and illustrate with a sketch. (a) lim x la f sxd ? L (b) lim x la1 f sxd ? L (c) lim x la2 f sxd ? L (d) lim x la f sxd ? ` (e) lim x la f sxd ? 2`
- Question : 13R - Describe several ways in which a limit can fail to exist. Illustrate with sketches.
- Question : 14R - What does it mean to say that the line x ? a is a vertical asymptote of the curve y ? f sxd? Draw curves to illustrate the various possibilities.
- Question : 15R - . State the following Limit Laws. (a) Sum Law (b) Difference Law (c) Constant Multiple Law (d) Product Law (e) Quotient Law (f) Power Law (g) Root Law
- Question : 16R - What does the Squeeze Theorem say?
- Question : 17R - (a) What does it mean for f to be continuous at a? (b) What does it mean for f to be continuous on the interval s2`, `d? What can you say about the graph of such a function?
- Question : 18R - (a) Give examples of functions that are continuous on f21, 1g. (b) Give an example of a function that is not continuous on f0, 1g
- Question : 19R - What does the Intermediate Value Theorem say?
- Question : 1TFQ - Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f is a function, then f ss 1 td ? f ssd 1 f std
- Question : 2TFQ - If f ssd ? f std, then s ? t.
- Question : 3TFQ - If f is a function, then f s3xd ? 3f sxd.
- Question : 4TFQ - If x1 , x2 and f is a decreasing function, then f sx1d . f sx2d
- Question : 5TFQ - A vertical line intersects the graph of a function at most once
- Question : 6TFQ - If x is any real number, then sx 2 ? x
- Question : 7TFQ - lim x l4S x 2 2 x 4 2 x 2 8 4D ? xlim l4 x 2 2 x 4 2 xlim l4 x 2 8 4
- Question : 8TFQ - lim x l1 x 2 1 6x 2 7 x 2 1 5x 2 6 ? lim x l1 sx 2 1 6x 2 7d lim x l1 sx 2 1 5x 2 6d
- Question : 9TFQ - lim x l 1 x 2 3 x 2 1 2x 2 4 ? lim x l 1 sx 2 3d lim x l 1 sx 2 1 2x 2 4d
- Question : 10TFQ - . x 2 2 9 x 2 3 ? x 1 3
- Question : 11TFQ - . lim x l 3 x 2 2 9 x 2 3 ? lim x l 3 sx 1 3
- Question : 12TFQ - If limx l 5 f sxd ? 2 and limx l 5 tsxd ? 0, then limx l 5 f f sxdytsxdg does not exist
- Question : 13TFQ - If lim x l5 f sxd ? 0 and limx l 5 tsxd ? 0, then limx l 5 f f sxdytsxdg does not exist
- Question : 14TFQ - If neither limx l a f sxd nor limx l a tsxd exists, then limx l a f f sxd 1 tsxdg does not exist.
- Question : 15TFQ - If limx l a f sxd exists but limx l a tsxd does not exist, then limx l a f f sxd 1 tsxdg does not exist
- Question : 16TFQ - If limx l 6 f f sxd tsxdg exists, then the limit must be f s6d ts6d.
- Question : 17TFQ - . If p is a polynomial, then limx l b psxd ? psbd.
- Question : 18TFQ - If limx l 0 f sxd ? ` and limx l 0 tsxd ? `, then limx l 0 f f sxd 2 tsxdg ? 0
- Question : 19TFQ - If the line x ? 1 is a vertical asymptote of y ? f sxd, then f is not defned at 1.
- Question : 20TFQ - If f s1d . 0 and f s3d , 0, then there exists a number c between 1 and 3 such that f scd ? 0.
- Question : 21TFQ - If f is continuous at 5 and f s5d ? 2 and f s4d ? 3, then limx l 2 f s4x 2 2 11d ? 2
- Question : 22TFQ - If f is continuous on f21, 1g and f s21d ? 4 and f s1d ? 3, then there exists a number r such that | r | , 1 and f srd ? .
- Question : 23TFQ - Let f be a function such that lim x l 0 f sxd ? 6. Then there exists a positive number such that if 0 , | x | , , then | f sxd 2 6 | , 1.
- Question : 24TFQ - If f sxd . 1 for all x and lim x l 0 f sxd exists, then lim x l 0 f sxd . 1.
- Question : 25TFQ - The equation x 10 2 10x 2 1 5 ? 0 has a root in the interval s0, 2d.
- Question : 26TFQ - If f is continuous at a, so is | f |.
- Question : 27TFQ - If | f | is continuous at a, so is f .
- Question : 1E - Let f be the function whose graph is given. y 1 x 1 f (a) Estimate the value of f s2d. (b) Estimate the values of x such that f sxd ? 3. (c) State the domain of f. (d) State the range of f. (e) On what interval is f increasing? (f) Is f even, odd, or neither even nor odd? Explain.
- Question : 2E - . Determine whether each curve is the graph of a function of x. If it is, state the domain and range of the function.
- Question : 3E - If f sxd ? x 2 2 2x 1 3, evaluate the difference quotient f sa 1 hd 2 f sad h
- Question : 4E - Sketch a rough graph of the yield of a crop as a function of the amount of fertilizer used.
- Question : 5E - 5
- Question : 6E - tsxd ? s16 2 x
- Question : 7E - . y ? 1 1 sin x
- Question : 8E - Fstd ? 3 1 cos 2t
- Question : 9E - Suppose that the graph of f is given. Describe how the graphs of the following functions can be obtained from the graph of f. (a) y ? f sxd 1 8 (b) y ? f sx 1 8d (c) y ? 1 1 2f sxd (d) y ? f sx 2 2d 2 2 (e) y ? 2f sxd (f) y ? 3 2 f sxd
- Question : 10E - The graph of f is given. Draw the graphs of the following functions. (a) y ? f sx 2 8d (b) y ? 2f sxd
- Question : 11E - 11
- Question : 12E - y ? 2sx
- Question : 13E - y ? x 2 2 2x 1 2
- Question : 14E - y ? 1 x 2 1
- Question : 15E - f sxd ? 2cos 2x
- Question : 16E - f sxd ?H1 1 1 1 x x 2 if if x x , > 0 0
- Question : 17E - Determine whether f is even, odd, or neither even nor odd. (a) f sxd ? 2x 5 2 3x 2 1 2 (b) f sxd ? x 3 2 x 7 (c) f sxd ? cossx 2d (d) f sxd ? 1 1 sin x
- Question : 18E - Find an expression for the function whose graph consists of the line segment from the point s22, 2d to the point s21, 0d together with the top half of the circle with center the origin and radius 1.
- Question : 19E - If f sxd ? sx and tsxd ? sin x, fnd the functions (a) f 8 t, (b) t 8 f , (c) f 8 f , (d) t 8 t, and their domains.
- Question : 20E - Express the function Fsxd ? 1ysx 1 sx as a composition of three functions.
- Question : 21E - Life expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth (in years) of males born in the United States. Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a male born in the year 2010. Birth year Life expectancy Birth year Life expectancy 1900 48.3 1960 66.6 1910 51.1 1970 67.1 1920 55.2 1980 70.0 1930 57.4 1990 71.8 1940 62.5 2000 73.0
- Question : 22E - A small-appliance manufacturer fnds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week. (a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent?
- Question : 23E - The graph of f is given. 0 x y 1 1 (a) Find each limit, or explain why it does not exist. (i) lim x l21 f sxd (ii) lim x l231 f sxd (iii) lim x l23 f sxd (iv) lim x l4 f sxd (v) lim x l0 f sxd (vi) lim x l22 f sxd (b) State the equations of the vertical asymptotes. (c) At what numbers is f discontinuous? Explain
- Question : 24E - Sketch the graph of an example of a function f that satisfes all of the following conditions: lim x l01 f sxd ? 22, lim x l02 f sxd ? 1, f s0d ? 21, lim x l22 f sxd ? `, lim x l21 f sxd ? 2`
- Question : 25E - 25
- Question : 26E - lim x l3 x 2 2 9 x 2 1 2x 2 3
- Question : 27E - lim x l23 x 2 2 9 x 2 1 2x 2 3
- Question : 28E - xlim l11 x 2 2 9 x 2 1 2x 2 3
- Question : 29E - lim h l0 sh 2 1d3 1 1 h
- Question : 30E - lim t l2 t 2 2 4 t 3 2 8
- Question : 31E - lim r l9 sr sr 2 9d4
- Question : 32E - vlim l 41 4 2 v | 4 2 v |
- Question : 33E - lim u l 1 u 4 2 1 u3 1 5u 2 2 6u
- Question : 34E - xlim l 3 sx x 31 263x 2 2 x
- Question : 35E - lim sl16 4 2 ss s 2 16
- Question : 36E - lim v l2 v 2 1 2v 2 8 v 4 2 16
- Question : 37E - lim x l 0 1 2 s1 2 x 2 x
- Question : 38E - 38. lim x l 1Sx 2 1 1 1 x 2 2 3 1x 1 2D
- Question : 39E - If 2x 2 1 < f sxd < x 2 for 0 , x , 3, fnd limx l1 f sxd.
- Question : 40E - Prove that limx l 0 x 2 coss1yx 2d ? 0
- Question : 41E - 41
- Question : 42E - lim x l 0 s 3 x ? 0
- Question : 43E - lim x l 2 sx 2 2 3xd ? 22
- Question : 44E - lim x l 41 2 sx 2 4 ? `
- Question : 45E - Let f sxd ?Hs 3 sx2 2 2 xx3d2 if if 0 if x x , . < 0 3 x , 3 (a) Evaluate each limit, if it exists. (i) lim x l01 f sxd (ii) lim x l02 f sxd (iii) lim x l0 f sxd (iv) lim x l32 f sxd (v) lim x l31 f sxd (vi) lim x l3 f sxd (b) Where is f discontinuous? (c) Sketch the graph of f.
- Question : 46E - Let tsxd ? 2x 2 x 2 2 2 x x 2 4 if 0 < x < 2 if 2 , x < 3 if 3 , x , 4 if x > 4 (a) For each of the numbers 2, 3, and 4, discover whether t is continuous from the left, continuous from the right, or continuous at the number. (b) Sketch the graph of t.
- Question : 47E - 47
- Question : 48E - . tsxd ? sx 2 2 9 x 2 2 2
- Question : 49E - 49
- Question : 50E - 2 sin x ? 3 2 2x, s0, 1d
- Question : 51E - Suppose that | f sxd | < tsxd for all x, where lim x l a tsxd ? 0. Find lim x l a f sxd
- Question : 52E - Let f sxd ? v x b 1 v2x b. (a) For what values of a does lim x l a f sxd ex (b) At what numbers is f discontinuous?
- Question : 1P - Solve the equation | 2x 2 1 | 2 | x 1 5 | ? 3.
- Question : 2P - Solve the inequality | x 2 1 | 2 | x 2 3 | > 5
- Question : 3P - Sketch the graph of the function f sxd ? | x 2 2 4| x | 1 3|.
- Question : 4P - Sketch the graph of the function tsxd ? | x 2 2 1 | 2 | x 2 2 4 |
- Question : 5P - Draw the graph of the equation x 1 | x | ? y 1 | y |.
- Question : 6P - Sketch the region in the plane consisting of all points sx, yd such that | x 2 y | 1 | x | 2 | y | < 2
- Question : 7P - The notation maxha, b, . . .j means the largest of the numbers a, b, . . . . Sketch the graph of each function. (a) f sxd ? maxhx, 1yxj (b) f sxd ? maxhsin x, cos xj (c) f sxd ? maxhx 2, 2 1 x, 2 2 xj
- Question : 8P - Sketch the region in the plane defned by each of the following equations or inequalities. (a) maxhx, 2yj ? 1 (b) 21 < maxhx, 2yj < 1 (c) maxhx, y 2j ? 1
- Question : 9P - A driver sets out on a journey. For the frst half of the distance she drives at the leisurely pace of 30 miyh; she drives the second half at 60 miyh. What is her average speed on this trip?
- Question : 10P - Is it true that f 8 st 1 hd ? f 8 t 1 f 8 h?
- Question : 11P - Prove that if n is a positive integer, then 7n 2 1 is divisible by 6.
- Question : 12P - Prove that 1 1 3 1 5 1 ??? 1 s2n 2 1d ? n2.
- Question : 13P - If f0sxd ? x 2 and fn11sxd ? f0s fnsxdd for n ? 0, 1, 2, . . . , fnd a formula for fnsxd.
- Question : 14P - (a) If f0sxd ? 1 2 2 x and fn11 ? f0 8 fn for n ? 0, 1, 2, . . . , fnd an expression for fnsxd and use mathematical induction to prove it. (b) Graph f0, f1, f2, f3 on the same screen and describe the effects of repeated composition
- Question : 15P - Evaluate lim x l1 s 3 x 2 1 sx 2 1
- Question : 16P - Find numbers a and b such that lim x l0 sax 1 b 2 2 x ? 1.
- Question : 17P - Evaluate lim x l 0 | 2x 2 1 | 2 | 2x 1 1 | x
- Question : 18P - The fgure shows a point P on the parabola y ? x 2 and the point Q where the perpendicular bisector of OP intersects the y-axis. As P approaches the origin along the parabola, what happens to Q? Does it have a limiting position? If so, fnd it.
- Question : 19P - Evaluate the following limits, if they exist, where v x b denotes the greatest integer function. (a) lim x l 0 v x b x (b) lim x l 0 x v1yx b
- Question : 20P - Sketch the region in the plane defned by each of the following equations. (a) v xb 2 1 v yb 2 ? 1 (b) v xb 2 2 v yb 2 ? 3 (c) v x 1 yb 2 ? 1 (d) v xb 1 v yb ? 1
- Question : 21P - Find all values of a such that f is continuous on R: f sxd ?Hx x 21 1 if if x x < . A
- Question : 22P - A fxed point of a function f is a number c in its domain such that f scd ? c. (The function doesn
- Question : 23P - If limx l a f f sxd 1 tsxdg ? 2 and limx l a f f sxd 2 tsxdg ? 1, fnd limx l a f f sxd tsxdg.
- Question : 24P - (a) The fgure shows an isosceles triangle ABC with /B ? /C. The bisector of angle B intersects the side AC at the point P. Suppose that the base BC remains fxed but the altitude | AM | of the triangle approaches 0, so A approaches the midpoint M of BC. What happens to P during this process? Does it have a limiting position? If so, fnd it. A B C M P (b) Try to sketch the path traced out by P during this process. Then fnd an equation of this curve and use this equation to sketch the curve
- Question : 25P - (a) If we start from 0

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