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- Question : EX1 - The radium in a piece of lead decomposes at a rate which is proportional th the amount present. If 10 percent of the radium decomposes in 200 yeans, what percent of the original amount of radium will be present in a piece of lead after 1000 years?
- Question : EX2 - Assume that the half life of the radium in a piece of lead is 1600 years. How much radium will be lost in 100 years?
- Question : EX3 - The following item appeared in a newspaper.
- Question : EX1 - Describe, in words, each of the following sets:
- Question : EX2 - Define the area A of a circle as a function of its radius r. Which is the de
- Question : EX3 - Under certain circumstances the pressure p of a gas and its volume V are related by the formula p F 3/2 = 1. Express each variable as a function of the other.
- Question : EX4 - Explain the difference between the function
- Question : EX5 - Let
- Question : EX6 - If . x2
- Question : EX7 - Why is the function defined in 6 not the same as the function defined by g(x') = x
- Question : EX8 - Determine the domain D for which each of the following formulas defines z as a function of x and y. {&) z
- Question : EX9 - Which of the domains in problem 8 are regions?
- Question : EX10 - If/(z,y) is a function of two independent variables x,y defined over a domain D, then the symbol f(b,y) for any element (b,y) in D, means the function of y obtained by replacing z by b; see Definition 2.65. Let /(z,y) = z2 + 2xy + log (xy), xy > 0. Find: (a) /(z ,l), f(x,b), /(z,0), /(z ,z 2). (b) /(a,y), /(0,y), /( z 2,y). (c) /(2,
- Question : EX11 - Draw the graphs of three different functions defined by x3 4" y3
- Question : EX12 - The following is a standard type of exercise in the calculus. dli/ c?t/ If x3 + y3
- Question : EX13 - Explain why the procedure followed in problem 12, applied to the relation x2 4" y2 4
- Question : EX14 - Find the function g(x) that is implicitly defined by the relation V ^~ -
- Question : EX15 - Explain why ?Ilin y V x2
- Question : EX16 - Can you apply the method of implicit differentiation as taught in the calculus to the function of problem 14, of problem 15?
- Question : EX17 - . Define a function of three independent variables xj, X2, X3; of n independent variables xi
- Question : EX1 - Determine the order of the following differential equations.
- Question : EX2 - Prove that the functions in the right-hand column below are solutions of the differential equations in the left-hand columns. (Be sure to state the common interval for which solution and differential equation make sense.)
- Question : EX3 - Show that the differential equation
- Question : EX4 - Determine whether, the equations on the right define implicit functions of z. For those which do, determine whether they are implicit solutions of the differential equations on the left.
- Question : EX1 - y = ci 4~ C2
- Question : EX2 - y
- Question : EX3 - y ~ cix 4
- Question : EX4 - y
- Question : EX5 - y ~ ci 4
- Question : EX6 - y
- Question : EX7 - x
- Question : EX8 - y ~ Ci cos 3x 4" C2 sin 3z.
- Question : EX9 - r = 9 tan (9 4" c).
- Question : EX10 - y = ex 4
- Question : EX11 - y
- Question : EX12 - y = cie***.
- Question : EX13 - y == z3 4
- Question : EX14 - y = eie2x + cse~2x.
- Question : EX15 - (y
- Question : EX16 - r = a(l
- Question : EX17 - log y = cix H~ C2.
- Question : EX18 - A family of circles of fixed radii and centers on the x axis.
- Question : EX19 - A family of circles of variable radii, centers on the x axis and passing through the origin.
- Question : EX20 - A family of circles with centers at (h,k) and of fixed radius.
- Question : EX21 - A family of circles with centers in the zy-plane and of variable radii. Hint. Write the equation of the family as z2 4r y2
- Question : EX22 - A family of parabolas with vertices at the origin and foci on the x axis
- Question : EX23 - A family of parabolas with foci at the origin and vertices on the x axi
- Question : EX24 - A family of parabolas with foci and vertices on the x axis.
- Question : EX25 - A family of parabolas with axes parallel to the x axis and with a fixed dis
- Question : EX26 - A family of equilateral hyperbolas whose asymptotes are the coordinate axes.
- Question : EX27 - A family of straight lines whose y intercept is a function of its slope.
- Question : EX28 - A family of straight lines that are tangents to the parabola y2 = 2z.
- Question : EX29 - A family of straight lines that are tangents to the circle x2 4
- Question : EX30 - Find a 1-parameter family of solutions of the differential equation dy = y dx and the particular solution for which y(3) - 1.
- Question : EX1 - Construct a direction field for the differential equation y' = 2x. Draw an integral curve.
- Question : EX2 - Construct a direction field for the differential equation X Where are its singular points? How many parameters are required to include all possible solutions? Draw the integral curve that goes through the points (
- Question : EX3 - Construct a direction field for the differential equation x ~ y x + y Draw an integral curve that goes through the point (1,1).
- Question : EX4 - Find the isoclines of the direction field and indicate the slope at each point where an integral curve crosses an isocline, (a) for problem 2, (b) for problem 3.
- Question : EX5 - Describe the isoclines of the direction field, if yf

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