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Calculus BC Supplemental Homework Problems for Chapter 3 1. Find the anti-derivative 2 (a) f(x) = 3x + sin(2x) – 5 (b) g(x) = 5 3( x 2) 5 x 2. Let cos(arcsin(x)) = (a) sin(arccos(x)) 1 x 2 . Give similar expressions for the following: (b) tan(arcsin(3x)) 3. (a) Find the slope of a tangent line to y = ln(x) at x = e2, as well as, the slope of a tangent line to y = ex at x = 2. (b) Explain the geometric relationship of your answers to (a). 4. By hand, complete the table for r = cos(2θ) and make a neat polar graph. θ 0 30 60 90 120 150 180 210 240 270 300 330 r (Should I have used radians? Yes. Was I too lazy to insert all of those fractions and π symbols? Yes.) Ditto says Dr. T. 5. Give the equation of the line tangent to the graph of y = x + cos(x) at the point (0,1). 6. If the graph of y = x3 + ax2 + bx – 4 has a point of inflection at (1, -6), what is the value of b? 7. Bacteria in a certain culture increase at a rate proportional to the number present. If the number of bacteria doubles in three hours, in how many hours will the number of bacteria triple? Answer exactly. 8. On [0,3], what is the maximum acceleration attainted by the particle whose velocity is given by v(t) = t3 – 3t2 + 12t + 4? x . x 1 9. Let f be the function given by f ( x) ln (a) What is the domain of f? (b) Find the value of the derivative of f(x) at x = - 1. (c) Write an expression for f -1(x). This denotes the inverse of the function f. 10. Let f be a function that is even and continuous on the closed interval [-3,3]. The function f and its derivatives have the properties indicated in the table below. x f(x) f ′ (x) f ′′ (x) 0 1 Undefined Undefined 0<x<1 Positive Negative Positive 1 0 0 0 1<x<2 Negative Negative Negative 2 -1 Undefined Undefined 2<x<3 Negative Positive Negative (a) Find the x-coordinate of each point at which f attains an absolute maximum value or an absolute minimum value. For each x-coordinate you give, state whether f attains an absolute maximum or an absolute minimum. (b) Find the x-coordinate of each point of inflection on the graph of f. Justify your answer. (c) In the xy-plane , sketch the graph of a function with all the given characteristics of f.