Calculus BC Supplemental Homework Problems for Chapter 3

1. Find the anti-derivative

(a) f(x) = 3x + sin(2x) – 5 (b) g(x) =

2. Let cos(arcsin(x)) = . Give similar expressions for the following:

(a) sin(arccos(x)) (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?

9. Let f be the function given by .

(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 / 0 / 0 < x < 1 / 1 / 1 < x < 2 / 2 / 2 < x < 3
f(x) / 1 / Positive / 0 / Negative / -1 / Negative
f ′ (x) / Undefined / Negative / 0 / Negative / Undefined / Positive
f ′′ (x) / Undefined / Positive / 0 / Negative / Undefined / 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.