Download Chapter 3 Supplementary Problems

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.