88AB6 WorkBook Integration Example AP Calculus Exam

1. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(AB1)
Let f be the function given by f(x) = x4 - 16x2 .
(a) Find the domain of f.
(b) Describe the symmetry, if any, of the graph of f.
(c) Find f ’(x).
(d) Find the slope of the line normal to the graph of f at x = 5.

2. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(AB2)
A particle moves along the x-axis so that its velocity at any time t ≥
≥
≥
≥ 0 is
given by v(t) = 1 - sin(2π
π
π
πt).
(a) Find the acceleration a(t) of the particle at any time t.
(b) Find all values of t, 0 ≤ t ≤ 2, for which the particle is at rest.
(c) Find the position x(t) of the particle at any time t if x(0) = 0.

3. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(AB3)
Let R be the region in the first quadrant enclosed by the hyperbola
x2 - y2 = 9, the x-axis, and the line x = 5.
(a) Find the volume of the solid generated by revolving R about the x-axis.
(b) Set up, but do not integrate, an integral expression in terms of a single
variable for the volume of the solid generated when R is revolved about the line
x = -1.

4. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(AB4)
Let f be the function defined by f(x) = 2xe-x for all real numbers x.
(a) Write an equation of the horizontal asymptote for the graph of f.
(b) Find the x-coordinate of each critical point of f. For each such x, determine
whether f(x) is a relative maximum, a relative minimum, or neither.
(c) For what values of x is the graph of f concave down?
(d) Using the results found in parts (a), (b), and (c), sketch the graph of y = f(x) in
the xy-plane provided below.

5. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(AB5)
Let R be the region in the first quadrant under the graph of y =
x
x2 + 2
for 0 ≤
≤
≤
≤ x ≤
≤
≤
≤ 6 .
(a) Find the area of R.
(b) If the line x = k divides R into two regions of equal area, what is the value of k?
(c) What is the average value of y =
x
x2 + 2
on the interval 0 ≤ x ≤ 6 .

6. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(AB6)
Let f be a differentiable function, defined for all real numbers x, with
the following properties.
(i) f ’(x) = ax2 + bx
(ii) f ’(1) = 6 and f ’’(1) = 18
(iii) ⌡
⌠
1
2
f(x)dx = 18
Find f(x). Show your work.

7. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(BC1)
Let f be the function defined by f(x) = (x2 - 3)ex for all real numbers x.
(a) For what values of x is f increasing?
(b) Find the x-coordinate of each point of inflection of f.
(c) Find the x- and y-coordinates of the point, if any, where f(x) attains its absolute
minimum.

8. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
0.5 1 1.5 2 2.5 3
x
1
2
3
4
y
(BC2)
Let R be the shaded region between the graphs of y =
3
x
and y =
3x
x2 + 1
from x = 1 to x = 3 , as shown in the figure above.
(a) Find the area of R.
(b) Find the volume of the solid generated by revolving R about the y-axis.

9. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
B
x y
A 100 C
(BC3)
The figure above represents an observer at point A watching balloon B
as it rises from point C. The balloon is rising at a constant rate of 3 meters
per second and the observer is 100 meters from point C.
(a) Find the rate of change in x at the instant when y = 50.
(b) Find the rate of change in the area of the right triangle BCA at the instant
when y = 50.
(c) Find the rate of change in θ at the instant when y = 50.

10. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(BC4)
Determine all values of x for which the series ∑
k = 0
∞
∞
∞
∞
2kxk
ln(k + 2)
converges.
Justify your answer.

11. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
0.25 0.5 0.75 1 1.25 1.5
x
0.2
0.4
0.6
0.8
1
y = 1 - sin x
(BC5)
The base of a solid S is the shaded region in the xy-plane enclosed by
the x-axis, the y-axis, and the graph of y = 1 - sin x, as shown in the figure
above. For each x, the cross section of S perpendicular to the x-axis at the
point (x, 0) is an isosceles right triangle whose hypotenuse lies in the xy-
plane.
(a) Find the area of the triangle as a function of x.
(b) Find the volume of S.

12. Calculus WorkBook 88 Name:_______________
Format © 1997 – 1999 MNA Consulting
calcpage@aol.com
(BC6)
Let f be a differentiable function defined for all x ≥
≥
≥
≥ 0 such that f(0) = 5
and f(3) = -1. Suppose that for any number b > 0 the average value of f(x) on
the interval 0 ≤
≤
≤
≤ x ≤
≤
≤
≤ b is
f(0) + f(b)
2
.
(a) Find ⌡
⌠
0
3
f(x)dx.
(b) Prove that f ’(x) =
f(x) - 5
x
for all x > 0.
(c) Using part (b), find f(x).