Stanford University
AP Calculus BC
1. Find the dimensions of the rectangle of√largest area that has its base lying on the x-axis and its other two vertices
above the x-axis on the semi-circle y = 4 − x2 .
√
√
Solution. 2 2 × 2
2. Let f (x) = x4 − 2x2 + 3.
(a) Find the intervals on which f is increasing or decreasing.
Solution. f is increasing in (1, ∞) and decreasing on (−∞, −1) and (0, 1)
(b) Find the local maximum and minimum values of f .
Solution. x = 0 is a local maximum and x = −1, 1 are local minima
√
(c) Find
the inflection
points of f . Solution. f is concave
up on (−∞, − 3/3) and
√ the intervals of concavity and √
√
√
( 3/3, ∞) and concave down on (− 3/3, 3/3). The inflection points are x = ± 3/3.
3. Find y 0 if
sin(x + y) = y 2 cos x.
Solution.
y0 =
y 2 sin x + cos(x + y)
.
2y cos x − cos(x + y)
4. Show that the sum of the x- and y-intercepts of any tangent line to the curve
√
√
√
x+ y = c
is equal to c.
Solution. An equation of the tangent line at (x0 , y0 ) is
√
− y0
y − y0 = √ (x − x0 ).
x0
√ √
√ √
The x-intercept is x0 + x0 y0 and the y-intercept is y0 + x0 y0 .
5. A man walks along a straight path at a speed of 4 feet per second. A searchlight is located on the ground 20 feet from
the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 15 feet from the
point on the path closest to the searchlight?
Solution. 16/125 = 0.128 radians per second
6. A water tank has the shape of an inverted circular cone with base radius 2 meters and height 4 meters. If water is
being pumped into the tank at a rate of 2 cubic meters per minute, find the rate at which the water level is rising when
the water is 3 meters deep.
8
Solution.
≈ 0.28 meters per minute.
9π
7. The position s of an object as a function of time t is given by s(t) = t3 − 12t − 5.
(a) Find the time intervals during which the object is traveling in the forward direction. Solution. (−∞, −2) and
(2, ∞)
(b) When is the acceleration of the object positive?
Solution. t > 0
(c) Find the total distance traveled by the object between t = 0 and t = 3.
Solution. 23
[email protected]
1
EPGY OHS
Stanford University
AP Calculus BC
8. The volume of a cube is increasing at a rate of 10 cm3 per minute. How fast is the surface area increasing when the
length of an edge is 30 cm?
Solution. 4/3 square centimeters per minute
9. Find the local linear approximation for the function f (x) =
√
3
1 + 3x at x = 0, and use it to approximate
√
3
1.03.
Solution. y = 1 + x; y(0.01) ≈ 1.01
10. Evaluate each of the following indefinite integrals.
Z
√
1
x + √ dx
(a)
x
√
2
Solution. x3/2 + 2 x + C
3
√
Z √
u u+ u
(b)
du
u2
Solution. −2u−1/2 + 2u1/2 + C
Z
(c)
3 sin x − π cos x dx
Solution. −3 cos x − π sin x + C
11. Find the solution of the initial-value problem
y(1) = −5.
dy
= 3x−2/3 ,
dx
Solution. y = 9x1/3 − 14
12. Find the solution of the initial-value problem
d2 y
= 2 − 6x, y 0 (0) = 4, y(0) = 1.
dx2
Solution. y = x2 − x3 + 4x + 1
13. Use right rectangular approximation with n = 4 rectangles to estimate the area between the graph of f (x) = x2 and
the x−axis from x = 1 to x = 5. Draw a picture that clearly illustrates how you are obtaining your approximation. Is
this approximation greater than the actual area or less than the actual area? Why?
Solution. R4 = 54. This approximation is greater than the actual area, as demonstrated by the graph.
14. The positions of two particles on a coordinate line are
s1 (t) = 3t3 − 12t2 + 18t + 5, s2 (t) = −t3 + 9t2 − 12t.
When do the particles have the same velocities?
Solution. t = 1 and t = 2.5
15. Suppose that the first derivative of y = f (x) is
y 0 = 6(x + 1)(x − 2)2 .
At what points, if any, does the graph of f have a local maximum, local minimum, or point of inlection?
Solution. Local minimum at x = −1. No local maximum. Points of inflection at x = 0, x = 2.
16. The sum of two non-negative numbers is 36. Find the number if (a) the difference of their square roots is to be as large
as possible and (b) the sum of their square roots is to be as large as possible.
Solution. (a) 0 and 36. (b) 18 and 18.
[email protected]
2
EPGY OHS
Stanford University
AP Calculus BC
17. (AP Exam)
Solution. (a) 0 < t < 1 and 3 < t < 6. (b) 0 < t < 1 and 3 < t < 4. (c) P is slowing down at time t = 3.
[email protected]
3
EPGY OHS