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AP Calculus BC
Summer Practice Set
Please complete the following problems on a SEPARATE SHEET OF PAPER. In preparation for the year to
come, your work should be clearly shown and neatly organized.
Functions
1. Find the domain and range of the following functions.
a) y 
2
x4
e) y | x  1 |
b) y  ln x
c) y  x 4  2 x 2  1
f) y  e x
g) y 
x2
x2  4
d) y 
h) y 
x 3
x
x 1
2. Identify all asymptotes of the graphs:
3x 2  13x  4
x4
c) f ( x)  2
2
x 9
x 1
1 x
3. Given the functions f ( x)  x 2 , g ( x)  4 x  3 , h( x) 
, find the following:
2x
a) y 
x
x3
a) f(-2)
b) f ( x) 
b) f(g(2))
c) g -1(x)
d) f(h(x))
e) f(g(h(x)))
4. Given f ( x)  x 2  2 x  1 , find f(x+3) – f(3)
5. Let f ( x)  x 2 . Describe in words what the following transformations would do to the graph of f.
a) f ( x)  2
b) f ( x  2)
c) f (2 x)
d) 2 f ( x)
e)  f (x)
f) f ( x)
6. Identify as even, odd, or neither. Show substitutions!
Even: f(-x) = f(x)
3
2
a) f ( x)  x  3x
b) f ( x)  x  x
7. Draw the graph of the piecewise function:
Odd: f(-x) = -f(x)
c) f ( x)  x 4  6 x 2  3
8. Write the equation of this piecewise function:
 x, x  2


f ( x)   x 2 , 2  x  1

1, x  1

Geometry
9. A piece of wire inches long is cut into two pieces. One piece is inches long and is to be bent into
the shape of a square. The other piece is to be bent into the shape of a circle. Find an equation for the
total area made up by the square and the circle as a function of .
Algebra Practice
10. Rationalize the denominator (a and b) and the numerator (c):
2
a)
3 2
b)
4
c)
1 5
x 1  2
x3
Hint: Multiply by the conjugate.
11. Solve for x:
a) 4 x 2  12 x  3  0
6
5
x
d) x 
2 5 1
 
3 6 x
x 1 x 1
f)

1
3
2
b) 3x  ( x  7)  4 x  5
e)
c)
x5 3

x 1 5
12. Solve for y: xy  z  1  y
13. Solve for a: V  2(ab  bc  ca)
14. Solve for p:
x y z
  1
p q r
15. Factor the following completely:
a) x 6  16x 4
b) 8 x 3  27
c) 9 x 2  3x  2
d) x 4  1
16. Simplify the expressions until they are in the form ax b y c where a, b, and c are numbers:
3x 
a)
2 4
b) 16xy 5
y
c)
x(2 / y )
3/ x
d)
xy  x
y2  y
17. Simplify the following expressions as much as possible:
a)
x3  9x
x 2  7 x  12
b)
( x  1) 3 ( x  2)  3( x  1) 2
( x  1) 4
c)
e*) ( x 2  1)1 / 2  ( x 2  1) 3 / 2
d*)  xe 2 x  x 2 e x  e x
2
3
x
1
1
x 1
Hint: Factor out any like
terms to simplify for d and e
18. Expand the following binomial expressions:
a) (
)
b) (
19. Find the fifth term in the expansion of (
)
c) (
)
) , when expanded in descending powers of .
20. TRUE OR FALSE. Are the following statements true? If not, explain in words why not.
a)
2x
x

2y  z y  z
d) 5
x 5x

y 5y
b)
1
1 1
 
ab a b
e) 5
x 5x

y
y
c)
mn m n
 
2
2 2
f) 5
p  q 5p  q

r
r
d) (
)
g)
3x  6
6
3x
h)
j) 2 x 2  3  (3  2 x 2 )
3x  6 x  2

3x
x
i)
2x  5
5  2x

3
3
l) x a  x b  x a b
k) (a  b) 2  a 2  b 2
21. Simplify the following expression using long division.
Log and Exponential Functions
22. Solve for x without using a calculator:
a) 4 ( x 3)  20
b) log 10 x  2
c) log 5 625  x
e) 2e
2 x
 10
g) 5  2 ln x  25
Remember you can rewrite an exponential
function in log form or rewrite a log in
exponential form.
d) e x  15
log b x  a if and only if
f) ln x  2
ba  x
h) log 6 ( x  3)  log 6 ( x  4)  1
23. Simplify the following expressions using the properties of natural log (without a calculator):
a) ln e
b) e ln 2
c) ln 1
d) ln 0
e) ln 4
f) ln 5  ln( x 2  1)  ln( x  1)
g) 2 ln 9  ln 3
h) 2 ln x  ln 10
24. Explain how the natural log function, y  ln x , and the exponential function, y  e x , are related. Discuss
their graphs and list all of their properties (like domain and range, asymptotes, etc.) in your explanation.
Trigonometry
25. Give the exact values of the following (do not use your calculator).
a) cos 0
b) sin 0
e) sin 
5
f) tan
3
c) tan

2
5
g) sin
6
d) cos

h) cos
4

4
26. Give the exact values of the inverse trig functions. Give your answers in radians.
a) arccos
3
2
d) cos 1 2
2
g) tan 1 3
b) arctan 1
c) arcsin 1
e) cos 1   2 
 2 
f) sin 1   1 

h) sin 1

3
2
 2
i) tan 1 0
Recall: there are two notations
for inverse trig functions.
arcsin x  sin 1 x
arccos x  cos 1 x
arctan x  tan 1 x
27. Which of the following expressions are identical?
cos 2 x , (cos x) 2 , cos x 2 , 2 cos x
28. Rewrite each of the following without the -1 exponent.
1
a) sin 1 x = ___________
b) sin x   ________
c) sin x 1 =___________
29. Solve each equation for x on the interval [0, 2π].
1
2
e) cos x  sin x  0
a) sin x 
b) 2 cos x  3  0
c) cos 2 x  cos x
f) 2 cos x sin x  cos x  0
g) 2 cos 2 x  cos x  1  0
d) 4 sin 2 x  1
30. Show that
(
(
31. Show that
32. Solve
)
)
on the interval
.
Calculus
33. Use the graph of g ' ( x) shown below to answer the following questions.
a) On what intervals is the graph of g(x) increasing? Decreasing? Justify your answer.
b) On what intervals is the graph of g(x) concave up? Concave down? Justify your answer.
c) Identify the x-values of all relative extrema on g(x).
d) Identify the x-values of all points of inflection on g(x).
g’(x)
34. Use the graph of f ' ' ( x) shown below to answer the following questions.
a) On what intervals is the graph of f ’(x) increasing? Decreasing? Justify your answer.
b) On what intervals is the graph of f(x) concave up? Concave down? Justify your answer.
c) Identify the x-values of all relative extrema on f ’(x).
d) Identify the x-values of all points of inflection on f(x).
f ’’(x)

2
2
2
2
35. Let F ( x) 

x
0
f (t )dt where f is the function graphed below (consisting of lines and a semi-circle).
Find the following:
a) F(0)
b) F(2)
d) F(-1)
e) F(-2)
g) F’(4)
h) F’(2)
c) F(4)
f) F(-3)
i) F’’(5)
36. The line y = 16x – 9 is a tangent to the curve y = 2x3 + ax2 + bx – 9 at the point (1,7). Find the values of a and b.
37. For what values of m is the line y = mx + 5 a tangent to the parabola y = 4 – x2?
38. A curve has equation x3 y2 = 8. Find the equation of the line normal to the curve at the point (2, 1).
39. If 2x2 – 3y2 = 2, find the two values of
dy
when x = 5.
dx
dy
dx .
41. Air is pumped into a spherical ball which expands at a rate of 8 cm3 per second (8 cm3 s–1). Find the exact rate
of increase of the radius of the ball when the radius is 2 cm. (hint: V  43 r 3 )
40. If x 2 ( x 2  y 2 )  y 2 , find
42. The radius of a circular oil patch is increasing at a rate of 1.2cm/min. Find the rate at which the surface area of
the patch is increasing when the radius is 25cm.
43. A man 1.8m tall is walking directly away from a street lamp 3.2m above the ground at a speed of 0.7m/s.
How fast is the length of his shadow increasing? (hint: think similar triangles)
44. Find two positive numbers whose product is 192 and their sum is a minimum.
(hint: set up your two equations – constraint and primary, once you have plugged the constraint into the primary
equation, differentiate and set equal to 0 to minimize.)
46. Evaluate the integral
47. Find

x
1 x  1 dx.
2
2
 x 

 dx.
2– x
k
48. Find the real number k > 1 for which 1  12  dx = 3 .
2
 x 
1

1

t3 1

1 
49. Let f(t) =  – 5  . Find f (t ) dt.


2t 3 

m
dx
50. Find
, giving your answer in terms of m.
0 2x  3


y
51. Calculate the area bounded by the graph of y = x sin (x2) and the x-axis, between x = 0 and the smallest
positive x-intercept.
52. The diagram shows part of the graph of y = 1 . The area of the shaded region is 2 units. Find the exact value of a.
x
0
1
a
x
53. Find the area of the region enclosed by the graphs of y = sin x and y = x2 – 2x + 1.5, where 0  x  .
54. Calculate the area enclosed by the curves y = lnx and y = ex – e, x > 0.
55. Find the total area of the two regions enclosed by the curve y = x3 – 3x2 – 9x +27 and the line y = x + 3.
56. The area between the graph of y = ex and the x-axis from x = 0 to x = k (k > 0) is rotated through 360°
about the x-axis. Find, in terms of k and e, the volume of the solid generated.
57. The area of the enclosed region shown in the diagram is defined by
y ≥ x2 + 2, y ≤ ax + 2,
where a > 0.
This region is rotated 360° about the x-axis to form a solid of revolution.
Find, in terms of a, the volume of this solid of revolution.
 x ln x dx.
(Hint: Integration by Parts is  udv  uv   vdu .)
58. Use integration by parts to find
2
59. Use integration by parts to evaluate ∫
.
dy
60. Solve the differential equation xy
= 1 + y2, given that y = 0 when x = 2.
dx
(Hint: separate variables, integrate, and find particular solution y = …)
y
2
0
a
x
1/--pages
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