Name _________________________________________
Date: _____________
AP Calculus Readiness Packet
Following are a selection of problems you may want to choose from to prepare a
summer packet for rising Calculus students.
Algebra
Exponents:
1.
(8x3yz)1/3(2x)3
4x1/3(yz2/3)–1
Factor
2.
9x2 + 3x – 3xy – y
3.
64x6 – 1
4.
42x4 + 35x2 – 28
5.
x3/2 + x – 12x1/2
6.
e2x + 2 + e–2x
7.
x–1 – 3x–2 + 2x–3
Rationalize the numerator or denominator
8.
9.
2 – 5x
2
Simplify the rational expression
11.
(x + 1)3(x – 2) + 3(x + 1)2
(x + 1)4
10.
x
4 – x – 2
x + 1 – 1
x
Solve algebraic equations and inequalities:
Use synthetic division to help factor the following, state all factors and roots.
12.
p(x) =x3 + 4x2 + x – 6
13.
p(x)= 6x3 – 17x2 – 16x + 7
14.
p(x) = 25x3 + 50x2 – 9x – 18
15.
p(x) = x4 – x3 – 2x – 4
16.
Explain why 3/2 cannot be a root of
4x + cx – dx + 5 = 0, where c and d are
integers
5
3
4
17.
2
Explain why x + 7x + x – 5 = 0 must
have a root in the interval [0, 1], (0 x 1)
Solve:
18.
(x + 3)2 > 4
19.
(x + 5)
<0
(x – 3)
20.
1
<1
x
21.
x(x – l)(x – 2) > 0
22.
x3 + 2x2 – 3x 0
23.
1<x
x
24.
x2 + 1 > 4x
25.
x2 – 9 > 0
x+1
26.
1
4
+
>0
x–1 x–6
2
512
Solve
27.
x < 2
30.
0 < x – 2 < 8
28.
2x + 1 < 1/4
31.
29.
5x – 1 > 9
0 < x < 1
Solve the system. (Check solution by graphing each function and tracing to find points
of intersection.)
32.
x–y+1=0
y – x2 = –5
4.
x2 – 4x + 3 = y
– 2
x + 6x – 9 = y
Graphing and Functions
Linear graphs: Write the equation of the line described below.
1.
–
Passes through the point (2, 1)
and has slope – 1/3.
2.
–
Passes through the point (4, 3)
and is perpendicular to 3x + 2y = 4.
3.
–
–
Passes through ( 1, 2) and is
parallel to y = 3/5x – I.
Conic Sections:
Write the equation in standard form and identify the conic.
4.
x2 + y2 – 4x + 2y – 4 = 0
5.
x = 4y2 + 4y – 3
6.
4x2 – 16x + 3y2 + 24y + 52 = 0
7.
x2 + 2x – 4y2 + 24y – 19 = 0
3
512
Functions:
Find the domain and range of the following.
Note: domain restrictions – denominator 0, argument of a log or ln > 0,
radicand of even index must be 0
range restrictions - reasoning, if all else fails, use graphing calculator
8.
y=
3
x–2
9.
y = log(x – 3)
y = 2x – 3
10.
y = x4 + x2 + 2
11.
12.
y= x – 5
13.
y = x2 + 1
x –1
domain only
14.
f(x) = 2 x – 3
x + 3x + 2
15. Given f(x) below on the domain [3,3], what is the
range?
f(x) =
x if x 0
1 if –1 x < 0
x – 2 if x –1
{
Find the composition and inverses as indicated; below.
16.
Let f(x) = x2 + 3x – 2 g(x) = 4x – 3 h(x) = lnx
g–1(x)
h–1(x)
w–1(x), for x 4
g(w(x))
g(h(x))
w(x) = x – 4
f(g(x))
h(g(f(1)))
4
512
17.
Let f(x) = 2x, g(x) = –x, and h(x) = 4, find
(f o g)(x)
(h o f)(x)
(g o h o g)(x)
18.
Find f-1(x) for each of the following.
f(x) = x + 6
f(x) = x + 2 , for x –2
f(x) = x/7
f(x) = x2 + 5, for x 5
f(x) = 6/(x + 1)
f(x) = x3
19.
Does 3x2 – 9 have an inverse function? Explain your answer.
20.
Let s(x) = 4 – x and t(x) = x2 find
the domain of s
the domain of t
(t o s)(x)
the domain of (s o t)(x)
(s o t)(x)
the domain of (t o s)(x)
Basic Shapes of Curves:
Sketch the graphs. You may use your graphing calculator to verify your graph, but you
should be able to graph by knowing the shape of the curve and plotting a few points.
21.
y = x
22.
y = lnx
23.
y = 1/x
24.
y = x – 2
25. y =
1
x–2
26. y =
x
x2 – 4
27. y = 4x + 3
2x – l
28.
y = 3x
Sketch the graphs. You may use your graphing calculator to verify your graph, but you
should be able to graph by knowing the shape of the curve and plotting a few points.
29.
y = 2–x
30.
y = 9 – x2
31.
y = 3sin2(x – /6)
5
512
Sketch the graphs
32.
x2 + y2 – 4x – 32 = 0
33.
f(x) =
{
25 – x2 for x < O
x2 – 25
for x 0, x 5
x–5
0 for x = 5
For each function identify any asymptotes.
34. y =
1
x–I
38.
y = Ixl + 1
35. y = x5 + x3 + x
39. y =
36.
x2 – y2 = 1
3x2
2x – 3x + 3
40.
2
Even, Odd, Tests for Symmetry:
Identify as odd, even, or neither: (Even: f(x) = f(–x)
37.
y=
y=
x2
x –1
2
5x2 – 5x + 1
x–1
Odd: f(x) = –f(–x))
41.
f(x) = x3 + 3x – 4
42.
f(x) = x4 – 6x2 + 3
43.
3
f(x) = x –2 x
x
44.
f(x) = sin 2x
45.
f(x)=x2 + 1
46.
f(x) = x(x2 – 1)
47.
f(x) = x5 + 1
48.
49.
f(x) = cos2x
51.
What type of function results from the product of two
even functions?
odd functions?
50.
f(x) = 1 + 2x
x
f(x) = ex
6
512
Test for symmetry.
Symmetric to y-axis: replace x with –x and relation remains the same.
Symmetric to x-axis: replace y with –y and relation remains the same.
Origin symmetry: replace x with –x, y with –y and the relation is equivalent.
52.
y = x4 + x2
53.
56.
x = y2 + 1
57.
60.
y = x5 – 3x3 + x
y=
–
y = sinx
54.
y = cosx
55. y =
x
x + x3
x
x +1
58.
y = x3
59.
1
x–2
61.
x2 – y2 = 1
2
y=
5
GEOMETRY
Pythagorean Theorem: Area and Volume Formulas: Similar Triangles:
1. A rectangle is inscribed in the upper half of the unit circle. Express the area of the
rectangle as a function of x only. i.e. as A(x).
y
y = 1 – x2
x
2. A cone is inscribed inside a sphere of fixed radius, k. Given that the volume of a
cone is (1/3)r2h, express the volume of the cone as a function of r only. i.e. as V(r).
r
7
512
3. A cylinder has a fixed volume of 20 cubic units. Express the surface area of the
cylinder as a function only of the radius of its base. i.e. as A(r).
(Use the fact that the volume of a cylinder is given by V = r2h and the surface area of a
cylinder is given by A = 2r2 + 2rh.) .
r
h
4. Repeat problem 3, expressing the surface area of the cylinder as a function only
of its height, i.e. as A(h).
5. The area of rectangle A'B'C'D' is 20 square units. Using the dimensions shown
in the figure drawn below, express the area of rectangle ABCD as a function of
(a) x only
(b) y only
A
C
D
LOGARITHMIC AND EXPONENTIAL FUNCTIONS
Simplify Expressions:
1.
log4(1/16)
3.
log327
4.
log125(1/5)
5.
logww45
6.
lne
7.
In 1
8.
In e2
2.
3log3 • 3/4log381 + 1/3log3(1/27)
8
512
Solve equations:
9. log6(x + 3) + log6(x + 4) = 1
10. logx2 – log100 = log1
11.
3x+1 = 15
Sketch graphs:
12.
y = lnx
13.
y = 3x
Find the inverse: Find f–1(x) for:
14.
f(x) = 7X
15.
f(x) = log3x
16.
f(x) = In(x + 5)
TRIGONOMETRY
Unit Circle:
Find the following without using your calculator.
1..
sec(–/6 )
5.
cot 8
2.
tan 9/4
6.
3.
cos 11/3
sin (–5/6)
7.
4.
sin 11/4
–
tan (–3/4)
State the domain and range and period of each function?
8.
sin x
9.
cos x
10.
tan x
11.
cot x
12.
sec x
13.
csc x
9
512
Identities:
Simplify
14.
(tan2x)(csc2x) – 1
(cscx)(tan 2x)(sinx)
15.
1 – cos2x
16.
1 + tan2 x
Prove these identities:
17.
(1 – sin2x)( 1 + tan2x) = 1
18.
sin4x – cos4x = sin2x – cos2x
19.
(1 – tanx)2 = sec2x – 2tanx
Solve the Equations
20.
cos2x = cosx + 2, 0 x 2
21.
4cos2x = 1, x 22.
secx(sin2x) = tanx, 0 x < 2
23.
4sinxcosx = 3, 0 x 2
25.
tan(/2 – 2) = –1, 0 < x < 24.
26.
cos2x + sinx + 1 = 0, –/2<x<3/2
2sin2x = 1, 0 < x < 2
10
512
Inverse Trig Functions
Evaluate:
Arcsin1
27.
31.
Arctan3/3
28. Arcsin(–2/2)
29. Arccos(3/2)
30.
sin(Arccos(3/2))
32.
Arctan(–1)
33.
tan(Arcsin(–1/2))
34.
Arcsin(tan5/4)
What is the domain and range for:
y = Arcsin x
35.
36.
y = Arccos x
Right Triangle Trig:
Find the value of x.
37.
50o
38.
39.
x
12
x
7
10
40.
A worker looks out of a second story window and sees the top of the
flagpole at an angle of elevation of 22 degrees. The worker is 18 feet above the
ground and 50 feet from the flagpole. Find the height of the flagpole.
11
512
41.
The roller coaster car shown in the diagram below takes 23.5 sec. to go up
the 23 degree incline segment AH and only 2.8 seconds to go down the drop from
H to C. The car covers horizontal distances of ISO feet on the incline and 60 feet
on the drop. How high is the roller coaster above point B1 Find the distances AH
and HC. How fast (in ft/sec) does the car go up the incline? What is the
approximate average speed of the car as it goes down the drop?
(Assume the car travels along HC.) Is your approximate answer too big or too
small?
(Advanced Mathematics. Richard G. Brown, Houghton Mifflin, l994, pg 336)
H
A 180 ft
B 60 ft
C
Graphs:
Identify the amplitude and period of these functions
42.
y = –2sin(–2x)
43.
y = –cos(/2x + )
44.
y = 3/4 sin(1/2 x – 3)
45.
y = 1/4 cos(4x + 12)
12
512