Calculus 1
Review- Precalculus Material
Name:____________________________________
Date:_____________________________________
This packet is a list of things you should be able to do by the time you get into Calculus 1. It’s all those
things that your precalc teachers said you would need to have memorized and know how to do! Imagine that!
These are all examples of the topics that you should know and understand. You should be able to do all the
material in this review packet without a calculator. If you are not fluent in this material you will be hampered
in learning the new stuff. Additional practice can be found in chapter 1 of the calculus book as well. Do all
work on separate paper.
1. Use the binomial theorem or Pascal’s triangle to expand each of the following polynomials:
a. (a + b)4
c. (1 – y)6
b. (x – 3)3
d. (2x + 1)5
2. a. Knowing the polynomial 2x4 – 5x3 – 6x2 + 19x – 10 has a double root at x = 1,
use synthetic division to find the other roots.
b. Use the rational zero theorem and synthetic division to find the roots of
x3 + 3x2 – 4x – 12.
3. Find the value of each expression without the use of a calculator
a.
b.
3
84
d.
253
e.
c. 5 328
4. Factor each expression completely
f.
3
2162
4
2563
493
a. x3 + 8
c. 8x3 + 64
b. x3 – 27
d. 27x3 – 125y3
5. Answer each of the following questions about logs and exponentials:
a. What is the domain of y = ln x?
b. What is the range of y = ln x?
c. What is the y-intercept of y = ln x?
d. What is the x-intercept of y = ln x?
e. What is the domain of y = ex?
f. What is the range of y = ex?
g. What is the y-intercept of y = ex?
h. What is the x-intercept of y = ex?
i. Solve ex = 0
j. ln 1 = ?
k. ln e = ?
l. elnx = ?
m. ln ex = ?
n. ln 0 = ?
o. Rewrite 2lnx using properties
p. Explain how ex+ln3 = 3ex
6. What does it mean for a function to be odd? even? Answer these questions not only with what it means
for the function but what it means with regards to its symmetry.
7.
Which trig functions are even and which are odd? Which basic polynomials are even and which are
odd?
8. Knowing which trig functions are even and odd should aid you in rewriting the following expressions in
terms of x:
a. cos(-x) =
b. sin(-x) =
9. TRIG!!! You’re going to love this part. You need to have each of the following values at your
fingertips. There will be a timed quiz as part of the beginning of the year test on these values. You will
also have surprise quizzes throughout the year.
a. sin 0
b. sin
c. sin
d. sin
e. sin

6

4

3

2
f. cos 0
k. tan 0

g. cos
6

h. cos
4

i. cos
3

j. cos
2

6

m. tan
4

n. tan
3

o. tan
2
l. tan
10. Based on your incredible skills from the previous problem you should be able to find the value of
expressions like these:

2

b. csc 
3
2
a. sec
c. sec
3
d. cot

2
e. csc2 

4

h. cot
3

i. csc
4
g. sec2

6
f. cot 0
11. On top of being able to evaluate trig expressions you should know stuff about them too…
a. What is the range of sin-1x?
b. What is the range of cos-1x?
c. What is the range of tan-1x?
d. When you evaluate sin-1x, what quadrants does it give you the answer in?
e. When you evaluate cos-1x, what quadrants does it give you the answer in?
f. When you evaluate tan-1x, what quadrants does it give you the answer in?
g. Why is sine positive in quadrants I and II but negative in III and IV?
h. Why is cosine positive in quadrants I and IV but negative in II and III?
i. Why is tangent positive in quadrants I and III but negative in II and IV?
12. You should be able to do problems like this as well:
Solve the trig equation in the given interval:
a. tan x = 1
0  x  2
b. sec x = -2
x 
c. sin x = -0.5
  x  
13. You should also be able to do trig stuff like this (this means you need to have the identities memorized!
You won’t get reference tables.):
20 sin5x cos5x is equivalent to:
*(this is a multiple choice question)*
a.
b.
c.
d.
e.
10sin10x
20sin5x
10sin5x
4cos5x
20cos5x
14. Simplify each of the following expressions. I will allow radicals in the denominator and unexpanded
polynomial expressions ((2x-3)2 for example). I do not allow you to have rational or negative exponents
unless you start with them, and I do not allow double fractions.
6
2x  5
2x  5
2x  5 
a.
c.
x3
3
3
3x  2
1
3
8
x
4
x
x
x2 
1  x2 
b.
x 5

5 x
x
1
5
d.
15. What do you know about a function and its inverse? (2 related things)
16. You should be able to graph each of these basic functions without the aid of a calculator:
a. y = x2
b. y = x3
c. y = x
d. y = x
e. y = sin x
f. y = cos x
17. Graph each of these equations using your knowledge of the graphs of basic functions and their
transformations without the aid of a calculator (think a, h, k!)
a. y = 2x2 – 4
b. y = 2(x-4)2
c. y = sin(2x) + 1
d. y = (x+1)3 – 3
x 5 3
f. y = 3 x  1
e. y =


g. y = 2cos  x 

 -5
4
18. You also need to be able to write a piecewise function. Be able to write a piecewise function for each of
the following:
a.
b.
19. You should also be able to graph a piecewise function. Graph each of the following:
 1 x0
a. f ( x)  
 x x0
b.
 x2
x0
 3
f ( x)   x 0  x  1
2 x  1 x  1

20. It is important too that you are able to understand and be able to write absolute value functions as a
piecewise function.
a.
f ( x)  x
b.
f ( x)  x  3