The following review sheet is intended to help you study. It does not contain every type of problem you
may see. It does not reflect the distribution of problems on the actual midterm. It probably has mistakes
(if you find one, tell us!). But, it can give you some extra practice with most of the main concepts covered
in the course.
Functions
You should know what a function is, how to graph basic functions and how to read graphs of functions.
You should know the different ways to combine functions, most especially by composing them. You should
know what inverse functions are. You should know how to model some situations using functions – real-life
situations modeled by functions in this class so far include supply and demand, cost, revenue, profit, interest
and exponential growth.1
(1) Let f (x) = x3 + 4x2 + 5 and g(x) = 2x2 − 3. Express the following in terms of x:
(a) (f + g)(x)
(b) (f − g)(x)
(c) 3f (x)
(d) (f g)(x)
(e) (f /g)(x)
(f) (f ◦ g)(x)
(g) (g ◦ f )(x)
(2) If f (x) =
√
x − 2 and g(x) = x2 , what is f ◦ g(−3)? What is the domain of f ◦ g?
(3) Plot points and sketch the graphs of the following functions: h(t) = t3 , g(z) = z1 , f (x) = 2x .
(4) Which of the following points does the graph of log5 x pass through? There may be more than one.
(a) (5,1)
(b) (5,0)
(c) (0,5)
(d) (1,0)
(e) (5,5)

x

if x < 1
e
(5) Sketch the graph of h(x) = 3x − 4 if 1 ≤ x ≤ 2 . You can use a calculator to help with this

 −x
2
if 2 < x
problem.
(6) Say f (3) = 5, f (5) = 8, f (2) = 3, and f (8) = 1. Say g(x) is the inverse of f . What is g(3)? What
is g(8)?
(7) Simplify ex ln 17 .
Exponents and Logarithms
Some new, very important functions introduced in this class are exponential functions and logarithmic
functions. You should know what those are, their basic properties, how to manipulate them and how to
solve equations that involve them.
(1) Simplify
√
x3 y(x2 )5 yz
.
z −3 x
(2) What is log4 16? What is log4 2? What is log4 8?
1
On Midterm I, you will not be tested on supply and demand or interest or exponential growth. However, those could appear
on the final exam.
1
(3) Why does it not make sense to write log3 by itself?
(4) What is the definition of logarithm?
(5) True or false?
log3 A
log3 B
(6) True or false?
log3 5x
log3 x2
= log3 (A − B)
=
log3 5
log3 x .
(7) Simplify 10log10 3 and log10 (107 ) (notice that log10 (107 ) is not the same as (log10 10)7 ).
(8) Write 2 log x + log y − log z as a single logarithm.
(9) Solve the following equation for x: 2 log10 (x + 7) + 3 = 0
(10) Solve the following equation for x: 3 · 52−x = 4
(11) Solve the following equation for x: 4 log9 (x + 5) = 10
(12) Solve the following equation for x: ln e(2x) + e = log2 1 − e(2x)
Limits
Limits are an important tool. Most especially for this class, they appear in the definition of the derivative.
You should know what a limit is conceptually, as well as how to compute them by plugging in values, by
looking at a graph, and algebraically. You should know the rules of limits, and how limits relate to continuity
of functions.
(1) Find the following limits algebraically:
(a) limx→1
√
x x+2
x−3 x2 +1
(b) limh→0
(2+h)2 −4
h
(c) limt→4
x2 +8x+16
x2 −16
2
−x−2
(d) limx→2 x x−2
(
x if x > 0
(2) Let f (x) = 1
. Find limx→0− f (x), limx→0+ f (x), and limx→0 f (x).
if x ≤ 0
x

3

2x + 1 if x < −2
(3) Suppose f (x) = 3x + 4
if − 2 < x ≤ 5 . Graph f (x). Evaluate the following limits using the


3x − 5
if x > 5
graph of f (x).
(a)
(b)
(c)
(d)
limx→4− f (x) =
limx→5− f (x) =
limx→2− f (x) =
limx→0− f (x) =
(4) Find the limit limx→2
limx→4+ f (x) =
limx→5+ f (x) =
limx→2+ f (x) =
limx→0+ f (x) =
|x−2|
x2 −4 .
limx→4 f (x) =
limx→5 f (x) =
limx→2 f (x) =
limx→0 f (x) =
Rates of Change
You should know what is meant by an average rate of change as well as how to compute an average
rate of change. You should know how to interpret your computations. Same goes for instantaneous change.
You should know how to interpret average and instantaneous change on the graph of a function, and the
difference between them.
(1) Find the average rate of change of the function h(x) = 2x2 − 2x + 1 on the intervals below:
(a) (0,3)
(b) (1,10)
(c) (-2,4)
(2) A train leaves Stockholm at 7:00 in the morning. It’s distance from Stockholm is given by the funct
tion f (t) = 3t2 − 21 + 1, where f (t) = 0 (so 7:00 corresponds to no time having passed). By 3:00 pm,
how far is it from Stockholm? What was its average speed during that time? How fast is it going at
3:00 pm?
(3) Use limits to find the instantaneous rate of change of the function f (x) = 2x2 − x − 1 at x = 2.
(4) Consider the following graph. Which is greater – the average rate of change over the interval (-1.5,0)
or the instantaneous rate of change at x = 2?
Derivatives
You should know the definition of the derivative, what it means, the various ways of interpreting it as the
slope of the tangent line or the instantaneous rate of change. You should be able to compute it from the
definition, or using rules. You should know all of the rules for basic derivatives, products, quotients, and the
chain rule. You should know how the value of the derivative can be related to the graph of a function and
vice versa. You should know how derivatives can be interpreted in real-world problems. You should know
how to use derivatives to find the equation of a tangent line, to be used in a linear approximation problem.
(1) Define the derivative of a function!
(2) Find the derivative of f (x) =
2
3x−1
using the limit definition.
(3) Find the slope of the tangent line to f (x) = 5x2 − 3 at x = 2. Find the equation of the tangent line
at x = 2. Approximate f (2.3) using the tangent line.
(4) Find the derivatives of the following functions:
(a) a(x) =
3
x2
− 12x7 + 3
(b) b(t) = 3t3 − t−3 + 3
(c) c(x) = 2x5 + 4ex −
(d) d(t) =
√
x−
√
8π
(3x2 −5x+7)4
2
(e) e(x) = (x3 )4 + ln x
(f) f (x) =
5x+4
3−x
(g) g(z) = 5ez ln z
(h) h(x) = x2 ln 3 − 55x
(i) i(q) =
q 2 +1)eq
eq
(j) j(z) = (x3 − 3x2 + 2)5 (x − 1)2
(k) k(x) =
2 +3
ex
(x3 −2)3
ln x
(l) `(x) = f (g(x)), where f (x) = x3 + x and g(x) = x6 .
(m) m(y) = 2−3y + 5y log2 y
(n) n(x) = 3(2x ) − log5 (x2 )
(5) If revenue is given by R(x) = 43x2 − 22x and costs are given by C(x) = 20x + 58, approximate
marginal profit when 500 units are sold.
(6) Match each of the graphs (first line of graphs below) with their derivative (second line of graphs
below).
(7) Draw the graph of a function whose derivative is always equal to zero.
(8) In the picture below, one of the curves is a function and the other curve is its derivative. Which is
which? Explain why your answer is correct.