MATH 112, Section 9
Study Guide for Test 1
The first test will cover all of the material from Chapter 11 of the text
book. For this test, you should
• know what a function and its domain is,
• know what the graph of a function is (see page 581),
• know how to identify values of a function from its graph,
• know how to graph a function by plotting points,
• know how to find the sum, difference, product, and quotients of functions either from their equations or their graphs,
• know how to calculate the composition of two functions,
• know how to interpret sums, differences, products, quotients, and compositions of functions in word problems (see applied examples from
11.2),
• know how to interpret functions in mathematical modeling contexts,
• know what a limit is,
• know how to evaluate limits from equations (both at a point a and at
infinity),
• know how to evaluate a limit from a graph (both at a point a and at
infinity),
• know how to evaluate one-sided limits,
• be able to tell if a function is continuous (from equation or graph),
• know the properties of continuous functions,
• know the statement of the intermediate value theorem,
• be able to apply the intermediate value theorem,
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• know the definition of the derivative of a function (this will be the first
question on the test,
• be able to interpret a difference quotient in applications,
• be able to interpret the derivative of a function in applications, and
• be able to calculate derivatives from the definition.
This list is not exhaustive, but should serve to give you the main topics from
the test. The test will consist of two parts. The first part will be no calculator
no study sheet, and on the second part you will be allowed to use both a
graphing calculator (or scientific calculator) and you may have one sheet of
notebook paper (both sides) as a study sheet. You may not use a cell phone
on the test (even as a calculator). Both parts of the test are handed out at
the start of the period, and you may work on both sections for as long as
you like without the use of calculator or study sheet. Once you turn in the
first section, then you may pull out your calculator and study sheet to finish
the second part. You should be able to work all homework problems that we
have seen in the class. The next pages are sample questions for each part.
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Part 1
No Study Sheet, No Calculator
1. State the definition of the derivative of a function. (This should be as
in the box on page 663)
2. State the intermediate value theorem.
3. If f (x) = 3x2 − 1 and g(x) = x + 2, find the equations for:
(a) f /g(2)
(b) f + g(x)
(c) f ◦ g(x)
x2 − −5x + 6
x→2
x2 − 4
4. lim
5. lim−
x→1
1
1−x
6. Fill in the blank: The graph of a function is the set of all points
(x, y) in teh xy-plane such that x is in the
of f and
.
y=
3
Part II
Calculator and Study Sheet Allowed
1. True or false
(a) The derivative of a function is the tangent line at a point of the
function.
(b) Let f and g be two functions with domain the entire real numbers.
Then f ◦ g = g ◦ f .
(c) If the lim f (x) exists, then lim− f (x) = lim+ f (x) = f (a).
x→a
x→a
x→
2. A company’s total sales (in millions of dollars) are approximately linear
as a function of time (in years). Sales in 2001 were $2.4 million, whereas
sales in 2006 amounted to $7.4 million.
(a) Find an equation that gives the company’s sales as a function of
time where t = 0 corresponds to the year 2000.
(b) What were the sales in 2004?
3. Economists define the disposable annual income for an individual by
the equation D = (1 − r)T , where T is the individual’s total income
and r is the net rate at which he or she is taxed. What is the disposable
income for an individual whose income is $80,000 and whose net tax
rate is 25%?
4. Suppose we have the functions f (x) and g(x), both of which are differentiable. Suppose further that we have the following chart:
0 1 2 3 4 5 6
f (x) 5 2 −3 1 2 6 3
g(x) −1 1 0 4 7 −2 2
(a) If h(x) = f (x)g(x), find h(6).
(b) If h(x) =
f (x)
,
g(x)
find h(3).
(c) If h(x) = f (g(x)), find h(2).
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5. Use the definition of the derivative to calculate the derivative of the
function y = f (x) = x2 + x.
6. Given the graph of y = f (x) below, find
2.0
1.5
1.0
0.5
0.5
1.0
1.5
2.0
2.5
3.0
-0.5
-1.0
(a) Find limx→1 f (x).
(b) Find limx→2− f (x).
7. Let f (x) = x2 − 1
(a) Use the definition of the derivative to find the slope of the line
tangent to the curve y = f (x) at the point (2, 3).
(b) Write the equation of the tangent line to the curve at (2, 3).
8. The total cost (in dollars) incurred by Aloha company in manufacturing
x surfboards a day (in hundreds) is given by the function C(x). What
is the meaning of
C(x + 2) − C(x)
?
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C(x + h) − C(x)
(b) lim
?
h→0
h
(a)
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