Practice Test #3
MAC2311
Find any critical numbers of the function:
4x
f ( x) 2
1.
x 1
2-4 Locate the absolute extrema on the closed interval:
f ( x) x 3 12 x
[0,4]
2.
f ( x) sec x
4.
x 2 2 x 1, 0 x<3
f ( x)
3 x 7, 3 x 5
5.
[
, ]
6 3
3.
[0,5]
What is

Practice Test #4
MAC2311
1-6 Find the Indefinite Integral:
3
1.
(4 x 3x 5)dx
3
2.
x
4
3.
t
(4t 3 3t )dt
4.
dx
5.
1 cos
6.
sec x(tan x sec x)dx
2
dx
cos x
2
x
dx
7-9 Evaluate by using the properties of summations:
10
7.
(i
2
4)
i 1
20
8.
(i
2) 2
i

Practice Test #1
MAC2311
1-4 Find the Limit:
x3
lim 2
1.
x 3 x x 12
2.
3.
lim
x 1
lim
x0
3 x 2
x 1
sin 2 x
x
tan x
x0
x
5-9 Find the limit (draw the graph if necessary):
lim f ( x)
where f ( x) 4 x 2
5.
4.
lim
x 3
6.
7.
lim f ( x)
x 3
lim
x 3
4 x, x 5
w

Practice Test #2
MAC2311
Find the slope of the tangent line to the graph of the function at the given point:
f ( x) 5 4 x; (2, 3)
1.
2-3 Find the derivative by the limit process.
f ( x) x 4;
2.
3.
f ( x) 4 x 2 3x 5
Find an equation of the tangent line to

Practice Test #5
MAC2311
1-2 Sketch the graph of the function and state its domain:
f ( x) 2 ln x
1.
2.
f ( x) 2 ln( x 4)
3-5 Find the derivative of the function:
ln x
f ( x)
3.
x
x2
x2
4.
f ( x ) ln 3
5.
f ( x ) ln sec x tan x
6.
Use implicit differenti

Module 12
SECTION 6.2: VOLUMES
The concept of volume provides the second application of a definite integral. Now
lim f ( xi ) xi
is replaced by
lim A( xi ) xi
where A( xi ) is the area of a cross-section of a solid. Examples of cross-sections are
shown in

Module 5
SECTION 3.4: THE CHAIN RULE
We have noted that the Power Rule can be applied to x 4 , x 2 , and x 4 , where each base
3
is x. But if the base is, say, x 3 + 1 as in ( x 3 + 1) , the same pattern will not give a
2
correct answer. The derivative of

Module 3
SECTION 2.6: LIMITS AT INFINITY; HORIZONTAL ASYMPTOTES
The distinction between y (or f ( x ) ) and x is significant. The infinite
limits considered earlier (pages 93 and 115) involved y and vertical asymptotes
because a denominator was approachin

Module 1
A PREVIEW OF CALCULUS
In this section in your text, the author presents an overview of calculus. Read it as a story
that exposes you to some of the ideas in calculus that will be covered in more detail later.
The major thread presented in the fiv

MATH 16A, SUMMER 2008, QUIZ SOLUTIONS
BENJAMIN JOHNSON
Quiz 1, Thursday, June 26
(1) Is the point (2, 12) on the graph of the function f (x) = x(5 + x)(4 x)?
f (2) = 2 (5 2) (4 (2) = 2 3 6 = 36 6= 12. NO.
(2) Assume that the amount of effort (in hours per

MATH 16A, SUMMER 2008, REVIEW SHEET FOR MIDTERM EXAM
BENJAMIN JOHNSON
The midterm exam will be held Thursday, July 17, from 8:10AM to 9:40AM in 3 Evans. The
exam will cover chapters 0, 1, and 2 of the textbook.
To do well on this exam you should be able t

Module 9
SECTION 4.7: OPTIMIZATION PROBLEMS
If you have been wondering where you might use calculus other than in graphing
functions, the answer is near at hand. We now look at some practical applications of
calculus, but first a word of warning. Most stu

Module 10
SECTION 5.1: AREAS AND DISTANCES
We now begin the study of the branch of calculus called integral calculus. Read page 360
carefully. In particular, it is not so easy to find the area of a region with curved sides.
We all have an intuitive idea o

Module 11
SECTION 5.4: INDEFINITE INTEGRALS AND THE NET CHANGE THEOREM
The symbols
f ( x ) dx are used to represent the antiderivative F ( x ) without having to
evaluate F (b) F (a) . Then:
x6
x5
( x x ) dx = 6 5 + C
5
4
1
(x
1
while
5
0
x ) dx =
4
x6

Module 4
SECTION 1.5: EXPONENTIAL FUNCTIONS
We begin with an outline of the review section for exponential functions.
1. How do we interpret an irrational exponent? The power 32 means 3 times 3
1
and 3 2 is 3 but what does 3 2 mean?
2. When does the graph

Module 7
SECTION 3.9: RELATED RATES
In an earlier module, we noted that implicit differentiation required an adjustment in
using the Power Rule.
d 3
d 3
x = 3x 2 1 but
y = 3y 2 y
dx
dx
Related rate problems require a similar kind of adjustment. In this ty

Module 6
SECTION 1.6: INVERSE FUNCTIONS AND LOGARITHMS
The first part of this section can be skimmed with just a brief look at the different ways of
seeing the relationships between inverse functions. A major point is the algebraic method
for finding an i

Module 2
SECTION 2.1: THE TANGENT AND VELOCITY PROBLEMS
MORE ON NOTATION
In the discussion of the tangent line, it is important to have a clear concept of the tangent
y y1
line. Remember the slope of a line is 2
where y2 y1 is the length of a vertical
x2

Module 8
SECTION 4.3: HOW DERIVATIVES AFFECT THE SHAPE OF A GRAPH
We start with a brief summary of this section.
1. A function is increasing when the derivative is positive. In a graph the curve is
rising.
2. A function is decreasing when the derivative i