Math 108 Topics for Exam #3
Optimization (Sections 3.3 and 3.4)
Meaning of optimizing
Word problems on revenue, profits, prices and optimizing them.
You will not be asked to solve elasticity questions on the exam
There are a number of basic types of word
2.3. Product and Quotient Rules; Higher-Order
Derivatives
The Product Rule
If f (x) and g(x) are differentiable at x, then so is their product
and
d
d
d
[f (x)g(x)] = f (x) [g(x)] + g(x) [f (x)]
dx
dx
dx
or equivalently
(fg) = fg + gf
Example
Differentia
1.5. Limits
Behavior of f (x) as x approaches c
Consider the behavior of f (x) =
as x approaches 1.
x
f (x)
0.8
-1.2
0.9
-1.1
0.99
-1.01
x 2 3x + 2
(x 1)(x 2)
=
x 1
x 1
1
undened
1.01
-0.99
1.1
-0.9
1.2
-0.8
As x approaches 1, f (x) approaches 1.
Denition
1.2. The Graph of a Function
Denition
The graph of a function f consists of all points (x, y) where x is
in the domain of f and y = f (x); that is, all points of the form
(x, f (x).
Example
Graph the function f (x) = x 2 .
x and y intercepts
Denition
The
1.1. Functions
Loosely speaking, a function consists of two sets and a rule
that associates elements in one set with elements in the other.
Denition
A function is a rule that assigns to each objects in a set A
exactly one object in a set B.
The set A is c
1.6. One-sided Limits and Continuity
One-sided Limit
If f (x) approaches L as x tends toward c from the left (x < c),
we write
lim f (x) = L.
xc
Likewise, if f (x) approaches M as x tends toward c from the
right (x > c), then
lim+ f (x) = M.
xc
Example
F
2.2. Techniques of Differentiation
The Constant Rule
For any constant c,
d
[c] = 0
dx
The Power Rule
For any real number n,
d n
[x ] = nx n1
dx
Example
Differentiate the function y =
x 5.
The Constant Multiple Rule
If c is a constant and f (x) is diffenen
2.1 The Derivative
The derivative of a function
The derivative of the function f (x) with respect to x is the
function f (x) given by
f (x + h) f (x)
.
h
h0
f (x) = lim
The process of computing the derivative is called differentiation,
and we say that f (
4.2. Logarithmic Functions
If x is a positive number, then the logarithm of x to the base
b(b > 0, b = 1), denoted logb x, is the number y such that
by = x; that is,
y = logb x
if and only if
Example
Evaluate log10 1, 000.
Example
Solve the equation log4
4.1 Exponential Functions
Denition of bn for rational values of n (and b > 0)
Integer Powers: If n is a positive integer,
bn = b b b
n factors
Fractional Powers: If n and m are positive integers,
m
m
bn/m = ( b)n = bn
Negative Powers: bn =
Zero Power: b0
4.4. Additional Exponential Models
General Procedure for Sketching the Graph
Step 1. Find the domain of f (x).
Step 2. Find and plot all intercepts.
Step 3. Determine all vertical and horizontal asymptotes and
draw them.
Step 4. Find f (x) and determine t
3.4. Optimization
Absolute Maxima and Minima of a function
Let f be a function dened on an interval I containing the
number c. Then
f (c) is the absolute maximum of f on I if f (c) f (x) for all
x in I.
f (c) is the absolute minimum of f on I if f (c) f (
3.3. Curve Sketching
Vertical Asymptotes
The vertical line x = c is a vertical asymptote of the graph of
f (x) if either
lim f (x) = + (or )
xc
or
lim f (x) = + (or )
xc +
Vertical Asymptotes
Example
Determine all vertical asymptotes of the graph of
g(x)
2.4. The Chain Rule
If y = f (u) is a differentiable function of u and u = g(x) is in
turn a differentiable function of x, then the composite function
f (g(x) is a differentiable function of x whose derivative is given
by the product
dy
dy du
=
dx
du dx
o
2.6. Implicit Differentiation and Related Rates
Example
Find
dy
1
if x + = 4.
dx
y
Implicit Differentiation
Suppose an equation denes y implicitly as a differentiable
function of x. To nd the derivative of y,
1. Differentiate both sides of the equation wi
1.3. Linear Functions
Denition
A linear function is a function that changes at a constant rate
with respect to its independent variable.
The graph of a linear function is a straight line.
The equation of a linear function can be written as
y = mx + b
wher
Math 108 Review for Exam #3
Optimization
1.
2.
3.
4.
5.
6.
#4 p. 269
#13 p. 271
#15 p. 271
#19 p. 272
#13 p. 288
#25 p. 290
Exponential Functions
1. Know the definition of the number e
2. #3, 5, 9, 10, 11 p. 319
3. #15 p. 319
4. #17-18 p. 319
5. #19 p. 32
Math 108, Solution of Midterm Exam 1
1 Specify the domain of each of the following functions.
x2 4x + 3
(a) f (x) = 2
x +x2
Solution. Since division by any nonzero number is possible, the domain of f is the set of all
numbers satisfying x2 + x 2 = 0. Sinc
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 1
1 Find all values of x such that f (g(x) = g(f (x), where f (x) = x2 + 2 and g(x) = x 1.
Solution. Replace x by g(x) = x 1 in the formula for f (x) to get
f (g(x)
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 0
1 Find the interval or intervals consisting of all real numbers x that satisfy the given inequality.
(a) |x 1| 3
Solution. Rewrite the given inequality as
3 x 1 3
Math 108, Solution of Midterm Exam 3
1 Find an equation of the tangent line to the curve x3 + y 3 = 2xy at the point (1, 1).
Solution. Dierentiating both sides of the given equation with respect to x, we get
d
x3 + y 3
dx
dy
3x2 + 3y 2
dx
dy
3x2 + 3y 2
dx
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 5
1 Let f be a function dened by f (x) = x5 5x4 3x + 2.
(a) Find intervals on which the graph of f is concave up and concave down.
Solution. The rst derivative of f
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 5
1 The rst derivative of a certain function () is given by
() = 3 (2 5)2 .
(a) Find intervals on which is increasing and decreasing.
5
The derivative of is continu
Math 108, Solution of Midterm Exam 2
1 List all values of x for which the following function f (x) is not continuous. Explain the reason.
2
x + 2x 3
2
if x < 1
x 4x + 3
f (x) =
if 1 x < 3
2x 4
2
x 2x 3 if x 3
x2 + 2x 3
is a rational function dened
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 2
1 Decide if the following function is continuous at x = 2. Explain the reason.
2
x x2
2
if x < 2
x 3x + 2
f (x) =
2
x x
if x 2
Solution. We need to verify the
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 3
1 Find the rate of change
3 + 1
for the function = 2
when = 1.
+1
Solution. By the quotient rule, the derivative of =
3 + 1
with respect to is given by
2 + 1
[
]
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 6
1 Find all real numbers that satisfy 22x = 83 .
Solution. The given equation is satised if and only if
22x = (23 )3
22x = 29
2x=9
x = 7
Thus, 22x = 83 if and only
Lecturer: Sangwook Kim
Oce : Science & Tech I, 226D
math.gmu.edu/ skim22
Solution of Quiz 3
1 Find the rate of change
dy
x3 + 3
for the function y = 2
when x = 1.
dx
x +1
Solution. By the quotient rule, the derivative of y =
x3 + 3
with respect to x is gi