STMATH 144c
Spring 2015
Quiz 7
Name:
1. Find the average value of ! f (x) = 7 x 2 on 3 ,1 .
!
2. Evaluate the following using u-substitution. Be sure to show all your work (i.e. identify u, du)
(ln x )
4
a.
!
x 2 x 3 5dx
3. Evaluate
!
b.
!
x
dx
xe3x dx us
STMATH 144c
Spring 2015
Quiz 6
Name:
1. If a flute manufacture finds that the demand function for flutes is !q = D(x) = 1000 2x , where q is
the number of flutes purchased at a price of x dollars.
a. Find the elasticity as a function of x. Simplify your a
STMATH 144c
Spring 2015
Quiz 5
Name:
d
dx
( )=
( f ( g(x) = f '( g(x) g'(x)
d
dx
(e ) = e
d
dx
!
N
D
DN 'ND'
D2
x
x
d
dx
d
dx
( f g) = f g'+ g f '
( x ) = kx
(a ) = (lna)a
k
d
dx
(ln x ) = 1
x
d
dx
x
P(t ) = P0e kt
ln2
k
T=
k1
x
d
dx
1
(log x ) = lna 1
x
STMATH 144c
Spring 2015
Quiz 4
d
dx
d
!dx
( )=
( f ( g(x) = f '( g(x) g'(x)
N
D
DN 'ND'
D2
Name:
d
dx
( f g) = f g'+ g f '
d
dx
( x ) = kx
k
k1
R(x) = x p(x)
P(x) = R(x) C(x)
1. Riverside Appliances is marketing a new refrigerator. It determines that in o
STMATH 144c
Spring 2015
Quiz 3
d
!dx
(x ) = kx
k
Name:
k1
d
dx
( f g) = f g'+ g f '
d
dx
( )=
N
D
DN 'ND'
D2
d
dx
(
)
(
)
f g(x) = f ' g(x) g'(x)
1. Find the indicated derivatives. Simplify your answer unless otherwise stated.
3
a. ! f '(x) , for ! f (x)
STMATH 144c
Spring 2015
Quiz 2
Name:
( )
(
d k
x = k x k1
dx
!
)
d
f g = f g'+ g f '
dx
1. Differentiate the following functions. Simplify your answer!
a. ! f (x) = 3x 4 + 3x 2 12x 1/3
b. y = 12x +
!
2 1
x3 x
2. Differentiate the following functions. Do n
Section 4.6: Integration by Parts & Section 4.7: TablesSolutions
Recall the Product Rule for differentiation:
= ! +
integrate both sides
(
(u v'+ v u') dx
u v = u v'dx v u'dx
u v + v u'dx = u v'dx
d
dx
[u v]) dx =
Then manipulate the notation so that v
BMATH 144c
Spring 2015
Section 4.6: Integration by Parts & Section 4.7: Tables
Recall the Product Rule for differentiation:
=
Integration by Parts Formula:
Example 1:
xe x dx
Example 2:
x ln x dx
Example 3:
x 2 e3x dx
Example 4:
x 7x 1 dx
Tips on Using I
Sections 2.3: Asymptotes & Rational Functions & Section 2.4: Using Derivatives to find
Absolute Max and MinSolutions
Definitions:
A rational function is a function that can be described by P(x) / Q(x)
where () and ()are polynomials and () is not the zero
BMATH 144b
Spring 2015
Sections 2.3: Asymptotes & Rational Functions
& Section 2.4: Using Derivatives to find Absolute Max and Min
Definitions:
A rational function is a function that can be described by _
where () and ()are polynomials and () is not the z
Sections 2.1 & 2.2 : Using First and Second Derivatives to Find Maximum and Minimum
Values and Sketch GraphsSolutions
Def: A function f is increasing over I if, for every a and b in I, if a < b, then f (a) < f (b).
Def: A function f is decreasing over I i
STMATH 144c
Spring 2015
Quiz 8
Name:
(
1. The demand function D(x) = x 10
!
)
2
is the price, in dollars per unit, that consumers are willing to
pay for x units of an item, and the supply function !S(x) = x 2 is the price, in dollars per unit, that
produc
Math 112, Spring 2014, Solutions to Midterm I
1. You do not have to simplify your answers but make sure they are clearly written with properly used
parentheses where necessary.
p
6
(a) Find f 0 (x) if f (x) = 7x3 5 x + .
x
5
6
p
f 0 (x) = 21x2
2 x x2
d (x
NAME: _
Student ID #: _
QUIZ SECTION:_
Math 112 A
Midterm I
April 26, 2011
Problem 1
12
Problem 2
12
Problem 3
12
Problem 4
14
Total:
50
You are allowed to use a calculator, a ruler, and one sheet of notes.
Your exam should contain 5 pages (including this
MATH 112 EXAM I Hints and Answers
Version Alphathe denominator in #1(a) is e4t
Winter 2013
1. (4 points each)
du
(a)
=
dt
(b)
e4t
3
2t+ 2 t1/2
2 +3 t
t
dy
1
= 10 ln x + x
dx
e
ln(t2 + 3 t)
(e4t )2
9
1 ex
x + e1
x
2. (4 points each)
3
(a) x1/2 + 25e2x + C
MATH 112 EXAM I Hints and Answers
Version Betathe denominator in #1(a) is e10t
Winter 2013
1. (4 points each)
du
(a)
=
dt
(b)
e10t
5
2t+ 2 t1/2
2 +5 t
t
dy
1
= 8 ln x + x
dx
e
ln(t2 + 5 t)
(e10t )2
7
1 ex
x + e1
x
2. (4 points each)
5
(a) x1/2 + 10e2x +
Name:
Section:
Math 112
Group Activity: Tangents and Secants
The graph below is of a function y = f (x).
30
y=f(x)
25
20
15
10
5
1
2
3
4
5
6
7
8
9
1. Draw a tangent line to the graph of f (x) at x = 2 and compute its slope. What is the functional notation
Name:
Section:
Math 112
Group Activity: Rate of Ascent Graphs
A Purple balloon and a Green balloon rise and fall. When we start watching, at t = 0, both balloons are 30 feet
above the ground. Their altitudes at time t minutes are given by functions P (t)
Name:
Section:
Math 112
Group Activity: Local vs. Global Optima
1. Below is the graph of altitude, A(t), for a balloon that is rising and falling.
15,000
altitude (feet)
A(t)
10,000
5000
1
2
3
4
5
time (minutes)
(a) List all critical values of the functio
Math 112
Midterm1 SOLUTIONS, v1
Spring 2013
1. (10 points) Compute the indicated derivatives and BOX your nal answer.
(a) If f (x) =
3
(x2 + 4)(2x3 5), compute f (x). Do NOT simplify your answer.
There are a few correct ways to proceed. One of them is:
f
NAME:
QUIZ Section:
STUDENT ID:
Math 112 Midterm 1
Spring 2013
HONOR STATEMENT
I arm that my work upholds the highest standards of honesty and academic integrity at the
University of Washington, and that I have neither given nor received any unauthorized
MATH 112
Exam I
January 31, 2013
Name
Student ID #
Section
HONOR STATEMENT
I arm that my work upholds the highest standards of honesty and academic integrity at the
University of Washington, and that I have neither given nor received any unauthorized assi
Name:
Section:
Math 112
Group Activity: The Vertical Speed of a Shell
A shell is red straight up by a mortar. The graph below shows its altitude as a function of time.
400
altitude (in feet)
300
200
100
1
2
3
4
5
time (in seconds)
6
7
8
9
10
The function
Name:
Section:
Math 112
Group Activity: Graphs of Derivatives
The graph below shows the function y = f (x).
9
8
7
6
5
4
3
2
1
-4
-3
-2
-1
y=f(x)
1
-1
-2
-3
-4
-5
-6
-7
-8
-9
2
3
4
1. Recall that f (a) gives the slope of the line tangent to f (x) at x = a.
Section 1.7 & 1.8: The Chain Rule and Higher-Order DerivativesSolutions
Practice with the Chain Rule:
1. Differentiate y = (3x 5)2 in two ways. One, expand the polynomial (multiply it out) first, then
differentiate. For the other method, use the chain rul