Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
MATH 125
Derivatives and Graphs
1. Consider a continuous function f (x) whose derivative is given by f 0 (x) = 6x
2/3 (4
x2 )(7
2x).
(a) Identify the critical values for f (x).
(b) Use the First Derivative Test clearly showing all relevant details! to det
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
MATH 125
Intro to Derivatives
1. Determine the value(s) of K for which each function below is continuous.
2
2
x + K, if x < 4
x + Kx,
(a) f (x) =
(b) f (x) =
10 2x, if x 4.
(x K)2 ,
if x 3
if x > 3.
2. Use the definition of derivative (fourstep process
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
MATH 125
Implicit Dierentiation/Related Rates
1. Find dy/dx:
5e3xy
2. Find dy/dx:
4 cos y
4y 2 = x + 2y
3yex = (5x
y)4
3. Find the equation of the line tangent to sin(xy) + 4y = 3
ln(x + y 2 ) + yex at the point (0, 1).
4. The surface area of a spherical
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 125
Dierentiation
1. Find y 0 if y = 174.
2. Find
dy
if y =
dx
11. Find y 0 if y = 4x7 + 22
4.
12. Find y 0 if
3. Find y 0 if y = x4 + 22.
4. Find y 0 if y = x8
5. Find y 0 if y = x3
6. Find
dp
if p = 3q 2
dq
1 2
x
4
10. Find
dx
if x = t12
dt
13. Fin
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 125
Exam #2 Fall 2012
1. Let f (x) and g(x) be dierentiable functions with
lim f (x) = 5
lim f (x) = 0
x!2
x!4
lim g(x) = 6
x!2
lim g(x) =
x!4
(a) Evaluate if possible: f (2) =
2
f 0 (2) = 3
g 0 (2) = 7.
g(2) =
(b) Evaluate if possible: (f g)0 (2) =
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 125
Exam #2 Fall 2013
1. Determine each limit algebraically. Numerical or graphical work to support your answer is not sufficient.
p
2
x
x2 4
lim
lim
x!4 x
x!2 1
4
3x2
2. Use the limit definition of the derivative (fourstep process) to determine f 0
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 125
Graphs and Derivatives
1. Assume the graph of a function f (x) is shown above. What can you say about f (x) (max/mins,
inc/dec, concavity, inflection points, etc.)? What can you say about f 0 (x)? What can you say
about f 00 (x)? Sketch possible
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
1. (15. 87) An apple orchard is infested with a horrible mildew. The percentage of apple trees infected
varies from a 10W .Of 1.0% an May 1 to a high of 70% six months later.
(3.) Use the general Sine function y = Asin[w(t 01.)] + C to model the percent
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
COUNTING & PROBABILITY 1
1)
A restaurant offers 7 entrees and 6 desserts. In how many ways
can a person order a twocourse meal?
1) _
2)
In how many ways can a girl choose a twopiece outfit from 5 2) _
blouses and 6 skirts?
3)
A restaurant offers a choic
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Spring 2012
The Longest Run of Heads
Mark F. Schilling
Mark F. Schilling is Associate Professor at California State
University, Northridge. He received his BA. and MA. in mathe—
matics at the University of California at San Diego and his
doctorate was earned in stati
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Spring 2012
MATH 247 HONORS, FALL 1999  PROBLEM SET 5
Do ve of the following six problems. Due Friday, Oct. 1, because of test on Wednesday. Test #1 Wednesday, Sept. 29, 7:309:30PM, 241 Altgeld Hall.
1. Let A1 ; A2 ; : : : be a sequence of sets, each of which is co
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Spring 2012
2.4.1. Example: Coupon Collector ’5‘ Problem
The coupon collector’s problem arises from the following scenario. Suppose that each
box of cereal contains one of n different coupons. Once you obtain one of every type
of coupon, you can send in for a prize.
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Spring 2012
Massachusetts Institute of Technology
6.042J/18.062J, Fall 05: Mathematics for Computer Science
Prof. Albert R. Meyer and Prof. Ronitt Rubinfeld
Course Notes, Week 14
December 2
revised December 8, 2005, 61 minutes
Missed Expectations?
In the previous not
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Spring 2012
Some practice problems for the nal
1. Let A, B, and C be events dened on a particular sample space . Write expressions for the following
combinations of events:
(a) All three events occur.
(b) At least one of the events occurs.
(c) A and B occur, but not
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Spring 2012
Exercises of Chapter 2
ChuangChieh Lin
Department of Computer Science and Information Engineering,
National Chung Cheng University, MingHsiung, Chiayi 621, Taiwan.
Exercise 2.6. Suppose that we independently roll two standard sixsided dice. Let X1 be
t
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Spring 2012
Answer key to practice problems for nal
1. (a) A B C (b) A B C (c) (A B) \ C
2. (a) cfw_H, T 200 (b) cfw_nonnegative integers (c) cfw_English words
3.
2
9
4.
1
512
5. log10/9 10 = 22
6.
10
2
4
( 1 )2 ( 5 )8 .
5
7. (a) 1 (
8.
90
10
/
98
50
100
50
/
)=
149
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 125
Final Review
1. Answer the following questions True or False by circling the appropriate letter.
T F
d
dx
T F
If f 0 (x0 ) exists and f (x) attains a relative maximum value at x = x0 , then the tangent line of f (x)
at x = x0 is horizontal.
T F
G
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
51
, ) _
. (c) y=BSec(0)+cos(62) (do; u.) : rsnn u.~ i
7 33609'm9 ~" SMLBLBHLEB
(W
ll
Sec have? . sinLBL)
(AM
' 4 Lv1 4;
1. (24) Differentiate. You do not need to simplify completely, but effort should be made to at least simplify
within each term
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
l. (30) Determine each integral. To receive full credit you must clearly indicate all substitutions used
(a) /($2l)dm t: jXS/LPXA A: inch! : U\V" S'Velm
2. (20) Determine each integral. To receive full credit you must clearly indicate all substitutions us
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Xdox. XN I Wm. H. fxa
_ U+nuunoudTT
.U .1 5mg N71
_.II_. I
3.1. .55an .
. a x u E, u:
x x . x s x .+xm.x 36;." :3;
3. E .a .9 . i3 3% . 5. a E
.13.: 323533 .3 33%.: .2330 v.25 :9. 32. :3 2:80. a. .EEE nun. . 3 85 .w ow: 33358.3 a. SEES. mice. 55 sp
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 126
Review, Test 1
1. Determine the following integrals.
Z
Z
(ln x)5
2
3x
(2x
e
)
dx
dx
(a)
(b)
2x
Z
(d)
Z
(g)
Z
2
(j)
1
Z
(m)
1
Z
(p)
2
1
4x1 + x2
2e3x
2
4
3
3x x
Z
dx
(h)
Z
dx
(k)
9x2 + 12x
dx
x3 + 2x2 2
/2
(q)
0
Z
x2 e4x dx
Z
2
x cos(x2 ) dx
(l
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 126
Spring 2011 Exam 1
1. Determine each integral. To receive full credit you must clearly indicate all substitutions used.
Z
(a)
x(x2 1) dx
Z
9x2 3
(b)
dx
x3 x
Z
p
(c)
24x 5 x2 dx
2. Determine each integral. To receive full credit you must clearly i
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 126
Integral Practice
Compute the following indefinite integrals.
Z
Z
2
2
x(x 1) dx
(x2 1)2 dx
1.
2.
Z
4.
Z
7.
Z
10.
2x3
dx
3
1 + x4
1
sin(ln x) dx
x
6ex
dx
ex ex
Z
8
(5 7x) dx
13.
Z
2
sec x tan x dx
16.
Z
3
cos x dx
19.
Z
22.
1 + ex
x
dx
e +x
Z
5.
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 126
Exam #1 Spring 2012
1. Determine each integral. To receive full credit you must clearly indicate all substitutions used.
Z
cos x
(a)
dx
x
Z
x 2 x dx
Z
10x2/3 (3 4x) dx
(b)
(c)
2. Determine each integral. To receive full credit you must clearly in
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 126
Exam #1 Spring 2013
1. Determine each integral. To receive full credit you must clearly indicate all substitutions used.
Z
4x 6
(a)
dx
2
(x 3x + 1)4
Z
(b) (1 x2 )2 dx
2. Determine each integral. To receive full credit you must clearly indicate al
Math and Calculus Applications for Life, Managerial, and Social Sci I
MATH 125

Fall 2014
Math 126
Review, Test 1
1. Determine the following integrals.
Z
(2x2 e3x ) dx
(a)
We use the basic rule for power functions in the first term and a simple usubstitution in the second with
u = 3x and du = 3 dx .
Z
Z
Z
1
2
3x
2
(2x e ) dx =
2x dx
eu du
3