Math 212 Quiz # 2 Review sheet of goodness!
February 26, 2014
1. Section 5.1: The Natural Logarithmic function, dierentiation. This section, along
with most of the others in Chapter 5, is really just a review of a basic function that
has a funny denition.
Section 5.2 Notes
Math 212
5
1
Logarithmic, Exponential, and Other Transcendental Functions
5.2
The Natural Logarithm Function: Integration
Example 1
d
[ln |u|] =?
dx
Theorem 5.5. Let u be a differentiable function of x. Then
Z
1
1.
dx = ln |x| + C
x
Z
1
Section 4.3 Notes
Math 212
4
4.3
1
Integration
Riemann Sums and Definite Integrals
In Section 4.2, the area of the region bounded by the graph of f , the x-axis, and the lines
x = a and x = b required that f be continuous and non-negative. The next defini
Section 5.6 Notes
Math 212
5
1
Logarithmic, Exponential, and Other Transcendental Functions
5.6
Inverse Trigonometric Functions: Differentiation
Much of this class analyzes the idea of inverse relationships. Ex: Derivatives and Antiderivatives/Integrals,
Math 212
5
Section 5.4 Notes
1
Logarithmic, Exponential, and Other Transcendental Functions
5.4
Exponential Functions: Differentiation and Integration
Example 1
1
Recall in Section 5.1, we defined e to be the number that gives us 1 unit of area under .
x
Section 4.4 Notes
Math 212
4
1
Integration
4.4
The Fundamental Theorem of Calculus
Review from Section 4.1
Something to remember: A function F is an antiderivative of f Z
on some interval I if F 0 (x) =
f (x) for every x in I. We denote the general antide
H
Spring 2017 Exam 1 Practice Problems Math 21206
Disclaimer: This is just an extra set of practice problems to do computations. It is by
no means a complete list of problems you should be comfortable solving. While it is good
to practice corrmutations, y
Section 4.2 Notes
Math 212
4
Integration
4.2
Area
Sigma Notation
a1 + a2 + a3 + . . . + an =
n
X
ai
i=1
i is the index. Most common letters used for index are i, j, k
i = 1 tells us when to start
n tells us when to stop
ai tells us what is being added
'7'?
‘ Math 212 W14 Examination 2 Good Luck to: % '
SHOW YOUR WORK to receive all possible credit.
1. Find the derivative of the following functions. Simplify your answer.
3- Y=1°81 X- Answer: y' b. y=4". Answer: y' #x
'l+ *l a4?) ix
c. y=5" *2“ 5 d. y=
-*
Good Luck to
Examination 1
Math 212 W14
SHOW YOUR WORK to receive all possible credit.
1. Use the limit process to find the area ofthe region between the graph of f(x) =6x +2 and the x-axis over
the interval [2,3]. No credit for any other method. Use t
Section 5.1 Notes
Math 212
5
1
Logarithmic, Exponential, and Other Transcendental Functions
5.1
The Natural Logarithm Function: Differentiation
Example 1
Z
Find each of the following:
Z
2
x dx
x
2
Z
dx
n
x dx
Z
x1 dx
Definition. The natural logarithm func
Math 212
4
4.5
Section 4.5 Notes
1
Integration
Integration by Substitution
Example 1
1
What is the derivative of f (x) = (x3 + 10)3 + 4?
3
Pattern Recognition
Example 2
Z
What is 3x2 (x3 + 10)2 dx?
Theorem 4.13. Let u = g(x) be a differentiable function w
Math 212 Quiz # 2 Solutions
March 10, 2014
Here are the solutions to the second quiz. Enjoy!
1. We use the chain rule to compute
d sec x
d
[e
] = esec x [sec x] = esec x sec x tan x.
dx
dx
2. Let u = ex . Then du = ex dx, and
tan ex + C .
ex sec2 ex dx =
MATH 212 QUIZ # 1 SOLUTIONS
Here are the solutions to the rst quiz. Enjoy!
(1) Integrating, we have y = 4x 8dx = 2x2 8x + C . Imposing the initial condition
y (1) = 2, we conclude
2 = y (1) = 2(1)2 8(1) + C,
so C = 8, and y = 2x2 8x + C.
1
(2) (a) x4 + 2x
MATH 212 QUIZ # 1 REVIEW SHEET OF FUN!
Hi everyone, here is a review sheet for the upcoming quiz. Enjoy!
(1) Section 4.1: Antiderivatives. The big deal from this section is to understand what
antiderivatives are, and why a certain functions antiderivative
MATH 212 FINAL EXAM REVIEW
Hi everyone, here is your review sheet for the nal exam. Remember, the nal exam will
be cumulative to the beginning of the termthe sections listed below are just those that Corey
hasnt written anything about yet. Also remember t
MATH 212 MIDTERM REVIEW SHEET
Hi everyone! Here is a sheet which reviews the remaining sections that the exam will
cover. That means that the exam will cover the sections that the quiz review reviewed in
addition to those below. Good luck!
(1) Section 4.4
Exam Solutions!
January 30, 2014
Here are the solutions to the midterm. Enjoy!
1. (a) We break the interval up into 4 pieces, each with width x = 1 = .25. So, for a
4
right approximation, we would have inputs x1 = 1.25, x2 = 1.5, x3 = 1.75, x4 = 2. So
f (
Section 8.2 Notes
Math 212
8
1
Integration Techniques, LH
opitals Rule, and Improper Integrals
8.2
Integration by Parts
Integration by parts is useful when there are products of functions in the integrand, but
u-substitution does not work.
Examples of pro
Section 5.7 Notes
Math 212
5
1
Logarithmic, Exponential, and Other Transcendental Functions
5.7
Inverse Trigonometric Functions: Integration
Example 1
u
i
d h
arcsin
+C
Find
dx
a
Example 2
u
i
d h
arccos
+C
Find
dx
a
Theorem 5.17. Let u be a differentiab
Math 212 W14
Quiz 3
Good Luck
to: ^W
SHOW YOURWORK to receive all possible credit.
1. Find the indefinite integral: J
V
jL d%
U Cs) "
dx.
. \ dn 5 J U '
j- (uh-s*\+ c
x +2
2. Given is f(x) =
-, x >-3, a =5. Verify that f(x) has an inverse. Use the functio