Math 113
Review for Final Exam
On the final exam, I will focus on Calculus topics, though you will still need to be
able to do algebra in order to solve for critical points, simplify, etc.
Since I will write your exam, studying your tests 1 – 3 is wise!
Chapter 1
Sec. 1.1—The Cartesian Plane and the Distance Formula
(Algebra)
Sec. 1.2—Graphs of Equations
(Algebra)
Sec. 1.3—Lines in the Plane and Slope
(Algebra)
1.4—Functions
(Algebra)
1.5–Limits



be able to evaluate limits by direct substitution (when it works), by simplifying first
and then substituting, by making a table, or by looking at the graph as needed
understand one-sided limits
be familiar with the properties of limits (pg 51) as needed
1.6–Continuity


be able to determine the intervals on which a function is continuous
identify whether a discontinuity is removable or not
Chapter 2 (sections 2.1 – 2.2)
2.1–Definition of the Derivative



be able to find the derivative using the limit definition (one question on the test
will REQUIRE use of the limit definition—shortcuts will earn 0 pts on this question)
understand what derivatives tell us
be able to write an equation for a tangent line
2.2—Rules for Differentiation



be able to differentiate using shortcuts
practice re-writing functions so that shortcuts can be applied
be able to write an equation for a tangent line
Review for Final Exam
Math 113
Formulas you need to know (from test 1 material):
a2  b2  c2
y y2  y1
m

x x2  x1
f ( x)  lim
h0
f ( x  h)  f ( x )
(may use x in place of h)
h
d
c  0, (c is constant )
dx
d n
x  nx n 1
dx
d
cf ( x)  cf ( x) (c is a constant)
dx
d
 f ( x)  g ( x)  f ( x)  g ( x)
dx
 
Chapter 2 (sections 2.3 – 2.7)
2.3 –Rates of Change: velocity and marginals



know how to find instantaneous rate of change and average rate of change
understand the relationship between position, velocity and acceleration
be able to find units of a derivative function
2.4 –Product and Quotient Rules


know them
be able to use them and recognize when to use them
2.5 –Chain Rule


know it
be able to use it and recognize when to use it
2.6 –Higher Order Derivatives


be able to calculate them (& understand meaning)
be comfortable with notation
2.7 –Implicit Differentiation
dy
dx

be able to find

don’t forget to use product rule or quotient rule when necessary
implicitly
Math 113
Review for Final Exam
Chapter 3 (sections 3.1 – 3.4)
3.1 –Increasing/Decreasing Functions



be able to find where a function is increasing or decreasing (understand what the
first derivative tells us about the original function)
know what critical numbers are and how to find them
Critical numbers occur when__________________________________________
3.2 –Extrema and the 1st Derivative Test


know what extrema are (both relative and absolute)
be able to find them using the first derivative test
3.3 –Concavity and the 2nd Derivative Test



understand what the second derivative tells us about the original function
know what “concavity” is and what “inflection points” are & how to find them
be able to use the second derivative test to find extrema
3.4 –Optimization


be able to optimize any quantity using either 1st or 2nd derivative test
you should be able to write your own function if necessary (see homework &
suggested problems for examples)
Formulas from test 2 (not given)
d
 f ( x)  g ( x)  f ( x) g ( x)  g ( x) f ( x)
dx
d  f ( x)  g ( x) f ( x)  f ( x) g ( x)

dx  g ( x) 
( g ( x)) 2
d
 f ( g ( x))  f ( g ( x))  g ( x)
dx
you should also know things like the Pythagorean theorem, formulas involving
rectangles and/or rectangular boxes, etc. I’ll provide formulas for circles, spheres,
cylinders, cones, etc. if needed.
Chapter 3 (sections 3.6 – 3.8)
3.6 – Asymptotes

Relationship between asymptotes (both horizontal and vertical) and limits
3.7 – Curve Sketching: A Summary

This section pulls together continuity, differentiability, extrema, concavity, inflection
points, and asymptotes (see page 231 for sections referenced)
3.8 – Differentials and Marginal Analysis

Compute differentials and use them to approximate error.
Review for Final Exam
Math 113

Formula: dy  f ( x)dx (recall dx = x)
Chapter 4 (sections 4.1—4.5)
4.1—Exponential Functions
(Algebra)
4.2—Natural Exponential Functions
(Algebra)
4.3—Derivatives of Exponential Functions


Be able to find derivatives of exponential functions.
Formulas to know:
d ex  ex
dx
d
dx
 
e   e
g ( x)
g ( x)
 g ( x)
4.4—Logarithmic Functions
(Algebra review) – properties of logs can make differentiation easier!
4.5—Derivatives of Logarithmic Functions


Be able to find derivatives of logarithmic functions.
Formulas to know:
d
ln( x)  1
dx
x
d
ln g ( x)   1  g ( x)
dx
g ( x)
d
log b ( x)  1  1
dx
x ln b
x
x
d
b  b  ln b
dx
 
4.6—Exponential Growth and Decay

Review problems from this section. It’s mostly algebra review, but there are some
questions involving calculus concepts.
Chapter 5 (sections 5.1—5.5)
5.1—Antiderivatives and indefinite integrals


Know what “antiderivatives” and “indefinite integrals” are.
f ( x)dx  F ( x)  C
Know the notation:

Basic rules you need to know: (next page)

Review for Final Exam
Math 113
 kdx  kx  C , k is a constant
 kf ( x)dx  k  f ( x)dx
 [ f ( x)  g ( x)]dx   f ( x)dx   g ( x)dx

* * x n dx 
x n1
C
n 1
**This last one is the “simple power rule”. Notice that it does not work for n = -1 (section
5.3 tells us how to deal with that).
5.2—The general power rule

Know and be able to use the general power rule:

g ( x)  g ( x)n dx 
g ( x)n1  C
n 1
(Again, this does not work for n = -1 —see section 5.3).
5.3—Exponential and Logarithmic integrals

Know and be able to use the rules for exponential integrals:
 e dx  e  C
g ( x)
g ( x)
 g ( x)  e dx  e  C
x

x
Know and be able to use the rules for logarithmic integrals:
1
 x dx   x
1
dx  ln x  C
1
 g ( x)  g ( x) dx  ln g ( x)  C
5.4—Area and the fundamental theorem of calculus

Be able to find area under a given graph. Recall that the area under f(x) between
b
x = a and x = b is given by
 f ( x)dx
a
b

Know that
 f ( x)dx  F (b)  F (a) , and be able to use this to find definite integrals.
a
Math 113
Review for Final Exam
5.5—The area of a region bounded by two graphs

Be able to find the area between two graphs. You may need to find points of
intersection first.
Chapter 6 (sections 6.1 and 6.2)
6.1—Integration by substitution


Be able to integrate by substitution.
Be able to solve definite integrals by substitution.
6.2—Integration by parts


Be able to integrate by parts.
I will give you the formula: udv  uv  vdu on the cover page of your exam.

Remember, if you try integration by parts and it makes your problem worse, try a
different choice for u and dv.


Chapter 7 (7.1, 7.3 – 7.5?) – coverage will depend on what we are able to finish in
class. We’ll discuss this further on the last day of class (review day).
7.1—The Three-Dimensional Coordinate System



Finding distance between points in 3-dimensional space
Finding midpoint between points in 3-dimensional space
Finding equations of spheres
7.3 – Functions of Several Variables


Evaluating functions of several variables
Reading contour maps and associating them with 3D functions
7.4 – Partial Derivatives
z

 f x ( x, y )  z x   f ( x, y )  , etc. (see text
x
x

Finding partial derivatives; notation:


pg 484)
Evaluating partial derivatives (i.e. plugging in a point)
Finding second partial derivatives
7.5 – Extrema of Functions of Two Variables



Critical points of functions of two variables
Second-partials test for relative extrema (pg 498)
(Note: the First-partials test on pg 495 requires that you visualize the graph of the
function in 3 dimensions; the Second-partials test does not require this).
Chapter 6 Review questions: pg 450-451 #1 – 34
Chapter 7 Review questions: pg 544-546 #1 – 10, 13 – 14, 27 – 28, 39 – 54, 63 –
70.