Section 5.8
Inverse Trigonometric Functions:
Integration
Formulas
du
u
= arcsin + C
a2 u2
a
du
1
u
a 2 + u 2 = a arctan a + C
u
du
1
u u 2 a 2 = a arc sec a + C
How do you determine if you are
dealing with arcsinu or arcsecu?
Dont key on whether or not
Steps to Solve Related Rate Problems Section 3.9
1. Draw a picture.
2. Identify all given quantities and the quantities to be determined. Remember, the rate of change is the
derivative with respect to time.
3. Write an equation relating the variables. You
Calculus I Prerequisite Worksheet
1. For each of the following, find the equation of the line satisfying the given information. Write your answers
in point-slope form.
a. Line through the point (2,3) with slope m = 1/3.
b. Line through the points (2,3) an
Calculus I, Departmental Final Exam Review
tv f/06
This review should be used as a guide only. These problems are intended to represent the types of
skills that you should have mastered in Calculus I. The final exam will require you to show mastery of
the
Calculus Review for Exam #3, fall 2005
3
2
1. a. Evaluate 2 cos x + csc x dx
x
1
dx .
b. Evaluate
1 2x
x
dx
c. Evaluate
x 1
2
d.
3
x
1dx
2
1
e.
x (x
f.
cos
2
3
sin x
3
)
4
+ 5 dx
x
dx
2.
Set up, but do not evaluate, the limit to find the area of the
Calculus I, Exam #3
Primarily Chapters 4 and 5
Fall 2005
Name_
Please show all work to receive full credit. Answers without appropriate supporting work
will not receive credit. Each problem is worth 10 points. Give exact answers unless
indicated otherwise
Calculus I, Exam #2 Tuesday, M h , 2011
Chapter 3 Name
You must show all work to receive credit. Answers without appropriate supporting work will not receive credit, even if
correct. There are 11 problems worth 10 points each. Only your best 10 attempts
Calculus I Exam #2
Fall 2005
Name _
You must show all your work on this paper. Solutions without correct supporting work will not be accepted. All problems
must be worked manually.
1. Locate the absolute extrema of
f ( x ) = 4 x 5 5 x 4 on [ 0,2] .
2. Det
Chapter 3 review
1.
a.
c.
Use the definition of derivative to find f (x) for each of the following: (Section 3.1)
b.
2. Find for each of the following: (Sections 3.2 3.6)
a)
b)
c)
d)
e)
f)
g)
h)
i) j)
k)
l)
m) sinx=x(1+tany)
n)
o)
p)
q)
3. For each of the
1.
Locate the absolute extrema of the function on the closed interval. (Section 4.1)
a)
[-1, 2]
b)
[ 0, 3 ]
c)
2. Determine whether Rolles Theorem can be applied to f on the closed interval [a,b]. if
Rolles Theorem can be applied, find all values of c in
Section 5.5
Integration by Substitution
With what we know now, how
would we
Integrate
( 2 x 1) 3dx
Rewriting in this manner is tedious and
sometimes impossible so we need a better
way. This new method is called u-substitution.
The goal of u-substitution i
Section 5.4
The Fundamental Theorem of
Calculus
How are indefinite and definite
integrals related?
Guidelines for Using the
Fundamental Theorem of Calculus
The Mean Value
Theorem for
Integrals
Average Value of a
Function
Section 4.5
Limits at Infinity
Definition of a Horizontal
Asymptote
The line y = is a horizontal asymptote of the graph of f
L
if
lim f ( x) = L
x
or
lim f ( x ) = L
x
Guidelines for Finding Limits at
Infinity of Rational Functions
Section3.6
DerivativesofInverseFunctions
DerivativesofInverseTrigFunctions
d
u'
[ arcsin u ] =
dx
1 u2
d
u'
[ arctan u ] = 2
dx
1+ u
d
u'
[ arc sec u ] =
dx
u u2 1
d
u'
[ arccos u ] =
dx
1 u2
d
u'
[ arc cot u ] = 2
dx
1+ u
d
u'
[ ar csc u ] =
dx
u u2 1
Section 3.4
The Chain Rule
One of THE MOST POWERFUL
Rules of Differentiation
The chain rule allows you to take derivatives of
compositions of functions that may be hard or
even impossible to differentiate with previous
rules only.
Examples:
f ( x ) = ( x
Basic Differentiation Rules and Rates of Change
The Constant Rule
The derivative of a constant function is 0. In other words, if
c is a real number, then
d
[ c] = 0
dx
The Power Rule:
If n is a rational number, then the function d n
f ( x) = x n
x = nx n
Assignment 4
Section 3.1
The Derivative and Tangent Line
Problem
The Basic Question is
How do you find the equation of a line that
is tangent to a function y=f(x) at an
arbitrary point P?
To find the equation of a line you need:
a point and a slope
How
Section 3.3
The Product and Quotient Rules
and Higher-Order Derivatives
The Product Rule
d
[ f ( x) g ( x)] = f ( x) g ' ( x) + g ( x) f ' ( x)
dx
Note: The derivative of a product is NOT the
product of the derivatives. NO PARTIAL
CREDIT for committing th
g.
h.
i.
j.
3. Find the limits using any appropriate method.
a.
b.
4.
Find the x-values (if any) at which f is not continuous. Identify any discontinuities as removable or nonremovable. (Section 2.4)
a)
b)
6. Find any vertical asymptotes for a)
b)
c)
7. U
Calculus I Exam #1
Fall 2005
Name _
Show all your work on this paper. Solutions without correct supporting work will not be accepted. All
solutions must be exact unless stated otherwise in the problem.
1. Use the graph of y = f ( x) given below to find th
Calculus I Section 4.2 Rolles Theorem and the Mean Value Theorem
Rolles Theorem
Let f be a continuous function on the closed interval [a, b] and differentiable on the open interval (a, b). If
f(a)=f(b) then there is at least one number c in (a, b) such th
9/13/2010
The Chain Rule (Section 3.5)
Chain Rule, Plane Curves,
and Parametric Equations
Calculus I
Section 3.5
Definition of a Plane Curve
If f and g are continuous functions of t on an
interval I, then the equations x = f(t) and y=g(t)
are called param