U-Substitution Exploration
Consider the following integrals. Use u-substitution to evaluate them. Can you find any general rules
for certain types of integrals?
3x e
2 x3
e
5x
dx
dx
3xe
x2
dx
4x3
1 x 4 dx
1
x 2dx
ln x
dx
x
3
2 xdx
x x
3
2
5
2 dx
Riemann Sums
Calculus
This section will help us understand how we will use the integral or antiderivatives. We begin by
exploring the following area problem.
Consider the curve y x 2 1 between the vertical bounds of x 0 and x 2 . If we draw this region
an
MA 181 Lecture
Chapter 10
College Algebra and Calculus by Larson/Hodgkins
Exponential and Logarithmic Functions
10.3) Derivatives of Exponential Functions
You can find a review of exponential functions in Sections 10.1 and 10.2.
Definition of the Number e
MA 180 Lecture
Chapter 7
College Algebra and Calculus by Larson/Hodgkins
Limits and Derivatives
7.2) Continuity
Consider the following graph of the function f(x). Use it to evaluate the limits.
lim f ( x)
x 3
lim f ( x)
lim f ( x)
x 1
x 4
lim f ( x)
x
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.1) Higher Order Derivatives
The derivative of f is the second derivative of f and is denoted f.
Equivalently,
d
f ' ( x) f ' ' ( x)
dx
Similarly, th
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.4) Increasing and Decreasing Functions
A function is increasing if its graph moves up as x moves to the right and is decreasing if its graph
moves do
Area and Other Applications of the Definite Integral
Given a non-negative function f(x) bounded by the x-axis and two vertical lines x=a and x=b, the definite
b
integral
f ( x)dx is exactly equal to the area of the region bounded by those curves.
a
b
We
MA 181 Lecture
Chapter 7
College Algebra and Calculus by Larson/Hodgkins
Limits and Derivatives
7.5) Rates of Change: Velocity and Marginals (Continued)
Rates of Change in Economics: Marginals
When we take the derivative of the cost, revenue, or profit fu
MA 181 Lecture
Chapter 9
College Algebra and Calculus by Larson/Hodgkins
Further Applications of the Derivative
9.1) Optimization
One of the most important applications of the derivative is optimization. It involves finding a value of x
where we can eithe
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.1) Higher Order Derivatives
The derivative of f is the second derivative of f and is denoted f.
Equivalently,
d
f ' ( x) f ' ' ( x)
dx
Similarly, th
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.2) Implicit Differentiation
We say that a function is in explicit form if it is of the form y=f(x). In other words, on variable is
explicitly defined
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.3) Related Rates
First we give special attention to notation. When we say
dy
d
or
, we are saying the derivative with
dx
dx
respect to the variable x
MA 181 Lecture
Chapter 9
College Algebra and Calculus by Larson/Hodgkins
Further Applications of the Derivative
9.2) Business and Economics Applications
In this section we expand the concept of optimization to business applications.
Example:
A company has
MA 181 Lecture
Chapter 10
College Algebra and Calculus by Larson/Hodgkins
Exponential and Logarithmic Functions
10.3) Derivatives of Exponential Functions
You can find a review of exponential functions in Sections 10.1 and 10.2.
Definition of the Number e
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.4) Increasing and Decreasing Functions
A function is increasing if its graph moves up as x moves to the right and is decreasing if its graph
moves do
MA 181 Lecture
Chapter 11
College Algebra and Calculus by Larson/Hodgkins
Integration and Its Application
11.1) Antiderivatives and Indefinite Integrals
This chapter begins a new process called antidifferentiation that is considered the inverse applicatio
MA 181 Lecture
Chapter 11
College Algebra and Calculus by Larson/Hodgkins
Integration and Its Application
11.2) Integration by Substitution and the General Power Rule
Recently we used the Simple Power Rule to do some basic antidifferentiation.
n
x dx
x
MA 181 Lecture
Chapter 7
College Algebra and Calculus by Larson/Hodgkins
Limits and Derivatives
7.6) The Product and Quotient Rules
We continue to find short cuts to finding derivatives without using the limit definition of the derivatives.
Consider that
MA 181 Lecture
8.2) Implicit Differentiation
We say that a function is in explicit form if it is of the form y=f(x). In other words, on variable is
5 xe x
explicitly defined in terms of the other. Examples include y = 2 x + 4 or y =
.
x +1
Some functions
MA 181 Lecture
Chapter 9
College Algebra and Calculus by Larson/Hodgkins
Further Applications of the Derivative
9.1) Optimization
One of the most important applications of the derivative is optimization. It involves finding a value of x
where we can eithe
Practice exam questions. Note there won't be this many questions, About 6 or 7 questions and 2 or 3
tables.
2. (8.3). A pebble is dropped into a calm pool of water, causing ripples in the form of concentric circles.
The radius r of the outer ripple is inc
MA 181 Lecture
Chapter 11
College Algebra and Calculus by Larson/Hodgkins
Integration and Its Application
11.1) Antiderivatives and Indefinite Integrals
This chapter begins a new process called antidifferentiation that is considered the inverse applicatio
MA 181 Lecture
Chapter 9
College Algebra and Calculus by Larson/Hodgkins
Further Applications of the Derivative
9.2) Business and Economics Applications
In this section we expand the concept of optimization to business applications.
Example:
A company has
MA 181 Lecture
Chapter 7
College Algebra and Calculus by Larson/Hodgkins
Limits and Derivatives
7.5) Rates of Change: Velocity and Marginals
Previously we learned two primary applications of derivatives.
1. Slope The derivative of f is a function that giv
Exam One*MA 181*Fall 2012*
Name_
_ of 80 points
Questions #1-6 are 6 points each. Show all work for questions in boxes.
A simplified answer with no work receives no credit. For questions 1-6 DO NOT SIMPLIFY!
Do not use the limit definition of the derivati
MA 181 Lecture
Chapter 7
College Algebra and Calculus by Larson/Hodgkins
Limits and Derivatives
7.1) Limits
The take-away
1.) Saying that the limit of f(x) approaches L as x approaches c means that the value of f(x) may be
made arbitrarily close to the nu
MA 181 Lecture
Chapter 10
College Algebra and Calculus by Larson/Hodgkins
Exponential and Logarithmic Functions
10.5) Derivatives of Logarithmic Functions
You can find a review of exponential functions in Sections 10.3.
We can use implicit differentiation
MA 181 Lecture
Chapter 11
College Algebra and Calculus by Larson/Hodgkins
Integration and Its Application
11.2) Integration by Substitution and the General Power Rule
Recently we used the Simple Power Rule to do some basic antidifferentiation.
n
x dx
x
MA 181 Lecture
Chapter 7
College Algebra and Calculus by Larson/Hodgkins
Limits and Derivatives
7.7) The Chain Rule
So far we have learned some basic differentiation rules that help us take the derivatives of constant
multipliers, sums and differences, an
MA 181 Lecture
Chapter 8
College Algebra and Calculus by Larson/Hodgkins
Applications of the Derivative
8.6.) Concavity and the Second-Derivative Test
In this section we begin our discussion about concavity of a graph, that is, the curving upward or curvi