Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
MAS 100 Midterm Exam  Solution
1. Find the limit.
lim x sin 1
x
x0
Solution. Since  sin t 1 we have
x x sin 1 x
x
for all x
1. Since
lim x = lim x = 0.
x0
x0
It therefore follows from the s
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
10
Parametric Equations
and Polar Coordinates
10.2
Calculus with Parametric Curves
Tangents
3
Tangents
Suppose x = f(t) and y = g(t) are differentiable functions and we want to
find the tangent line a
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
8
Further Applications
of Integration
8.2
Area of a Surface of Revolution
Area of a Surface of Revolution
A surface of revolution is formed when a curve is rotated about a line.
Such a surface is the
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
8
Further Applications
of Integration
8.1
Arc Length
Arc Length
What do we mean by the length of a curve? We might think of fitting a
piece of string to the curve and then measuring the string against
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
6
Inverse Functions
6.4
Derivatives of Logarithmic
Functions
Derivatives of Logarithmic Functions
We start with the natural logarithmic function y = ln x.
We know that it is differentiable because it
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
7
Techniques of Integration
7.1
Integration by Parts
Integration by Parts
Every differentiation rule has a corresponding integration rule. For
instance, the Substitution Rule for integration correspon
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
6
Inverse Functions
6.2
Exponential Functions and
Their Derivatives
Exponential Functions and Their Derivatives
The function
f(x) = 2x
is called an exponential function.
It should not be confused with
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
6
Inverse Functions
6.1
Inverse Functions
Inverse Functions
An important difference is that f never takes on the same value twice,
whereas g does take on the same value twice.
3
Inverse Functions
If a
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
6
Inverse Functions
6.3
Logarithmic Functions
Logarithmic Functions
For a > 0 and a 1, the exponential function f(x) = ax has an inverse
function, which is called the logarithmic function with base a
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
5
Applications of
Integration
5.5
Average Value of a Function
Average Value of a Function
The average value of finitely many numbers y1, y2, . . . , yn:
But how do we compute the average temperature d
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
5
Applications of
Integration
5.1
Areas Between Curves
Areas Between Curves
Consider the region S that lies between two curves
y = f(x) and y = g(x)
and between the vertical lines x = a and x = b, whe
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
5
Applications of
Integration
5.2
Volumes
Volumes
In trying to find the volume of a solid we face the same type of problem
as in finding areas.
We have an intuitive idea of what volume means, but we m
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
5
Applications of
Integration
5.3
Volumes by Cylindrical Shells
Volumes by Cylindrical Shells
Lets consider the problem of finding the volume of the solid obtained
by rotating about the yaxis the reg
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
4
Integrals
4.2
The Definite Integral
The Definite Integral
We have seen that a limit of the form
arises when we compute an area.
It turns out that this same type of limit occurs in a wide variety of
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
4
Integrals
4.4
Indefinite Integrals
Indefinite Integrals
3
Indefinite Integrals
Because of the relation given by the Fundamental Theorem between
antiderivatives and integrals, the notation
is traditi
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
4
Integrals
4.3
The Fundamental
Theorem of Calculus
The Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus is appropriately named
because it establishes a connection between the two b
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
4
Integrals
4.5
The Substitution Rule
The Substitution Rule
Because of the Fundamental Theorem, its important to be able to find
antiderivatives.
But our antidifferentiation formulas dont tell us how
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
4
Integrals
4.1
Areas
The Area Problem
3
The Area Problems
We begin by attempting to solve the area problem: Find the area of the
region S that lies under the curve y = f (x) from a to b.
This means t
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
3
Applications of
Differentiation
1
3.8
Newtons Method
Newtons Method
Consider the equation
48x(1 + x)60 (1 + x)60 + 1 = 0
We can find an approximate solution by plotting the left side of the
equation
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
3
Applications of
Differentiation
3.7
Optimization Problems
Optimization Problems
3
Example
A farmer has 2400 ft of fencing and wants to fence off a rectangular
field that borders a straight river. He
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
3
Applications of
Differentiation
3.5
Summary of Curve Sketching
Guidelines for Sketching a Curve
3
Guidelines for Sketching a Curve
A. Domain Its often useful to start by determining the domain D of
Korea Advanced Institute of Science and Technology
Differential Equations and Applications
MATHEMATIC 201

Fall 2014
3
Applications of
Differentiation
3.4
Limits at Infinity; Horizontal Asymptotes
Limits at Infinity; Horizontal Asymptotes
Here we are interested in the socalled end behavior of a function f(x).
Examp