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 squeeze theorem that
lim x sin 1 = 0
x
x0
2. Find the li
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 at a point on the curve where y is also a
differentiable
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 lateral boundary of a solid of revolution.
We want to d
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 a
ruler. But that might be difficult to do with much a
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 is the inverse of the
differentiable function y = ex.
3
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 corresponds to the Chain
Rule for differentiation. The rule that
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 the power function
g(x) = x2
In general, an exponentia
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 horizontal line intersects the graph of f in more than
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 and is
denoted by loga.
Thus, if x > 0, then logax is t
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 during a day if
infinitely many temperature readings are
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, where f and g are
continuous functions and f (x) g(x) for
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 must make
this idea precise by using calculus to give an
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 region bounded by
y = 2x2 x3 and y = 0.
If we slice perpen
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
situations even when f is not necessarily a positive fu
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 traditionally used
for an antiderivative of f and is called an
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 branches of
calculus: differential calculus and integral
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 to evaluate
integrals such as
3
The Substitution Rule
O
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 that S, is bounded by the graph of a continuous function
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.
In addition to the solution x = 0, there is a solutio
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 needs no fence along the river.
What are the dimension
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 f,
that is, the set of values of for which f (x) is def
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).
Example.
3
Limits at Infinity; Horizontal Asymptotes
4
Limit