5.5
Substitution Rule
Substitution method works whenever we have an integral that we can write in
the form
In the composite function f (g(x) in equation, we identify the inner function
as u = g(x), which implies that du = g(x) dx. Making this identificati
2.6
Continuity
We noticed in Section 2.3 that the limit of a function as x approaches a can often be found
simply by calculating the value of the function at a. Functions with this property are called
continuous at a.
If any item in the continuity checkli
Chapter 2
Limits
Modern calculus was developed in 17th century Europe by Newton and Leibniz,
but the origins of calculus go back at least 2500 years to the ancient Greeks.
It has two major branches, differential calculus (concerning rates of change
and sl
4.5
Linear Approximations
and Differentials
We have seen that a curve lies very close to its tangent line near the point of tangency. In fact, by
zooming in toward a point on the graph of a differentiable function, we notice that the graph looks
more and
5.2
Definite Integrals
We introduced Riemann sums in Section 5.1 as a way to approximate the area of a region
bounded by a curve y = f (x) and the x-axis on an interval [a, b]. In that discussion, we
assumed f to be nonnegative on the interval.
Our next t
6.5 Length of Curves
The space station orbits Earth in an elliptical path. How far does it travel in one orbit?
This question deals with the length of trajectory or, more generally, with arc length.
As you will see, the answer can be found by integration.
4.9
Antiderivatives
A physicist who knows the velocity of a particle might wish to know its position at a
given time.
A biologist who knows the rate at which a bacteria population is increasing might
want to deduce what the size of the population will b
Ch 5. Review
Solve
Answer:
Answer:
1
ln e 2 x 1 C
2
Evaluate
Answer:
Answer:
Answer:
1
3
2
21
6.2
Regions Between Curves
We use integrals to find areas of regions that lie between the graphs of two
functions.
Answer:
A 2
Compound region
Answer:
1
A ln 2
2
3.9
Derivatives of Logarithmic and
Exponential Functions
We return now to developing rules of differentiation for the standard
families of functions.
First, we use implicit differentiation to find the derivatives of natural
logarithmic function. From ther
4.2
What Derivatives Tell Us
In the previous section, we saw that the derivative is a tool for finding critical points, which
are related to local maxima and minima. As we show in this section, derivatives (first and
second derivatives) tell us much more
4.9
Antiderivatives
A physicist who knows the velocity of a particle might wish to know its position at a
given time.
A biologist who knows the rate at which a bacteria population is increasing might
want to deduce what the size of the population will b
3.10
Derivatives of Inverse
Trigonometric Functions
In this section, we develop the derivatives of the six inverse trigonometric functions
Proof for derivative of
y sin 1 x
CLASS WORK
Compute the following derivative.
CLASS WORK
Solution:
We found the der
4.9
Antiderivatives
A physicist who knows the velocity of a particle might wish to know its position at a
given time.
A biologist who knows the rate at which a bacteria population is increasing might
want to deduce what the size of the population will b
2.4
Infinite Limits
Two limit scenarios are frequently encountered in calculus: Infinite limit and limit at infinity.
Infinite limit: function values increase or decrease without bound near a point.
limit at infinity occurs when the independent variable x
4.7
LHpitals Rule
Some limits, called indeterminate forms, cannot generally be evaluated using the techniques
presented in Chapter 2.
A powerful result called lHpitals Rule enables us to evaluate such limits with relative
ease.
CLASS WORK
convert the diff
5.5
Substitution Rule
Substitution method works whenever we have an integral that we can write in
the form
In the composite function f (g(x) in equation, we identify the inner function
as u = g(x), which implies that du = g(x) dx. Making this identificati
5.5
Substitution Rule
Substitution method works whenever we have an integral that we can write in
the form
In the composite function f (g(x) in equation, we identify the inner function
as u = g(x), which implies that du = g(x) dx. Making this identificati
3.11
Related Rates
The essential feature of related rates problems is that two or more variables,
which are related in a known way, are themselves changing in time.
After the first example, a general procedure is given for solving relatedrate problems.
4.6
Mean Value Theorem
Many of the results of this chapter depend on one central fact, which is
called the Mean Value Theorem.
But to arrive at the Mean Value Theorem we first need the following
result.
Our main use of Rolles Theorem is in proving the fol
3.11
Related Rates
The essential feature of related rates problems is that two or more variables,
which are related in a known way, are themselves changing in time.
After the first example, a general procedure is given for solving relatedrate problems.
2.3
Techniques for Computing Limits
Limits of Linear Functions
The graph of f (x) = mx + b is a line with slope m and y-intercept b.
Examples: Evaluating Basic Limits
1.lim 3 3
x5
2. lim x
2
(3) 2 9
x 3
3.lim 5 x 5(1)3 5
3
x 1
Example (1 of 2)
Find the f
2.5
Limits at Infinity
In Section 2.4 we investigated infinite limits and vertical asymptotes. There we
let x approach a number and the result was that the values of y became
arbitrarily large (positive or negative).
In this section we let x become arbitr
5.3
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 calculus.
Area Functions
The first part of the Fund
Chapter 5
Integration
5.1
Approximating Areas under
Curves
THE AREA PROBLEM
We begin by attempting to solve the area problem: Find the area of the region S that lies
under the curve f (x) from a to b.
Recall that in defining a tangent we first approximate
4.3
Graphing Functions
We have now collected the tools required for a comprehensive approach to graphing
functions. These analytical methods are indispensable, even with the availability of
powerful graphing utilities.
GUIDELINES FOR SKETCHING A CURVE
The
2.3
Techniques for Computing Limits
Limits of Linear Functions
The graph of f (x) = mx + b is a line with slope m and y-intercept b.
Examples: Evaluating Basic Limits
1.lim 3 3
x5
2. lim x
2
(3) 2 9
x 3
3.lim 5 x 5(1)3 5
3
x 1
Example (1 of 2)
Find the f
6.5 Length of Curves
The space station orbits Earth in an elliptical path. How far does it travel in one orbit?
This question deals with the length of trajectory or, more generally, with arc length.
As you will see, the answer can be found by integration.
6.4 Volume by Shells
Some volume problems are very difficult to handle by the methods of the preceding section.
For instance, lets consider the problem of finding the volume of the solid obtained by
rotating about the y-axis the region bounded by y 2 x 2
6.6 Surface Area
In Sections 6.3 and 6.4, we introduced solids of revolution and presented
methods for computing the volume of such solids. We now consider a related
problem: computing the area of the surface of a solid of revolution.
Here is an interesti
4.4
Optimization Problems
In optimization problems, the challenge is to find an efficient way to
carry out a task, where efficient could mean least expensive, most profitable,
least time consuming
To introduce the ideas behind optimization problems, thin
7.3 Trigonometric Integrals
At the moment, our inventory of integrals involving trigonometric functions is
rather limited.
For example, we can integrate sin ax and cos ax, where a is a constant, but
missing from the list are integrals of tan ax, cot ax, s
3.8
Implicit Differentiation
This chapter has been devoted to calculating derivatives of functions of the form
y = f (x), where y is defined explicitly as a function of x.
However, relations between variables are often expressed implicitly.
For example, t
3.7
The Chain Rule
DEFINITION:
The composed function f og , the composition of f and g,
is defined as
f og f ( g ( x)
3
2
f
(x)
x
g(x)
1
x
,
Example: For
and
Find ( f og )( x ) and ( g o f )( x).
( f og )( x) f ( g ( x)
2
f (1 x )
(1 x 2 )3
( g o f )( x)
2.2
Definition of Limits
Having seen in the preceding section how limits arise when we want to find the tangent to
a curve or area under the curve, we now turn our attention to limits in general and
numerical and graphical methods for computing them.
DEFI