Download Test #3 Topics

Topics for Test #3 (Sec 4.8, Chap 5 Sec. 5.1-5.5
CHAPTER 4 APPLICATIONS OF THE DERIVATIVE
4.8 Antiderivatives
Overview
We reverse the process of differentiation to find antiderivatives and lay the groundwork for
integration in Chapter 5. In addition, differential equations and initial value problems are
introduced.
Lecture
The rules for this section is given a function f, we seek a differentiable function F so that F′ = f.
This function F is called the antiderivative. Keep in mind that the antiderivatives are not unique.
See Theorem 4.15, Figure 4.70 and Example 1.
Watch video of Definitions of Antiderivatives at
http://www.youtube.com/watch?v=gvdl4XdeNDI&feature=player_embedded
Study the concept of the Indefinite Integral (and its notation) along with Theorem 4.16 and
Theorem 4.17 and Example 2.
Watch video of How to define an Indefinite Integral at http://www.wonderhowto.com/how-todefine-indefinite-integral-calculus-341498/
Study the Indefinite integrals of Trigonometric Functions, Table 4.5 and Example 3.
Notice that differentiating the solution provides an easy check to an indefinite integral
calculation.
We will conclude with a brief discussion of elementary differential equations and initial value
problems, and their connection to equations of motion. See Examples 4 and 5.
Watch video of Introduction to Differential Equations at
http://www.youtube.com/watch?v=C8mudsCSmcU&p=DAB9DDF56D818C0A&index=47&fea
ture=BF
Chapter 4 (Section 4.8) Key Terms and Concepts
Antiderivatives and indefinite integrals (Section 4.8)
Finding all antiderivatives (Theorem 4.15) (Section 4.8)
Power rule for indefinite integrals (Theorem 4.16) (Section 4.8)
Sum and Constant rules (Theorem 4.17) (Section 4.8)
Indefinite integrals (antiderivatives) for trigonometric functions (Section 4.8)
Initial value problems for velocity and position (Section 4.8)
CHAPTER 5 INTEGRATION
The material in this chapter is the climax of a first-semester calculus course. Riemann sums are
used to approximate the area under a curve, and the definite integral is defined as the limit of
Riemann sums. The marriage of limits, derivatives and integration is expressed in the beautiful
Fundamental Theorem of Calculus, the reward of a semester of hard work.
5.1 Approximating Areas under Curves
Overview
The area of a region bounded by the graph of a function over some interval (more simply, the
area beneath a curve) is approximated by summing the areas of rectangles whose heights are
given by the values of the function (that is, by evaluating Riemann sums). In this section, regions
are assumed to be bounded by the graph of a positive function.
Lecture
In Section 2.1, we used the idea of instantaneous velocity to introduce the concept of a limit. In
Section 3.1 (and again in Section 3.5) that idea was developed further to introduce the derivative.
To compute the distance traveled by an object moving along a straight line at constant velocity
(e.g. a car travels at 45 mi/hr for 2 hr on a straight road; how far has the car traveled?), the
answer (units included) is the area of the region between the graph of the velocity function and
the time axis over the interval [0,2].
We approximate the area under the graph of a function with sums of rectangles whose heights
are given by values of the function. These sums generally approach a limit as the number of
rectangles increases. See Example 1.
Study the Left, Right and Midpoint Riemann Sums. See its definition and Examples 2, 3 and 4.
Read the concept of Sigma Notation and the definition of Riemann Sums using sigma notation,
and do Example 5
Watch the video Approximating areas Using Rectangles at
http://www.wonderhowto.com/how-to-approximate-area-under-curve-using-rectangles-341505/
Watch the video of a Right Riemann Sum at
http://www.youtube.com/watch?v=uoipFxrjge4&NR=1
Watch the video (first 7.5 minutes) of a left and Right Riemann Sums at
http://www.youtube.com/watch?v=C58TZEzYSRs&feature=mfu_in_order&playnext=1&videos
=O5QxfxXvNNE
Watch the video (first 6 minutes) of a Midpoint Riemann Sum at
http://www.youtube.com/watch?v=gQqm4BkFqfw&feature=mfu_in_order&playnext=1&videos
=BOvau__CTe8
You can find a Riemann Sums applet at
http://www.math.tamu.edu/AppliedCalc/Classes/Riemann/index.html or at
http://www.scottsarra.org/applets/calculus/RiemannSums.html
5.2 Definite Integrals
Overview
A good deal of information is presented in this section. Regions above the x-axis make positive
contributions to a Riemann sum, while regions below the x-axis make negative contributions.
The definite integral is defined as the limit of Riemann sums (with a general partition), and
properties of definite integrals are explored. Finally, we present methods for evaluating definite
integrals.
Lecture Support Notes
Go over the concept of net area, and the definition of the Definite Integral.
Watch the video The Indefinite Integral-Understanding the Definition at
http://www.youtube.com/watch?v=LkdodHMcBuc&feature=related
Theorem 5.2 gives conditions under which a function is integrable; these conditions are met by
most of the functions in the text. Do Examples 3 and 4.
• Study the Properties of Definite Integrals, Table 5.3 and Example 5.
5.3 Fundamental Theorem of Calculus
Overview
A discussion of area functions and their properties leads to the Fundamental Theorem of
Calculus, where we discover the inverse relationship between differentiation and integration. The
FTC also gives an efficient method for evaluating definite integrals.
Lecture
Start with the area function A(x) (see Example 1) and construct A(x) for a linear function f (x)
(see Example 2)
Look at Theorem 5.3, The Fundamental Theorem of Calculus (FTC) Part 1 and Part 2.
Notice that the FTC links two of the most important processes in calculus, and that
differentiation and integration “undo” one another (see the Inverse relationship Between
Differentiation and Integration). Evaluate some integrals, and look at how they relate to areas
(see Examples 3 through 5).
You can watch the video on The Fundamental Theorem of Calculus FTC (Part 1)
http://www.youtube.com/watch?v=PGmVvIglZx8
You can watch the videos on The Fundamental Theorem of Calculus FTC (Part 2)
http://www.brightstorm.com/math/calculus/the-definite-integral/the-fundamental-theorem-ofcalculus
http://www.youtube.com/watch?v=Lb8QrUN6Nck&feature=fvw
You can also watch another video on The Fundamental Theorem of Calculus
http://www.math.armstrong.edu/faculty/hollis/calculusvideos/H.264/23-FTC-H264.mov
5.4 Working with Integrals
Overview
In this section, we exploit symmetry to make the evaluation of a definite integral easier and we
extend the idea of the average of a finite set of numbers to arrive at the definition for the average
value of a function over an interval.
Lecture
Review the properties of odd and even functions by studying the definition and Example 8 on
section 1.1
Read Theorem 5.4 and see figure 5.50. Do Example 1.
Read the definition of Average Value of a Function, and do Example 2.
5.5 Substitution Rule
Overview
Substitution (or change of variables) is one of the most important analytical techniques for
evaluating integrals.
Lecture
The most effective way to learn the method of substitution is to do as many problems as possible.
You need a lot of practice to see the important patterns. Begin with indefinite integral in
Example1. See Theorem 5.6 and Examples 2 through 4.
Practice substitution for definite integrals in Example 5.
Watch the Video Presentation (Indefinite Integrals and the Substitution Method) in the 5.5
Video Lecture link in the Interactive e-book.
Exponential Integration
Evaluate the following integrals. Use the chain rule if needed.
 e du  e
u
C
5.

e
6.
e
u
1.

3x
2.
 5e
3.
e
4.
e


 4e
2x
5
6x
sin(e3 x ) dx
1 e 2x dx
dx
tan(2x)
sec 2 (2x) dx

x
x
2
dx
dx
7.  aebx dx , where a and b constant

8.
 e (1  e ) dx
x
x
Answers:
4
5
5
1
cos(e 3x ) +C, 2.
(e 2x 1)3  C , 3. e 6x  C , 4. e tan(2x)  C , 5. 2e
3
3
6
2
2
a
1
7. e bx  C , 8. e x 1  C
b
2




1.


x
 C,
2
6. e x  C ,

Chapter 5 Key Terms and Concepts
Left/right/midpoint Riemann sum (Section 5.1)
 (Section 5.2)
Net area
Definite integral (Section 5.2)
Conditions for integrability (Theorem 5.2) (Section 5.2)
Properties of definite integrals (Section 5.2)
Area functions (Section 5.3)
Fundamental Theorem of Calculus, Parts 1 and 2 (Theorem 5.3) (Section 5.3)
Integrals of symmetric functions (Theorem 5.4) (Section 5.4)
Average value of a function (Section 5.4)
Substitution Rule (Change of Variables) for Indefinite Integrals (Theorem 5.6) (Section 5.5)
Substitution Rule (Change of Variables) for Definite Integrals (Theorem 5.7) (Section 5.5)
Integrals of Exponential Functions (handout)