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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)