Chap B-210915-Flecture notes 4 class 23562 PDF

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Marketing activity is bound by
legislation, regulation, and professional
guidelines all meant to protect
consumers and ensure fairer
competition in the marketplace. Marketing activity is bound by
legislation, regulation, and professional
guidelines all meant to pr...

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Chapter B from “Sandbox Examples for Intermediate Consumer and Producer Theory” by Peter A. Streufert1

Prerequisite Mathematics 15. This document’s title refers to a “sandbox” of examples. In real life, a sandbox is a place for a child to play with a small collection of safe toys. Similarly, this document is a place for you to learn economics with a small collection of mathematically simple examples. More precisely, the examples use a small portion of Grade 12 calculus. The first three sections of this chapter review this material, and the final section describes how this material will be used to construct the examples in the “sandbox”. B.1. Derivative rules for exponential, logarithm, and power functions 16. The theorems in steps 18–21, 28–29, and 36 are from the Ontario Grade 12 mathematics curriculum. In particular, these theorems are taught in MCV4U, which is the Ontario high school course called “Calculus and Vectors”. They are also taught in all university calculus classes, such as Western’s Mathematics 0110 and Calculus 1100 and Calculus 1500. You are presumed to know this material, and its presentation here is meant to serve only as a review and a reminder. 17. To be clear, a counting number is one of 1, 2, 3, and so on. An integer is a counting number, or the negative of a counting number, or zero. A real number is number from the real number line, which includes all the integers, but does not include the special numbers −∞ and ∞ discussed in step 6. High-school calculus works with integers and real numbers. Accordingly, you will not see the special numbers −∞ and ∞ in this chapter. 17B. Economics frequently uses logarithms and exponents. Accordingly, please review the following identities. They hold for any positive real numbers x and y, and for any real numbers a and b. (a) ln(xy) = ln(x)+ ln(y). (b) ln(xa ) = a ln(x). (c) xa+b = xa xb . (d) xab = (xa )b . (e) eln(x) = x. (f) ln(ea ) = a. (The positivity of x and y avoids nonsensical expressions.) 18. Power Rule for Positive Integer Exponents. Suppose b is a positive integer. Then for all (real) x, (d/dx) xb = bxb−1 . 19. Power Rule for Nonzero Exponents. Suppose b is a nonzero (real) number. Then for all positive (real) x, (d/dx) xb = bxb−1 . 20. Rule for Exponential Function. For all (real) x, (d/dx) ex = ex 1 This chapter, and the accompanying OWL questions, have been prepared for Economics 2150 at Western University during the 2021-22 academic year. Outside of their distribution to the teachers and students in this course at this time, the author retains all rights to this document and the OWL questions (  c Peter A. Streufert 2021). Plans for future distribution are pending. The author is very grateful for the talented and enthusiastic assistance of Cecilia Diaz Campo and Ali Kamranzadeh. (This version: September 15, 2021.)

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21. Rule for Logarithm Function. For all positive (real) x, (d/dx) ln(x) = 1/x. 22. Exercise. Please fill in the blanks. f (x) f (1) f ′ (1) f (2) f ′ (2) x−2 x−1 x−1/2 x1/2 x x2 ex ln(x) 23. Exercise. Notice that Rules 19 and 21 assume that x is positive. Economics does not work with negative numbers as often as positive numbers. For both reasons, there are fewer rows in the following table than there were in the corresponding table in Exercise 22. Please fill in the blanks. f (x) x x2 ex

f (−1) f ′ (−1) f (0) f ′ (0)

24. Exercise. Figure 24A has six graphs.2 For each of the following functions, determine which of these six graphs is most like the graph of the function. Since the figure has six graphs and the table has eight functions, some of the graphs will have to be associated with more than one function. f (x) graph number x−2 x−1 x−1/2 x1/2 x x2 ex ln(x) 25. Exercise.OWL Please fill in the blanks. f (x) f (1) f ′ (1) f (2) f ′ (2) x−1.5 x0.6 x1 x−0.2 x−2.2 x0.2

2Graph 1 is called a hyperbola, and Graph 4 is called a parabola. These special kinds of curves will appear again in several contexts.

B. Prerequisite Mathematics

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Figure 24A. The graphs of six functions f (x). In each case, the argument x is on the horizontal axis, and the value y = f (x) is on the vertical axis. 26. Exercise.OWL For each of the following functions, determine which graph in Figure 24A is most like the graph of the function. As in Exercise 24, some of the graphs may have to be associated with more than one function. f (x) x−1.5 x0.6 x1 −0.2 x x−2.2 x0.2

graph number

B.2. Derivative rules for constant terms and constant multipliers 27. As you know, the derivative of a function f is the function f ′ defined by f (x) = (d/dx) f (x).3 Often this is called the first derivative. The second derivative of the function f is defined by f ′′ (x) = (d/dx) f ′ (x). The third and still higherorder derivatives are defined similarly. A function f is said to be smooth if and only if you can find its first derivative f ′ , its second derivative f ′′ , and all higher-order derivatives. All the examples and exercises in this document are smooth. ′

28. Constant-Term Rule. Suppose f is a smooth function and c is a (real) number. Then (d/dx) (f (x)+c) = f ′ (x). 29. Constant-Multiplier Rule. Suppose f is a smooth function and a is a (real) number. Then (d/dx) (af (x)) = af ′ (x). 30. Exercise. The derivative of f (x) = 6 ln(x) + 2 is (1) f ′ (x) = 6 ln(x). (2) f (x) = 6ex . (3) f ′ (x) = 6/x. (4) f ′ (x) = 6/x + 2. (5) f ′ (x) = 6/x + 2x. (6) None of the above. ′

3For the mathematically sophisticated, the domain of f ′ is taken to be the interior of the domain of f .

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31. Exercise. The derivative of f (x) = −(1/x) is (1) f ′ (x) = 1/x2 . (2) f (x) = −1. (3) f ′ (x) = −1/x2 . (4) f ′ (x) = 2x−1 . (5) f ′ (x) = x−1 . (6) None of the above. ′

32. Exercise.OWL Please fill in the blanks. f (x) 2x 3 − 4 2x1/2 + 6 5 + 2x−1/2 (2.7)ex + 6.3 (1.3) ln(x)

f (1) f ′ (1) f (3) f ′ (3)

33. Exercise. Together, Figures 24A and 33A display 12 graphs. For each of the following functions, determine which graph is most like the graph of the function. A graph can be associated with zero, one, or many functions. Note that the function in Exercise 31 appears again here. f (x) −x−2 −1/x 2x−1/2 3x1/2 5x −7x −x2 −5ex 6 ln(x)

graph number

Figure 33A. The graphs of six functions f (x). In each case, the argument x is on the horizontal axis, and the value y = f (x) is on the vertical axis.

B. Prerequisite Mathematics

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34. Exercise.OWL Please fill in the blanks. f (x) 4x0.5 − 4 −2x1 + 6 5 − 3x−0.4 12 + .5x−1.5 −4x0.9 + 3.6 −0.5x−0.9 − 6.2

f (1) f ′ (1) f (3) f ′ (3)

35. Exercise.OWL Together, Figures 24A and 33A display 12 graphs. For each of the following functions, determine which graph is most like the graph of the function. A graph can be associated with more than one function. Note that the functions are like those in Exercise 34. f (x) 4x0.5 −2x1 −3x−0.4 0.5x−1.5 −4x0.9 −0.5x−0.9

graph number

B.3. The chain rule 36. Chain Rule. Suppose that f and g are smooth functions, and that each value of f is an argument of g. Then (d/dx) g(f (x)) = g ′ (f (x))f ′ (x). 37. Exercise. (a) The derivative of f (x) = (x−2)2 is [1] f ′ (x) = 2x−2. [2] f ′ (x) = 2x−4. [3] f ′ (x) = 4x+4. [4] f ′ (x) = −2(x−2). [5] None of the above. (b) The derivative of f (x) = −(x−2)2 is [1] f ′ (x) = −2x−2. [2] f ′ (x) = −2x−4. [3] f ′ (x) = −4x−4. [4] f ′ (x) = 2(x−2). [5] None of the above. 38. Exercise.OWL Please fill in the blanks f (x)

f (1) f ′ (1) f (2) f ′ (2) f (3) f ′ (3) 2

(x−2) −(x−2)2 (x−3)2 −(x−3)2 39. Exercise.OWL Together, Figures 24A, 33A, and 39A display 16 graphs. For each of the following functions, determine which graph is most like the graph of the function. As usual, a graph can be associated with more than one function. Note

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that some of the functions in Exercises 37 and 38 appear again here.

f (x) 3x2 (x−2)2 (x+1)2 −2x2 −(x−2)2 −3(x+5)2 −3ex −5(x−4)2

graph number

Figure 39A. The graphs of four functions f (x). In each case, the argument x is on the horizontal axis, and the value y = f (x) is on the vertical axis.

B. Prerequisite Mathematics

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40. The questions in Exercises 41 and 42 can be mechanically solved by the chain rule. The functions in the exercises are meant for negative x and have the form f (x) = (−x)b for a negative b. The graphs of these functions are like the graph in Figure 40A. These functions will not appear until Chapter H, and you need not study them carefully now. Their awkward combination of negative arguments, negative coefficients, and negative exponents make them much less intuitive than the other functions in this chapter.

Figure 40A. The graph of the function f (x) = −x−1 . The argument x is on the horizontal axis, and the value y = f (x) is on the vertical axis.

41. Exercise. (a) What is the derivative of (−x)−1 at x = −3? (b) What is the derivative of (−x)−1 at x = −2? (c) What is the derivative of (−x)−1 at x = −1? 42. Exercise. (a) The derivative of f (x) = (−x)−2 is [1] f ′ (x) = −2x−3 . [2] f ′ (x) = 2x−1 . [3] f ′ (x) = 3x−2 . [4] f ′ (x) = 2x−3 . [5] None of the above. (b) The derivative of f (x) = 6(−x)−1/2 + 6 is [1] f ′ (x) = −3x−3/2 . [2] f ′ (x) = −3(−x)−3/2 . [3] f ′ (x) = 3x1/2 . [4] f ′ (x) = −3x1/2 . [5] None of the above. B.4. The functions in the sandbox 43. This document is limited to a “sandbox”, that is, to a limited class of examples. All these examples can be manipulated with the mathematical tools that were reviewed in this chapter. To be more precise, all the examples in the sandbox will be defined by three functions called G, H , and f . Further, these three functions can take only certain forms. In particular, each of the functions can be (a) a power function of the form xb for a nonzero b (here x will be positive), (b) the logarithm function ln(x) (here x will be positive), (c) a quadratic function of the form (x−c)2 for a positive c (here x will be any number), (d) the exponential function ex (here x will be any number), (e) a power function of the form (−x)b for a negative b (here x will be negative), or (f) any of the preceding multiplied by a constant factor and/or summed with a constant term. Thus all the functions in the sandbox are smooth functions, and all the derivative rules in this chapter apply. (If (c) and (e) are unfamiliar, please see steps 38–42.)

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B.A. Selected Answers for Exercises For Exercise 22. f (x) x−2 x−1 x−1/2

f (1) 1 1 1

f ′ (1) −2 −1 −0.5

f (2) 0.25 0.5 0.707

f ′ (2) −0.25 −0.25 −0.177

x1/2 x x2 ex ln(x)

1 1 1 2.718 0

0.5 1 2 2.718 1

1.414 2 4 7.389 0.693

0.354 1 4 7.389 0.5

f (x) x x2 ex

f (−1) −1 1 0.368

For Exercise 23. f ′ (−1) 1 −2 0.368

f (0) 0 0 1

f ′ (0) 1 0 1

For Exercise 24. f (x) x−2

graph number 1

−1

1 1

1/2

2 3 4 5 6

x x−1/2 x

x x2 ex ln(x)

For Exercise 25. f (x) x−1.5

f (1) 1

f ′ (1) −1.5

f (2) 0.354

f ′ (2) −0.265

x0.6 x1 x−0.2

1 1 1

0.6 1 −0.2

1.516 2 0.871

0.455 1 −0.087

x−2.2 x0.2

1 1

−2.2 0.2

0.218 1.149

−0.239 0.115

For Exercise 26. f (x) x−1.5 x0.6 x1 x−0.2 x−2.2 x0.2

For Exercise 30.

3

For Exercise 31.

1

graph number 1 2 3 1 1 2

For Exercise 32. f (x) 2x3 − 4 1/2 2x +6

f (1) −2 8

f ′ (1) 6 1

f (3) 50 9.464

f ′ (3) 54 0.577

5 + 2x−1/2 (2.7)ex + 6.3 (1.3) ln(x)

7 13.639 0

−1 7.339 1.3

6.155 60.531 1.428

−0.192 54.231 0.433

B. Prerequisite Mathematics

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For Exercise 33. f (x) −x−2 −1/x

graph number 7 7

2x−1/2

1

3x1/2 5x −7x −x2 −5ex 6 ln(x)

2 3 9 10 11 6

For Exercise 34. f (x) 4x0.5 − 4 −2x1 + 6 5 − 3x−0.4 12 + .5x−1.5 −4x0.9 + 3.6

f (1) 0 4 2 12.5 −0.4

f ′ (1) 2 −2 1.2 −0.75 −3.6

f (3) 2.928 0 3.067 12.096 −7.152

f ′ (3) 1.155 −2 0.258 −0.048 −3.225

−0.5x−0.9 − 6.2

−6.7

0.45

−6.386

0.056

For Exercise 35.

For Exercise 37. (a) 2

f (x) 4x0.5 −2x1 −3x−0.4 0.5x−1.5 −4x0.9

graph number 2 9 7 1 8

−0.5x−0.9

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(b) 5

For Exercise 38. f (x)

f (1)

f ′ (1)

f (2)

f ′ (2)

f (3)

f ′ (3)

(x−2)2 −(x−2)2 (x−3)2

1 −1 4

−2 2 −4

0 0 1

0 0 −2

1 −1 0

2 −2 0

−(x−3)2

−4

4

−1

2

0

0

For Exercise 39.

For Exercise 41. (a) 0.111 For Exercise 42. (a) 1

f (x) 3x2 (x−2)2

graph number 4 14

(x+1)2 −2x2 −(x−2)2 −3(x+5)2 −3ex −5(x−4)2

13 10 16 15 11 16

(b) 0.25 (c)

(b) 5

1...