Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)
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This document summarizes sections 3.1-3.2 of a Calculus I course at New York University on exponential and logarithmic functions taught on October 20, 2010. It outlines definitions and properties of exponential functions, introduces the special number e and natural exponential function, and defines logarithmic functions. Announcements are made that the midterm exam is nearly graded and a WebAssign assignment is due the following week.
Lesson 13: Exponential and Logarithmic Functions (Section 041 handout)
- 1. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Notes Sections 3.1–3.2 Exponential and Logarithmic Functions V63.0121.041, Calculus I New York University October 20, 2010 Announcements Midterm is nearly graded. Should get it back in recitation. There is WebAssign due Monday/Tuesday next week. Announcements Notes Midterm is nearly graded. Should get it back in recitation. There is WebAssign due Monday/Tuesday next week. V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 2 / 33 Objectives for Sections 3.1 and 3.2 Notes Know the definition of an exponential function Know the properties of exponential functions Understand and apply the laws of logarithms, including the change of base formula. V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 3 / 33 1
- 2. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Outline Notes Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 4 / 33 Derivation of exponential functions Notes Definition If a is a real number and n is a positive whole number, then an = a · a · · · · · a n factors Examples 23 = 2 · 2 · 2 = 8 34 = 3 · 3 · 3 · 3 = 81 (−1)5 = (−1)(−1)(−1)(−1)(−1) = −1 V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 5 / 33 Fact Notes If a is a real number, then ax+y = ax ay ax ax−y = y a (ax )y = axy (ab)x = ax b x whenever all exponents are positive whole numbers. Proof. Check for yourself: ax+y = a · a · · · · · a = a · a · · · · · a · a · a · · · · · a = ax ay x + y factors x factors y factors V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 6 / 33 2
- 3. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Let’s be conventional Notes The desire that these properties remain true gives us conventions for ax when x is not a positive whole number. For example: ! an = an+0 = an a0 Definition If a = 0, we define a0 = 1. Notice 00 remains undefined (as a limit form, it’s indeterminate). V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 7 / 33 Conventions for negative exponents Notes If n ≥ 0, we want ! an · a−n = an+(−n) = a0 = 1 Definition 1 If n is a positive integer, we define a−n = . an Fact 1 The convention that a−n = “works” for negative n as well. an am If m and n are any integers, then am−n = n . a V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 8 / 33 Conventions for fractional exponents Notes If q is a positive integer, we want ! (a1/q )q = a1 = a Definition √ If q is a positive integer, we define a1/q = q a. We must have a ≥ 0 if q is even. √ q √ p Notice that ap = q a . So we can unambiguously say ap/q = (ap )1/q = (a1/q )p V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 9 / 33 3
- 4. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Conventions for irrational powers Notes So ax is well-defined if x is rational. What about irrational powers? Definition Let a > 0. Then ax = lim ar r →x r rational In other words, to approximate ax for irrational x, take r close to x but rational and compute ar . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 10 / 33 Graphs of various exponential functions y Notes y = ((21/2))xx (1/3)x y = /3 =y y = (1/10)xy = 10x= 3xy = 2x y y = 1.5x y = 1x x V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 11 / 33 Outline Notes Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 12 / 33 4
- 5. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Properties of exponential Functions Notes Theorem x If a > 0 and a = 1, then f (x) = a is a continuous function with domain R and range (0, ∞). In particular, ax > 0 for all x. If a, b > 0 and x, y ∈ R, then ax+y = ax ay ax ax−y = y negative exponents mean reciprocals. a xy (ax )y = a fractional exponents mean roots (ab)x = ax b x Proof. This is true for positive integer exponents by natural definition Our conventional definitions make these true for rational exponents Our limit definition make these for irrational exponents, too V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 13 / 33 Simplifying exponential expressions Notes Example Simplify: 82/3 Solution √ 3 √ 82/3 = 82 = 64 = 4 3 √ 2 8 = 22 = 4. 3 Or, Example √ 8 Simplify: 21/2 Answer 2 V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 14 / 33 Limits of exponential functions Notes Fact (Limits of exponential y y = (= 2(1/(2/3)x y = (y/10)10x3x 2xy = 1.5x y 1/ = 3)x y )x 1 = xy = y= functions) If a > 1, then lim ax = ∞ x→∞ and lim ax = 0 x→−∞ If 0 < a < 1, then lim ax = 0 and y = 1x x→∞ lim ax = ∞ x x→−∞ V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 15 / 33 5
- 6. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Outline Notes Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 16 / 33 Compounded Interest Notes Question Suppose you save $100 at 10% annual interest, with interest compounded once a year. How much do you have After one year? After two years? after t years? Answer V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 17 / 33 Compounded Interest: quarterly Notes Question Suppose you save $100 at 10% annual interest, with interest compounded four times a year. How much do you have After one year? After two years? after t years? Answer V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 18 / 33 6
- 7. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Compounded Interest: monthly Notes Question Suppose you save $100 at 10% annual interest, with interest compounded twelve times a year. How much do you have after t years? Answer V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 19 / 33 Compounded Interest: general Notes Question Suppose you save P at interest rate r , with interest compounded n times a year. How much do you have after t years? Answer V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 20 / 33 Compounded Interest: continuous Notes Question Suppose you save P at interest rate r , with interest compounded every instant. How much do you have after t years? Answer rnt r nt 1 B(t) = lim P 1 + = lim P 1 + n→∞ n n→∞ n n rt 1 =P lim 1+ n→∞ n independent of P, r , or t V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 21 / 33 7
- 8. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 The magic number Notes Definition n 1 e = lim 1+ n→∞ n So now continuously-compounded interest can be expressed as B(t) = Pe rt . V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 22 / 33 Existence of e See Appendix B Notes n 1 n 1+ We can experimentally verify n that this number exists and 1 2 is 2 2.25 3 2.37037 e ≈ 2.718281828459045 . . . 10 2.59374 100 2.70481 e is irrational 1000 2.71692 e is transcendental 106 2.71828 V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 23 / 33 Meet the Mathematician: Leonhard Euler Notes Born in Switzerland, lived in Prussia (Germany) and Russia Eyesight trouble all his life, blind from 1766 onward Hundreds of contributions to calculus, number theory, graph theory, fluid mechanics, optics, and astronomy Leonhard Paul Euler Swiss, 1707–1783 V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 24 / 33 8
- 9. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 A limit Notes Question eh −1 What is lim ? h→0 h Answer If h is small enough, e ≈ (1 + h)1/h . So eh − 1 ≈1 h eh − 1 In fact, lim = 1. h→0 h 2h − 1 This can be used to characterize e: lim = 0.693 · · · < 1 and h→0 h 3h − 1 lim = 1.099 · · · > 1 h→0 h V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 25 / 33 Outline Notes Definition of exponential functions Properties of exponential Functions The number e and the natural exponential function Compound Interest The number e A limit Logarithmic Functions V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 26 / 33 Logarithms Notes Definition The base a logarithm loga x is the inverse of the function ax y = loga x ⇐⇒ x = ay The natural logarithm ln x is the inverse of e x . So y = ln x ⇐⇒ x = e y . Facts (i) loga (x · x ) = loga x + loga x x (ii) loga = loga x − loga x x (iii) loga (x r ) = r loga x V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 27 / 33 9
- 10. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Logarithms convert products to sums Notes Suppose y = loga x and y = loga x Then x = ay and x = ay So xx = ay ay = ay +y Therefore loga (xx ) = y + y = loga x + loga x V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 28 / 33 Example Notes Write as a single logarithm: 2 ln 4 − ln 3. Solution 42 2 ln 4 − ln 3 = ln 42 − ln 3 = ln 3 ln 42 not ! ln 3 Example 3 Write as a single logarithm: ln + 4 ln 2 4 Answer ln 12 V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 29 / 33 Graphs of logarithmic functions Notes y x xx x y = yy = 3e = 2 10=y y = log2 x y = ln x y = log3 x (0, 1) y = log10 x (1, 0) x V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 31 / 33 10
- 11. V63.0121.041, Calculus I Sections 3.1–3.2 : Exponential Functions October 20, 2010 Change of base formula for exponentials Notes Fact If a > 0 and a = 1, then ln x loga x = ln a Proof. If y = loga x, then x = ay So ln x = ln(ay ) = y ln a Therefore ln x y = loga x = ln a V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 32 / 33 Summary Notes Exponentials turn sums into products Logarithms turn products into sums V63.0121.041, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 20, 2010 33 / 33 Notes 11