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Definitions and elementary properties of exponential and logarithmic functions.

Definitions and elementary properties of exponential and logarithmic functions.

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- 1. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 Sections 3.1–3.2 Notes Exponential and Logarithmic Functions V63.0121.021, Calculus I New York University October 21, 2010 Announcements Midterm is graded and scores are on blackboard. Should get it back in recitation. There is WebAssign due Monday/Tuesday next week. Announcements Notes Midterm is graded and scores are on blackboard. Should get it back in recitation. There is WebAssign due Monday/Tuesday next week. V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 2 / 38 Midterm Statistics Notes Average: 78.77% Median: 80% Standard Deviation: 12.39% “good” is anything above average and “great” is anything more than one standard deviation above average. More than one SD below the mean is cause for concern. V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 3 / 38 1
- 2. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 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. V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 4 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 5 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 6 / 38 2
- 3. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 Anatomy of a power Notes Deﬁnition A power is an expression of the form ab . The number a is called the base. The number b is called the exponent. V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 7 / 38 Fact Notes If a is a real number, then x+y x y a = a a (sums to products) x−y ax a = y (diﬀerences to quotients) a (ax )y = axy (repeated exponentiation to multiplied powers) (ab)x = ax b x (power of product is product of powers) 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 8 / 38 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, what should a0 be? We cannot write down zero a’s and multiply them together. But we would want this to be true: ! ! an an = an+0 = an · a0 =⇒ a0 = =1 an (The equality with the exclamation point is what we want.) Deﬁnition If a = 0, we deﬁne a0 = 1. Notice 00 remains undeﬁned (as a limit form, it’s indeterminate). V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 9 / 38 3
- 4. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 Conventions for negative exponents Notes If n ≥ 0, we want ! ! a0 1 an+(−n) = an · a−n =⇒ a−n = = n an a 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 10 / 38 Conventions for fractional exponents Notes If q is a positive integer, we want ! ! √ (a1/q )q = a1 = a =⇒ a1/q = q 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 11 / 38 Conventions for irrational exponents Notes So ax is well-deﬁned if a is positive and 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 . V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 12 / 38 4
- 5. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 Approximating a power with an irrational exponent Notes r 2r 3 3 2 =8 √ 10 3.1 231/10 = √ 31 ≈ 8.57419 2 314/100 100 3.14 2 = √ 314 ≈ 8.81524 2 1000 3.141 23141/1000 = 23141 ≈ 8.82135 The limit (numerically approximated is) 2π ≈ 8.82498 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 13 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 14 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 15 / 38 5
- 6. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 Properties of exponential Functions Notes Theorem x If a > 0 and a = 1, then f (x) = a is a continuous function with domain (−∞, ∞) and range (0, ∞). In particular, ax > 0 for all x. For any real numbers x and y , and positive numbers a and b we have ax+y = ax ay ax ax−y = y (negative exponents mean reciprocals) x y a xy (a ) = 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 16 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 17 / 38 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→−∞ V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 18 / 38 6
- 7. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 19 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 20 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 21 / 38 7
- 8. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 22 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 23 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 24 / 38 8
- 9. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 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 . V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 25 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 26 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 27 / 38 9
- 10. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 A limit Notes Question eh −1 What is lim ? h→0 h Answer e = lim → ∞ (1 + 1/n)n = lim (1 + h)1/h . So for a small h, e ≈ (1 + h)1/h . n h→0 So h eh −1 (1 + h)1/h −1 ≈ =1 h h eh − 1 It follows that 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 28 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 29 / 38 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 (x1 · x2 ) = loga x1 + loga x2 x1 (ii) loga = loga x1 − loga x2 x2 r (iii) loga (x ) = r loga x V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 30 / 38 10
- 11. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 Logarithms convert products to sums Notes Suppose y1 = loga x1 and y2 = loga x2 Then x1 = ay1 and x2 = ay2 So x1 x2 = ay1 ay2 = ay1 +y2 Therefore loga (x1 · x2 ) = loga x1 + loga x2 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 31 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 32 / 38 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 33 / 38 11
- 12. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 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 V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 34 / 38 Example of changing base Notes Example Find log2 8 by using log10 only. Solution log10 8 0.90309 log2 8 = ≈ =3 log10 2 0.30103 Surprised? No, log2 8 = log2 23 = 3 directly. V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 35 / 38 Upshot of changing base Notes The point of the change of base formula logb x 1 loga x = = · logb x = constant · logb x logb a logb a is that all the logarithmic functions are multiples of each other. So just pick one and call it your favorite. Engineers like the common logarithm log = log10 Computer scientists like the binary logarithm lg = log2 Mathematicians like natural logarithm ln = loge V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 36 / 38 12
- 13. V63.0121.021, Calculus I Sections 3.1–3.2 : Exponential Functions October 21, 2010 Summary Notes Exponentials turn sums into products Logarithms turn products into sums V63.0121.021, Calculus I (NYU) Sections 3.1–3.2 Exponential Functions October 21, 2010 38 / 38 Notes Notes 13

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