Lesson 15: Inverse Functions and Logarithms
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This document covers the concepts of inverse functions and logarithms, including their definitions, properties, and how to find inverse functions. It details the inverse function theorem, which states conditions under which an inverse function can be differentiable, and explains the relationship between logarithms and exponentials. Additionally, it provides examples of logarithmic operations and the change of base formula.
Lesson 15: Inverse Functions and Logarithms
- 1. Section 3.2 Inverse Functions and Logarithms V63.0121.027, Calculus I October 22, 2009 Announcements Quiz on §§2.5–2.6 next week Midterm course evaluations at the end of class . . Image credit: Roger Smith . . . . . .
- 2. Outline Inverse Functions Derivatives of Inverse Functions Logarithmic Functions . . . . . .
- 3. What is an inverse function? Definition Let f be a function with domain D and range E. The inverse of f is the function f−1 defined by: f−1 (b) = a, where a is chosen so that f(a) = b. . . . . . .
- 4. What is an inverse function? Definition Let f be a function with domain D and range E. The inverse of f is the function f−1 defined by: f−1 (b) = a, where a is chosen so that f(a) = b. So f−1 (f(x)) = x, f(f−1 (x)) = x . . . . . .
- 5. What functions are invertible? In order for f−1 to be a function, there must be only one a in D corresponding to each b in E. Such a function is called one-to-one The graph of such a function passes the horizontal line test: any horizontal line intersects the graph in exactly one point if at all. If f is continuous, then f−1 is continuous. . . . . . .
- 6. Graphing an inverse function The graph of f−1 interchanges the x and y f . coordinate of every point on the graph of f . . . . . . .
- 7. Graphing an inverse function The graph of f−1 interchanges the x and y f . coordinate of every point on the graph of f .−1 f The result is that to get the graph of f−1 , we . need only reflect the graph of f in the diagonal line y = x. . . . . . .
- 8. How to find the inverse function 1. Write y = f(x) 2. Solve for x in terms of y 3. To express f−1 as a function of x, interchange x and y . . . . . .
- 9. How to find the inverse function 1. Write y = f(x) 2. Solve for x in terms of y 3. To express f−1 as a function of x, interchange x and y Example Find the inverse function of f(x) = x3 + 1. . . . . . .
- 10. How to find the inverse function 1. Write y = f(x) 2. Solve for x in terms of y 3. To express f−1 as a function of x, interchange x and y Example Find the inverse function of f(x) = x3 + 1. Answer √ y = x3 + 1 =⇒ x = 3 y − 1, so √ f−1 (x) = 3 x−1 . . . . . .
- 11. Outline Inverse Functions Derivatives of Inverse Functions Logarithmic Functions . . . . . .
- 12. derivative of square root √ dy Recall that if y = x, we can find by implicit differentiation: dx √ y= x =⇒ y2 = x dy =⇒ 2y =1 dx dy 1 1 =⇒ = = √ dx 2y 2 x d 2 Notice 2y = y , and y is the inverse of the squaring function. dy . . . . . .
- 13. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open interval containing b = f(a), and 1 (f−1 )′ (b) = ′ −1 f (f (b)) . . . . . .
- 14. Theorem (The Inverse Function Theorem) Let f be differentiable at a, and f′ (a) ̸= 0. Then f−1 is defined in an open interval containing b = f(a), and 1 (f−1 )′ (b) = ′ −1 f (f (b)) “Proof”. If y = f−1 (x), then f (y ) = x , So by implicit differentiation dy dy 1 1 f′ (y) = 1 =⇒ = ′ = ′ −1 dx dx f (y) f (f (x)) . . . . . .
- 15. Outline Inverse Functions Derivatives of Inverse Functions Logarithmic Functions . . . . . .
- 16. Logarithms 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 ex . So y = ln x ⇐⇒ x = ey . . . . . . .
- 17. Logarithms 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 ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ . . . . . .
- 18. Logarithms 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 ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ (x) (ii) loga ′ = loga x − loga x′ x . . . . . .
- 19. Logarithms 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 ex . So y = ln x ⇐⇒ x = ey . Facts (i) loga (x · x′ ) = loga x + loga x′ (x) (ii) loga ′ = loga x − loga x′ x (iii) loga (xr ) = r loga x . . . . . .
- 20. Logarithms convert products to sums 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′ . . . . . .
- 21. Example Write as a single logarithm: 2 ln 4 − ln 3. . . . . . .
- 22. Example 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 . . . . . .
- 23. Example 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 . . . . . .
- 24. Example 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 . . . . . .
- 25. “ . . lawn” . . Image credit: Selva . . . . . .
- 26. Graphs of logarithmic functions y . . = 2x y y . = log2 x . . 0 , 1) ( ..1, 0) . ( x . . . . . . .
- 27. Graphs of logarithmic functions y . . = 3x= 2x y . y y . = log2 x y . = log3 x . . 0 , 1) ( ..1, 0) . ( x . . . . . . .
- 28. Graphs of logarithmic functions y . . = .10x 3x= 2x y y=. y y . = log2 x y . = log3 x . . 0 , 1) ( y . = log10 x ..1, 0) . ( x . . . . . . .
- 29. Graphs of logarithmic functions y . . = .10=3xx 2x y xy y y. = .e = y . = log2 x y . = ln x y . = log3 x . . 0 , 1) ( y . = log10 x ..1, 0) . ( x . . . . . . .
- 30. Change of base formula for exponentials Fact If a > 0 and a ̸= 1, then ln x loga x = ln a . . . . . .
- 31. Change of base formula for exponentials 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 . . . . . .