Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our User Agreement and Privacy Policy.

Slideshare uses cookies to improve functionality and performance, and to provide you with relevant advertising. If you continue browsing the site, you agree to the use of cookies on this website. See our Privacy Policy and User Agreement for details.

Like this presentation? Why not share!

15,000 views

Published on

Inverse functions in general, and the inverses of very important exponential functions

No Downloads

Total views

15,000

On SlideShare

0

From Embeds

0

Number of Embeds

119

Shares

0

Downloads

124

Comments

0

Likes

1

No embeds

No notes for slide

- 1. Section 3.2 Inverse Functions and Logarithms V63.0121, Calculus I March 4/9/10, 2009 . . 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 = y − 1, so 3 √ 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 √ x =⇒ y2 = x y= dy =⇒ 2y =1 dx dy 1 1 =√ =⇒ = dx 2y 2x d2 y , and y is the inverse of the squaring function. Notice 2y = 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 yxy 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 . . . . . .

No public clipboards found for this slide

×
### Save the most important slides with Clipping

Clipping is a handy way to collect and organize the most important slides from a presentation. You can keep your great finds in clipboards organized around topics.

Be the first to comment