2. Laws of Exponents (1) b 0 1 (2) b x b y b x y (3) bx b y. b x y (4) b n (5) b r s b rs (6) n b b 1/n Example: Solve the equations (a) e 2x

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1 7.1 Derivative of Exponential Function 1. Exponential Functions Let b 0 and b 1. f x b x is called an exponential function. Domain of f x :, Range of f x : 0, Example: lim x e x x2 2. Laws of Exponents (1) b 0 1 (2) b x b y b x y (3) bx b y b x y (4) b n b 1 n (5) b r s b rs (6) n b b 1/n Example: Solve the equations (a) e 2x e e x2 (b) 3 x 1 3 x 1 1

2 3. Derivative of f x b x From the graph of f x, we know f x exists everywhere. f f x h f x x lim h h b lim x b h 1 h lim h b x h b x h b h x lim h b h 1 h Compute numerically lim h e h 1 1 h Compute numerically lim h b h 1 ln b h d dx ex e x d b x b x ln b dx d dx eg x e g x g x d b g x b g x ln b g x dx Example: Find the equation of the tangent line to y e t2 at 0,1 Example: Find f x where (a) f x tan te t 2

3 4. Integrals of b x : e x dx e x b x dx 1 ln b bx C Example: cos e 2x e 2x dx e x x dx Example: Find the area between y e x and y e x for 0 x 1. 3

4 7.2 Inverse Functions 1. Definition: Let D and R be domain and range of f x. g x is the inverse of f x if the domain and range of g x are R and D, respectively, and (1) f g x x and (2) g f x x. The inverse of f is denoted as f 1 x. Note that the graphs of f and f 1 are symmetric about the line y x. 4

5 2. Existence of inverse function f x is invertible if and only f is an 1-1 function. Graphically, any horizontal line intersects with y f x only once - the Horizontal Line Test Note that an 1-1 function is either strictly increasing or strictly decreasing Example: f x x 2n 1 are 1-1 for n 1,2,..., f x x 2n are 1-1 if x 0 or x 0 for n 1,2,... Example: f x sin x is 1-1 for x in,, f x cos x is 1-1 for x 2 2 in 0,. 5

6 3. Steps to compute f 1 x : (1) Let y f x (2) Solve x in teams of y. (3) Exchange x and y. Example: Let f x x 4 10, for x in 0,. Find f 1. Example: Let f x 2x x 3 1. Find (1) domain and range of f x ;(2) all asymptotes of y f x ; and (3) find f 1 (check its domain, range, and asymptotes). Example: Let f x x 3 9. Find (1) domain and range of f x ; and (2) find f 1 (check its domain, and range). Example: Let f x x 3 x 1. Show that f 1 exists. Can we find f 1 exactly? What are values of f 1 1, f 1 3, f 1 4? 6

7 y f x 3x So, f is increasing x Let f 1 1 a. Then f a 1. Graphically, f 1 1. a 1 Example: Let f x x 2 2x. Determine a domain on which f 1 exists and find a formula for f 1 in this restricted domain. y x -2 f : Domain: 1,, Range: 1, f 1 : Domain: 1,, Range: 1, (1) Let y x 2 2x. x 2 2x y 0 (2) x 1 1 y, x 1 x 1 1 y (3) f 1 x 1 1 x 7

8 3. Derivative of Inverse Function Derivation: Know that f f 1 x x d dx f f 1 x d dx x, f f 1 x d dx f 1 x 1 d dx f 1 x 1 f f 1 x d dx f 1 b 1 f f 1 b 1 f a Example: f x 4x 3 1. Compute (i) Compute f 1 x : (1) Let y 4x 3 1. (2) x y 1, x (3) f 1 x x 1 y 1 where f a b d dx f 1 x. f x 12x 2, d f 1 x 1 dx 12 1 x 1 2/3 4 8

9 Example: Let f x 4x 3 2x. Compute d dx f 1 x x 2. We cannot find f 1 x exactly. (i) Evaluate f 1 2. Find a such that f a 2. Observe that f So, f (ii) f x 12x 2 2. f (iii) d dx f 1 x x

10 7.3 Logarithms and Their Derivatives 1. Logarithmic functions: Let f x log b x. f x is the inverse function of g x b x. Hence, f g x log b b x x and g f x b log b x x. When b e, log b ln. Domain of f x log b x is 0, and range is,. lim x 0 log b x, lim x log b x log b x ln x ln b 2. Laws of Logarithms (1) log b 1 0 (b 0 1), log b b 1 (b 1 b) (2) log b xy log b x log b y (3) log x b y log b x log b y (4) log b x r r log b x 10

11 3. Derivative of log b x Derivation: Because e ln x x, d dx e ln x d dx x e ln x d ln x 1, d ln x 1 1 dx dx e ln x x. d dx log b x dx d ln x 1 ln b xln b Combining with the Chair Rule: d dx ln h x h x h x Logarithmic Differentiation: Example: y x 1 2x 3. Compute y by the Lorithmic Differentiation. xe sin x (1) ln y ln x 1 ln 2x 3 ln x sin x (2) y y x 1 2x 3 x cos x y y x 1 2x 3 x cos x x 1 2x xe sin x x 1 2x 3 x cos x 11

12 Example: y x x, y x cos x, y tan x x 12

13 4. Logarithm as an Integral: 1 x dx ln x C h x h x dx ln f x C Example: 1 xln x dx, ln ln x xln x dx, 0 1 x 3 2x 4 1 dx, 13

14 7.4 Exponential Growth and Decay Let y P 0 e kt. Observe that y P 0 ke kt ky. So, y P 0 e kt is the general solution of the differential equation y ky (the rate of change of the population is proportional to the population. The function y P 0 e kt is called a population function. When k 0, the population is increasing (a growth model) and when k 0, the population is decreasing (a decay model). k is called the growth/decay constant of the model. In this model, the parameters P 0 y 0 and k y t k can be obtained by given t 1, y 1 and t 2, y 2. Special models: (1) Doubling time: t ln2 k (k 0, y t 2P 0 ) (2) Half time: t ln2 k (k 0, y t 1 2 P 0) Example: Suppose we know that penicillin leaves a person s bloodstream at a rate proportional to the amount present. (a) Express this statement as a differential equation. 14

15 (b) Find the decay constant if 50 mg of penicillin remains in the bloodstream 7 hours after an initial injection of 450 mg. (c) At what time, the penillin was 200 mg? 15

16 7.7 L Hopital Rule Assume that both f and g are differentiable. Suppose that Then lim x a f a 0 (or ), g a 0 (or ) and g a 0. f x g x lim x a f x g x. Indeterminat Forms: 0, 0 0,1 Example: lim x /2 cos 2 x 1 sin x Example: lim x 0 xln x Example: lim x 0 sin x x Example: lim x 1 2 x x 16

17 7.8 Derivatives of Inverse Trigonometric Functions 17