Presentation on theme: "Calculus Notes 7.1 Inverse Functions & 7.2 Exponential Functions and Their Derivatives Start up: 1.(AP question) A continuous f(x) has one local maximum."— Presentation transcript:
Calculus Notes 7.1 Inverse Functions & 7.2 Exponential Functions and Their Derivatives Start up: 1.(AP question) A continuous f(x) has one local maximum and two local minima. Which of the following is true about its inverse function, ? (A) has one local maximum and two local minima. (B) has two local maxima and one local minimum. (C) has three local extrema; there is not enough information to tell if they are maxima or minima. (D) has no local extrenum. (E) does not exist. 2.(AP question) If your wealth was dollars after years, after how many years would you become a billionaire? Answers: (1)E (2).
Calculus Notes 7.1 Inverse Functions Definition: A function f is called a one-to-one function if it never takes on the same value twice; that is, Horizontal Line Test: a function is one-to-one if and only if no horizontal line intersects its graph more than once. Definition: Let f be a one-to-one function with domain A and range B. Then its inverse function has domain B and range A and is defined by for any y in B. How to Find the Inverse Function of a One-to-one Function f: Step 1: Write y=f(x) Step 2: Solve for x in terms of y (if possible) Step 3: interchange x and y and write The graph of is obtained by reflecting the graph of f about the line y=x.
Calculus Notes 7.1 Inverse Functions Theorem: If f is a one-to-one continuous function defined on an interval, then its inverse function is also continuous. Theorem: If f is a one-to-one differentiable function with inverse function then the inverse function is differentiable at a and Calculus Notes 7.2 Exponential Functions and Their Derivatives Theorem: If a>0 and a≠1, then is a continuous function with domain and range. In particular, for all x. If 0 1, f is an increasing function. If a,b>0 and then
Calculus Notes 7.2 Exponential Functions and Their Derivatives Definition: e is the number such that Derivative of the Natural Exponential Function: Properties of the Natural Exponential Function: The exponential function is an increasing continuous function with domain and range. Thus, for all x. Also. So the x-axis is a horizontal asymptote of.
Calculus Notes 7.1 Inverse Functions Example 1: Given (a)Is f one-to-one? What is the Domain and Range of f and its inverse? (b)Find its inverse. (c)Graph the function, its inverse, and y=x. Example 2: Given find
Calculus Notes 7.2 Exponential Functions and Their Derivatives Example 3: Starting with the graph of find the equations of the graph that results from: (a) Reflecting about the line y=4.(b) Reflecting about the line x=2. Example 4: Find the domain of each function. (a). (b).
Calculus Notes 7.2 Exponential Functions and Their Derivatives Example 6: Differentiate the function Example 5: Find the limit of Letas soBy 11
Calculus Notes 7.2 Exponential Functions and Their Derivatives Example 7: The flash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the flash is set off. The following data describe the charge Q remaining on the capacitor (measured in microcoulombs, ) at time t (measured in seconds). t 0.000.020.040.060.080.10 Q 100.0081.8767.0354.8844.9336.76 (a)Use a graphing calculator to find an exponential model for the charge. (b)The derivative Q’(t) represents the electric current (measured in microamperes, ) flowing from the capacitor to the flash bulb. Use part (a) to estimate the current when t=0.04s. Compare with the result of Example 2 in Section 2.1 (-670 ) Close to the estimate from chapter 2. 7.1 pg.420 #2, 3, 6, 9, 12, 20, 21, 22, 23, 26, 27, 32, 33, 42, 44 (15) 7.2 pg.431 #3, 4, 7, 8, 11, 13, 15, 17, 24, 29, 31, 32, 38, 45, 48, 60, 75, 78,81 (19)