Breakout Session 13 Solutions

Transcription

1 Problem True or False: If f = 2, then f = 2 False Any time that you have a function of raise to a function of, in orer to compute the erivative you nee to use logarithmic ifferentiation or something equivalent Correct erivative of f: P Outcome: Recognize the ifference between a base an a eponent f = 2 = f = e ln 2 = f = e ln 2 ln = f = 2 ln If f =, then f = ln False Same as part Correct erivative of f: f = = f = e ln = f = e ln ln + = f = ln + Problem 2 Fin the erivatives of the following functions: a f = e + 7 f = problem is to fin e e + 7 = e + 7 So the real Recognize the ifference between a base an a eponent Ientify situations where logs can be use to help fin erivatives

2 e = e ln e = e e ln = e e ln Thus, f = e e ln + e + 7 e ln + e = e e ln + e b g = ln + 9 sec4 g = ln + 9 sec4 = e sec4 lnln +9 = e sec4 lnln +9 4 sec 4 tan 4 lnln sec 4 ln + 9 = ln + 9 sec4 4 sec 4 tan 4 lnln sec4 ln + 9 c h = cos Rewrite h using properties of logarithms: ln h = ln cos = 5 ln lncos 2 5 Derivative of h: ln h = h h = cos 2 5 sin = sin2 5 cos 2 5 = tan2 5 0 = h = h = h = cos tan tan2 5 2 Take erivatives of functions raise to functions Take erivatives of logarithms an eponents of all bases

3 Problem A table of values for f an f is shown below Suppose that f is a one-to-one function an f is its inverse I Evaluate f f at = f f P Outcome: Fin erivatives of inverse functions in general P Outcome: Unerstan how the erivative of an inverse function relates to the original erivative b is the correct answer: a b c 6 4 e DNE f None of the previous answers II Evaluate ff at = f f = f 4 is correct answer: [ ] ff = f f f = ff = f f f = = f 4 5 = 5 = 5 a 6 b 25 c 5 5 e DNE f None of the previous answers III Evaluate lnf at = =

4 c is the correct answer: a /4 b 5 c 5/4 /5 e DNE lnf = f f = f None of the previous answers IV Evaluate f at = [ ] lnf = f = f = 5 4 b is the correct answer: a 4 b c / 5 e DNE f None of the previous answers V Evaluate f at = f = = f is the correct answer: f = f f = [ f = f = 4 ] = = f f a b 4 c /5 /4 4

5 e 5 f None of the previous answers Problem 4 Fin the erivatives of the following functions: a f = sec f = 2 2 = 2 P4 Outcome: Take erivatives which involve inverse trig functions b g = lnsin g = c h = tan sin 2 h = tan Problem 5 Fin the erivative of f at the following points without solving for f a f = 2 + for 0 at the point 5, 2 f 5 = f 2 Since f = 2, f 2 = 4 Thus, f 5 = 4 b f = 2 2 for at the point 2, f 2 = 2 = 8 Thus, f 2 = 8 f Since f = 2 2, f = 6 5

6 Etra Problems for Personal Practice Eplain what each of the following means: Problem 6 a sin This enotes the inverse function to sin, sometimes enote by arcsin P6 Outcome: Recall the meaning an properties of inverse trig functions b sin c sin This means sin raise to the power, ie This means sin sin f e f This enotes the inverse function of f This means f f f This means f raise to the power, ie f Problem 7 Suppose that f is a ifferentiable function which is one-to-one Given the table of values below, fin the value of f 7 7 f 7 f f 7 = f f 7 Since f = 7, f 7 = Thus f 7 = f = 6 P7 Outcome: Fin erivatives of inverse functions in general P7 Outcome: Unerstan how the erivative of an inverse function relates to the original erivative 6

7 Problem 8 Fin the slope of the tangent line to the curve y = f at 4, 7 if the slope of the tangent line to the curve y = f at 7, 4 is 2 Note that the statement the slope of the tangent line to the curve y = f at 7, 4 is 2 specifically means that f 7 = 2 The slope of the tangent line to the curve y = f at 4, 7 is f 4, an so we use the formula for the erivative of the inverse function to compute: f 4 = f 7 = 2 = 2 P8 Outcome: Fin erivatives of inverse functions in general P8 Outcome: Unerstan how the erivative of an inverse function relates to the original erivative 7