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5.3 Inverse Function Review for Algebra Two: Definition of Inverse Function A function g is the inverse function of the function f if f g x x for each x in the domain of g and g f x x for each x in the domain of f. The function g is denoted by f 1 (read “f inverse”) Example: f x x 3 , g x is its inverse function, which is g x f 1 x x 3 Find the inverse functions. Verify them by the definition of inverse function I do: f x 3x 2 We do: f x x 3 You do: f x 2 x 3 1 We all do: f x 2 x 3 Use TI 84 to graph an inverse function http://www.tc3.edu/instruct/sbrown/ti83/drawinv.htm Continuity & Differentiability of Inverse Functions Let f be a function whose domain is an interval I. If f has an inverse function, then the following statement are true.. 1. If f is continuous on its domain, then f 1 is continuous on its domain. 2. If f is increasing on its domain, then f 1 is increasing on its domain. 3. If f is decreasing its domain, then f 1 is decreasing on its domain. 4. If f is differentiable on an interval containing c and f ' c 0 , then f 1 is differentiable at f (c). The derivative of an Inverse Function Let f be a function that is differentiable on an interval I. If f has an inverse function g, then g is differentiable at any x for which f ' g x 0 . 1 Moreover, g ' x , f ' g x 0 f ' g x I do: Let f x 1 3 x x 1 4 a) What is the value of f 1 b) What is the value of f x when x = 3? ' x when x = 3? 1 We do: Let f x x 3 1 a) What is the value of f 1 x when x = 26? b) What is the value of f 1 ' x when x =26? Let f x 5 2 x 3 c) What is the value of f 1 x when x = 7? d) What is the value of f 1 ' x when x = 7? Homework: Text P. 350 #73-to 79 odd 5.6 Inverse Trigonometric Functions: Differentiation Definitions of Inverse Trigonometric Functions Function Domain y = arcsin x iff sin y = x 1 x 1 y = arccos x iff cos y = x y = arctan x iff tan y = x 1 x 1 x y = arccot x iff cot y = x y = arcsec x iff sec y = x x Range y 2 2 0 y y 2 2 0 y x 1 0 y , y y = arccsc x iff csc y = x x 1 Evaluate each function. I do: 1 arcsin 2 2 y 2 We do: arccos 0 Use TI 84 to verify your answer. Properties of Inverse Trigonometric Functions If 1 x 1 and y , then sin arcsin x x and sin arcsin y y 2 2 Similar properties hold for the other inverse trigonometric functions. arctan 2 x 3 4 We do: arcsin 3x 1 2 2 , y0 You do: arctan 3 I do: You do: arctan 2x 5 1 Derivatives of Inverse Trigonometric Functions Let u be a differentiable function of x. d d arcsin u u' 2 arccos u u' 2 dx dx 1 u 1 u d d arctan u u ' 2 arc cot u u '2 dx dx 1 u 1 u d u' d u' arc sec u arc csc u dx dx u u 2 1 u u 2 1 I do: d arcsin 2 x dx We do: d arctan 3x dx You do: d arcsin dx x We all do: d arc sec e 2 x dx We all do: y arctan x x 1 x 2 Find y’ Homework: Text P. 379 #5 to 11 odd #39, #43 to 53 odd 5.7 Inverse Trigonometric Functions: Integration Since the derivatives of the six inverse trigonometric functions fall into three pairs, we only need to remember one from each pair for the integration. Let u be a differentiable function of x, and let a > 0, 1 dx arcsin x C 1 x2 1 1 x 2 dx arctan x C 1 x x 2 1dx arc sec x C 1 du arcsin a2 u2 1 1 u a 2 u 2 du a arctan a C u 1 1 du arc sec C u u2 a2 a a *Only need to remember the right hand columns. I do: 1 4 x2 dx You do: 1 x 4x 2 9 You do: 1 e 2x 1 dx dx u C a We do: 1 2 9x 2 dx We all do: x2 4 x 2 dx Completing the Square 2 2 b b x bx c x bx c x 2 2 2 2 2 I do: x 2 1 dx 4x 7 We do: Find the area of the region bounded by the graph of 1 f x 3x x 2 3 9 The x axis, the line x , and the line x 2 4 You do: 2 dx 0 x 2 2x 2 Review: P. 386 When we have a integral with fraction 2 b b c 2 2 Power Rule Log Rule Inverse Trig rules Can not find this integral using the techniques you have learned Homework: Text P. #3 to 13 odd #25 to #29 odd #41