Example 1a ~ Like # 1-39 f(x) = A. The domain is {x x 2 1 0} = {x x 1} DOM: (, 1) ( 1, 1) (1, ) B. The x- and y-intercepts are both 0. C. Since f( x) = f(x), the function f is even. The curve is symmetric about the y-axis. D. For H.A. : We note =2 Thus: H.A. is y = 2 For V.A. : We note f(x) D.N.E. when denom. = 0, when x = 1, So we compute the following limits: Thus: V.A. are x = 1 and x = -1 1
Example 1b ~ Like # 1-39 f(x) = E. Intervals of INCR/DECR: F. Local Max & Min. : C.P. When f ' (x) = 0...When x = 0 When f '(x) D.N.E....When x = 1, -1-1 0 1 Test Values : -2-1/2 1/2 2 Sign of f '(x) : POS POS NEG NEG Nature of f(x) : INCR INCR DECR DECR f(x) must have: MAX @ x =0 2
Example 1c ~ Like # 1-39 f(x) = G. Concavity & P.I. C.P. When f '' (x) = 0...Never Happens When f ''(x) D.N.E....When x = 1, -1-1 1 Test Values : -2 0 2 Sign of f '' (x) : POS NEG POS Nature of f(x) : C.U. C.D. C.U. f(x) must have: No Points of Inflection, since f(x) D.N.E. @ x = ± 1 H. Using the information in E G, we sketch f(x): 3
Example 2a ~ Like # 1-39 A. The domain is {x x + 1 > 0} = {x x > - 1} DOM: ( 1, ) B. The x- and y-intercepts are both 0. C. Symmetry: None D. For H.A. : We note Thus: No H.A. For V.A. : We note f(x) D.N.E. when denom. = 0, when x = -1, So we compute the following limit: Thus: V.A. is x = -1 4
Example 2b ~ Like # 1-39 E. Intervals of INCR/DECR: F. Local Max & Min. : C.P. When f ' (x) = 0...When x = 0, -4/3 When f '(x) D.N.E....When x = -1 ** But the only # in the DOM f(x) is x = 0 ** [ -1 0 Test Values : -1/2 1 Sign of f '(x) : NEG POS Nature of f(x) : DECR INCR f(x) must have: MIN @ x =0 5
Example 2c ~ Like # 1-39 G. Concavity & P.I. C.P. When f '' (x) = 0...Never Happens (By Analyzing the Quadratic) When f ''(x) D.N.E....Never Happens [-1 Test Values : 0 Sign of f '' (x) : POS Nature of f(x) : C.U. f(x) must have: No Points of Inflection, since f(x) C.U. on entire DOM H. Using the information in E G, we sketch f(x): 6
Example 3a ~ Like # 1-39 A. The domain is DOM: (, ) B. The y-intercept is ( 0, 1/2). The x-intercepts are when cosx = 0: C. Symmetry: None Periodicity: With a period of 2π, we only need to consider the graph on [0, 2π]...And then extend the pattern D. No H.A. and No V.A. 7
Example 3b ~ Like # 1-39 E. Intervals of INCR/DECR: F. Local Max & Min. : C.P. When f ' (x) = 0...When sinx = -1/2 On [0, 2π]: x = 7π/6, 11π/6 When f '(x) D.N.E....Never Happens [0 7π/6 11π/6 2π] Test Values : π/2 3π/2 23π/24 Sign of f '(x) : NEG POS NEG Nature of f(x) : DECR INCR DECR f(x) must have: MIN MAX @ x =7π/6 @ x =11π/6 Note: 8
Example 3c ~ Like # 1-39 G. Concavity & P.I. C.P. When f '' (x) = 0...When cosx = 0 (x = π/2, 3π/2)...When sinx = 1 (x = π/2 ) When f ''(x) D.N.E....Never Happens [0 π/2 3π/2 2π] Test Values : π/4 π 7π/4 Sign of f '' (x) : NEG POS NEG Nature of f(x) : C.D. C.U. C.D. f(x) has : P.I. @ P.I. @ x = π/2 x = 3π/2 H. Using the information in E G, we sketch f(x) on [0, 2π]: Periodicity gives us: 9
Slant Asymptotes Some curves have asymptotes that are oblique (i.e. neither horizontal nor vertical. ) If where m 0, then the line y = mx + b is called a slant asymptote NOTE: The vertical distance between the curve y = f(x) and the line y = mx + b approaches 0 (A similar situation exists if we let x.) May 7-10:42 AM 10
Example 4a ~ Like # 1-39 A. The domain is DOM: (, ) B. The x- and y-intercepts are both 0 C. Since f( x) = - f(x), the function f is odd. The curve is symmetric about the origin. D. No V.A. (Notice the denominator x 2 +1 is never = 0 ) No H.A. (Notice f(x) as x and f(x) as x ) For S.A. (Slant Asymptote), we do some Long Division: Let's see what happens as x ± By our Definition of S.A. since Lim [ f(x) - x ] = We have S.A. y = x Lim [ ] = 0, 11
Example 4b ~ Like # 1-39 E. Intervals of INCR/DECR: F. Local Max & Min. : C.P. When f ' (x) = 0...When x = 0 When f '(x) D.N.E....Never Happens 0 Test Values : -1 1 Sign of f '(x) : POS POS Nature of f(x) : INCR INCR f(x) must have: No Loc. Min. or Max. No Abs. Min. or Max. 12
Example 4c ~ Like # 1-39 G. Concavity & P.I. C.P. When f '' (x) = 0...x = 0, ± 3 When f ''(x) D.N.E....Never Happens - 3 0 3 Test Values : -2-1 1 2 Sign of f '' (x) : POS NEG POS NEG Nature of f(x) : C.U. C.D. C.U. C.D. f(x) has : P.I. @ P.I. @ P.I. @ x = - 3 x = 0 x = 3 H. Using the information in E G, we sketch f(x) on : C P.I.... A: B: C: (0, 0), A B S.A. y = x 13
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