By Richard. Abbatt

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Intercept y = 1. Local maximums at x = (2k+1)π for integers k. Local minimums at 2 x = kπ for integers k. Vertical asymptotes at x = (2k+1)π for integers k. 5 -2 -3 kπ 4 22. Intercept at y = 0 and x = for integers k. No extrema. Vertical asymptotes at x = (2k+1)π for integers k 8 3 25. Intercept y = 4 (no x-intercepts). No extrema. Left horizontal asymptote y = 0. 2 y 1 10 0 -2 -1 0 1 8 2 x -1 6 -2 y 4 -3 2 0 -3 23. Intercept at y = 2 and from the amplitude/phase shift form f (x) = √ √ 5 sin x + sin−1 (2/ 5) , we could write down all the intercepts only at considerable√inconvenience.

F (0) = 4 ⇒ a = 4. Then f (2) = 2 gives 4e2b = 2, so 2b = ln 12 and 1 1 b = 1 ln 1 . So f (x) = 4e( 2 ln 2 )x . 2 2 50. f (0) = 5 ⇒ a = 5. Then f (1) = 2 gives 5eb = 2, so and b = ln 25 . 2 So f (x) = 5e(ln 5 )x . ex + e−x 51. We know that cosh x = . To show 2 that cosh x ≥ 1 for all x is the same as showing that cosh x − 1 ≥ 0 for all x. So we ask when ex + e−x is the expression cosh x − 1 = −1 2 greater than or equal to 0? We have: ex + e−x − 1 ≥ 0 if and only if 2 x −x e +e −2 ≥ 0 if and only if 2 x −x e + e − 2 ≥ 0 if and only if ex + 1 − 2e−x ≥ 0 if and only if * 28 CHAPTER 0.

Local minimum at x = −1. No asymptotes. 5 x -2 10 -1 0 1 2 0 -5 5 -10 y -6 -4 0 -2 0 2 4 6 x -5 -10 16. 53, and √ y = 1. Local maximum at x = − 2. Local √ minimum at x = 2. No asymptotes. 19. Intercept at y = 0 and at x = 0. No extrema. Horizontal asymptote y = 4. Vertical asymptote x = −2. 20 15 10 80 y 5 40 0 -10 0 -4 -2 0 2 4 -5 0 5 10 x -5 x -40 -10 -80 17. Intercepts at x = −1 and 1, and y = 1. Local minimum at x = 1 and at x = −1. Local maximum at x = 0. No asymptotes. 20. Intercept at y = 1.

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A treatise on the calculus of variations by Richard. Abbatt


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