121616949-math.117 - x 1 p 5 x 2 35 12 y = x 5-x 13 y = 6 x...

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5.5 Asymptotes and Other Things to Look For 103 If there are any points where the derivative fails to exist (a cusp or corner), then we should take special note of what the function does at such a point. Finally, it is worthwhile to notice any symmetry. A function f ( x ) that has the same value for - x as for x , i.e., f ( - x ) = f ( x ), is called an “even function.” Its graph is symmetric with respect to the y -axis. Some examples of even functions are: x n when n is an even number, cos x , and sin 2 x . On the other hand, a function that satisfies the property f ( - x ) = - f ( x ) is called an “odd function.” Its graph is symmetric with respect to the origin. Some examples of odd functions are: x n when n is an odd number, sin x , and tan x . Of course, most functions are neither even nor odd, and do not have any particular symmetry. Exercises Sketch the curves. Identify clearly any interesting features, including local maximum and mini- mum points, inflection points, asymptotes, and intercepts. 1. y = x 5 - 5 x 4 + 5 x 3 2. y = x 3 - 3 x 2 - 9 x + 5 3. y = ( x - 1) 2 ( x + 3) 2 / 3 4. x 2 + x 2 y 2 = a 2 y 2 , a > 0. 5. y = xe x 6. y = ( e x + e - x ) / 2 7. y = e - x cos x 8. y = e x - sin x 9. y = e x /x 10. y = 4 x + 1
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Unformatted text preview: x + 1) / p 5 x 2 + 35 12. y = x 5-x 13. y = 6 x + sin 3 x 14. y = x + 1 /x 15. y = x 2 + 1 /x 16. y = ( x + 5) 1 / 4 17. y = tan 2 x 18. y = cos 2 x-sin 2 x 19. y = sin 3 x 20. y = x ( x 2 + 1) 21. y = x 3 + 6 x 2 + 9 x 22. y = x/ ( x 2-9) 23. y = x 2 / ( x 2 + 9 24. y = 2 √ x-x 25. y = 3 sin( x )-sin 3 ( x ), for x ∈ [0 , 2 π ] 26. y = ( x-1) / ( x 2 ) For each of the following five functions, identify any vertical and horizontal asymptotes, and identify intervals on which the function is concave up and increasing; concave up and decreasing; concave down and increasing; concave down and decreasing. 27. f ( θ ) = sec( θ ) 28. f ( x ) = 1 / (1 + x 2 ) 29. f ( x ) = ( x-3) / (2 x-2) 30. f ( x ) = 1 / (1-x 2 ) 31. f ( x ) = 1 + 1 / ( x 2 ) 32. Let f ( x ) = 1 / ( x 2-a 2 ), where a ≥ 0. Find any vertical and horizontal asymptotes and the intervals upon which the given function is concave up and increasing; concave up and...
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