ENM 500 Mental Calculus Review    1. If f(x) = 2x2 + x + 1, then f(0) + 2 f(1) = . Ans. 9.   2. The solution set for x + 1 > x - 1 is . Ans. {x: x  R). 3. = Ans. 1/2. 4. If f(x) = e2, then f(2x) = . Ans. e2. 5. If ln x = 2 ln 3 - 3 ln 2, then x = . Ans. 9/8. 

6. For f(x) = x2 + 1, = Ans. 17.   7. d/dx ( ) = ______. Ans. x2 + 1. 8. If e2x = 4, then x = . Ans. ln 2. 9. The slope of the curve y = |x| at x =  is . Ans. 1. If = 5, then a = . Ans. 9/2.

11. The rate of change of the surface area S of a cube with respect to x the length of a side is . Ans. S = 6x2 => dS/dx = 12x. 12. (f + g2)’ (x) = when f(x) = x2 - 2x and g(x) = x. Ans. 4x - 2. 13. If f ‘(x) = ex and f(0) = 4, then f(1) = Ans. e + 3. If d(5x2 + x3) = f(x)dx, then f(2) = Ans. 32. 15. The area of the region bounded by y = (sqrt x), the x-axis from x = 0 to x = 1 is ____. Ans. 2/3.  

16. The area bounded by |x| + |y| = 1 is . Ans. 2.   17. = Ans. 1. 18. lim (1 + x)1/x = Ans. e. x -> 0 If f(1) = 1 and f(x) = f(x - 1) + 2 for all x, then f(3) = Ans. 5.  20. The inverse function g of f(x) = 1/x is Ans. 1/x

21. If x = Ln xy, then y’ = Ans. y’ = ex(x - 1)/x2.   22. If z = 2y2 + 2y + 7 and y = 2x2 - 6, then at x = 2, dz/dx = Ans. 80. 23. = Ans. 104. 24. = Ans. x3 - x0. 25. For y = 2x, y’ = . Ans. 2x ln 2.

26. The slope of the curve y = x ln x at x = e is Ans. 2. 27. d(uv) = Ans. u dv + v du. 28. d(u/v) = Ans. (v du - u dv)/ v2. 29. the function f(x) = 3x2 - 12x + 5 is increasing over the interval . Ans. (2, ).   30. If f(u) = u3 and g(x) = 5x2 - x, then f[g(x)] = . Ans. (5x2 - x)3. 31. The rate of change of the area of a circle with respect to its diameter when the circumference of the circle is 5 is Ans. 5/2

32. For y = (Ln x)2, y’ = Ans. (2 Ln x) / x.   33. = Ans. 1 - e-2. 34. The only real root of e5x + e4x - 6e3x = 0 is . Ans. Ln 2. 35. Why do tan2 x and sec2x have the same derivative? Ans. They differ by a constant (1).

True or False 36. The equation x99 - 39x88 + 33 = 0 has a root between 0 and 1. Ans. T.  37. Ln (1/x) = - Ln x. Ans. T.  38. The equation x = Ln x has no solution. Ans. T.  39. Ln (a + b) = Ln a + Ln b. Ans. F.  40. y = e2x is a solution to y” - 3y’ + 2y = 0. Ans. T.  41.. The function y = x4 + 1 is increasing for x < 0. Ans. F.  42. A horizontal asymptote of y = (x2 - 4)/(x2 + 3x) is y = 1. Ans. T.  43. The function f(x) = ln x has a relative minimum at x = e. Ans. F.  44. Limits are never indeterminate. Ans. T.

Integration 45. Find the area under the curve y = x2 bounded by the x-axis for x on [0,1]. Ans. 1/3 1/3.   1 46. Ans. 1

Write Taylor’s and Maclaurin’s Series f(x) = Maclauin’s Series is written about c = 0.

Define e, the base of the natural logarithms. e = lim (1 + x)1/x x  0

Limits lim 1 – cos x lim ex – e-x x  0 sin x lim sin x x  0 cx