1. 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).
= Ans. 1/2.
4. If f(x) = e2, then f(2x) = Ans. e2.
5. If ln x = 2 ln ln 2, then x = . Ans. 9/8. 

2. 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.

3. 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.

4. 16. The area bounded by |x| + |y| = 1 is . Ans. 2.
= 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

5. 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.
= Ans
= Ans. x3 - x0.
25. For y = 2x, y’ = Ans. 2x ln 2.

6. 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

7. 32. For y = (Ln x)2, y’ = Ans. (2 Ln x) / x.
= Ans 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).

8. True or False
36. The equation x x = 0 has a root between 0 and 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 = 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.

9. 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
Ans. 1

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

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

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