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. 

1 1 3

0 1[ ]x t dt dx

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.

 

10. If = 5, then a = . Ans. 9/2.

2

2( 1)xt dt

22adx

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.

14. 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

 

19. 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

1

1edtt

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.

100

1

2( 1/ 2)i

i

3

11

( )i ii

x x

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).

2

0

1xdx

e

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

 

1

02xdx

46. Write Taylor’s and Maclaurin’s Series

f(x) =

Maclauin’s Series is written about c = 0.

( )

0

( )( )

!

kk

k

f cx c

k

47. Define e, the base of the natural logarithms.

e = lim (1 + x)1/x x 0

2 3

12! 3! !

nx x x xe x

n

Limits

lim 1 – cos x

x 0 x2

lim ex – e-x

x 0 sin x

lim sin x

x 0 cx