101, Math Spring 2008, Practice Exam 2 1) a) State the first part of the fundamental theorem of calculus. b) Use the fundamental theorem of calculus to find the derivative of the function x4 cos( t)dt g(x) = t3/2 + 7t 4 c) Use the fundamental theorem of calculus to find the derivative of the function
x
h(x) =
5
(t5000 e7t )dt
3
2)Find the following limits: a) limx ( x - 7 - x) x e -1 b) limx0 ex -e-x 7x c) limx1 x-1 - x21 -1 d) limx0 xsin x 2 5x4 e) limx -3x-3x +7 4 +7x-2 3) Show that the function f (x) = x3 satisfies the hypotheses of the mean value theorem on the interval [0, 4]. Find all numbers c in that interval that satisfy the conclusion of the mean value theorem. 4) Find the definite integral: 2 x sec2 dx a) 2 0 4 dx b) 2 1 x 4 4dx c) x 2
ln 4
d)
0
3e 2 dx
x
5) Consider the function f (x) = x2 + x + 1 on the interval I = [0, 5]. a) Using a partition of I into 5 regular subintervals, use a right-endpoint sum to compute the Riemann sum for f (x). b) Without completing part c, is this an overestimate or an underestimate of the integral?
5
c) Using the Riemann sum definition of the integral, find
0
f (x)dx.
1
dy 6) a) Find dx when y is defined implicitly by the equation 2y x2 = + sin y. b) Find the first two derivatives of y with respect to x when y is defined implicitly by the equation 2x3 - 3y 2 = 7.
7) Let f (x) = x4 + 2x3 . a) Find and classify all critical points of the graph of y = f (x). b) Where is f (x) increasing and decreasing? c) Sketch a graph of f (x). Label local extrema and inflection points. Be sure to include behavior "at infinity" and any asymptotes. 8) x2 - 3 2x - 4 2(x - 3)(x - 1) (2x - 4)2 8 (2x - 4)3
f (x) f (x) f (x)
= = =
Use this information to sketch a graph of x. Label local extrema and inflection points. Be sure to include behavior "at infinity" and any asymptotes. 9) A 13-foot ladder is leaning against a house when its base starts to...