217
Spring Math 2005
Exam 1 Review
In addition to working old homework problems, quizzes, and end-of-chapter review exercises, you might want to do the following problems as you review for the exam. The last page of this review sheet is the same as the last page of the exam. Solutions to these problems will be available on the course web site on Sunday February 20.
1. Find each of the following sums.
16
(a)
k=2
1 (k+1)2
1 k2
10
(b)
k=1
1 ( 2 )k
3
2. Use the denition of the denite integral to evaluate
0
(x3 6x) dx.
3. Find each of the following indenite integrals. (a) (b) x dx 1 x2 3 t2 t3 + 1 dt (c) (d) tan3 x sec2 x dx 3 t5 t3 + 1 dt
4. Evaluate each of the following denite integrals.
3 4
(a)
4
sec2 d
(c)
4
4
tan x dx cos x + 1 cos x dx 1 sin x
4
(b)
1
1 x2
1+
1 dx x
(d)
0
Math 217 Review 5. In each of the following, sketch the region R and nd its area. (a) R is the region enclosed by y = x + 1, y = 9 x2 , x = 1, and x = 2. (b) R is the region enclosed by y 2 = x and x 2y = 3.
2
6. Find the volume of the solid described in each of the following. (a) The region bounded by y = x2 and x = y 2 is rotated about the x-axis. (b) The base is the region bounded by y = x2 and y = 1, and the cross sections perpendicular to the y-axis are squares.
cos x
7. Let f (x) =
0
1 + sin t dt.
(a) Find f ( ). 2 (b) Find f ( ). 2 8. A ball is thrown upward with a speed of 48 ft/s from the edge of a cli 432 ft above the ground. When it will hit the ground? 9. A particle moves in a straight line and has acceleration given by a(t) = 6t + 4. Its initial velocity is v(0) = 6 cm/s and its initial displacement is s(0) = 9 cm. Find its position function s(t). 10. A particle is moving along a line with velocity v(t) = t2 2t 8 (t 0) (a) What is the total distance traveled by the particle in the rst 6 seconds? (b) What is the displacement of the particle after 6 seconds?
11. How much work is done by a weightlifter in raising a 60-kg barbell from the oor to a height of 2 m? 12. Find the length of the arc of the curve y =
x3 3
+
1 4x
from x = 1 to x = 3.
13. Let f0 (x) = x2 and fn+1 (x) = f0 (fn (x)). Find a formula for fn (x) and use mathematical induction to prove it.
Math 217 Review
3
More Formulas Than You Will Ever Need
This page will be the last page of the exam. You do not need to memorize anything on ...