Sheet Review For the Third Midterm
Warnings. These problems are not intended to be completely and utterly representative of the questions you will see on the test. The problems you will see on the exam will not be exactly like the problems you see here. This review sheet is not intended to replace general studying for the exam it is instead meant to supplement whatever studying you might do otherwise. Being able to do all these problems well is not a guarantee that you will do well on the exam (but it does suggest that you understand the material well). Attempt all the problems first before looking at the answers (which are also available from the website). If you have to explain yourself, use COMPLETE ENGLISH SENTENCES. 1. Evaluate the following indefinite integrals. Don't forget the constant of integration (+C)! Assume that k is a real number. In some cases you will need to use substitution it is up to you to determine for which problems this is necessary. (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) t13 + 4.8t7 dt 5t2 - 3t + 2 dt xk dx k x dx e2x+5 dx (2x + 5)e dx 1 3 2 - 2 + -t dt t t e 3t2 + 1 dt 4 3 t +t ee dt x- 1 dx (Hint: split this into two integrals before doing substitution) x-1
2. Evaluate the following definite integrals. Give an exact value for your answer, not a decimal approximation. Simplify your answer as much as possible.
2
(a)
1 1
t3 dt x6 + 3x5 dx
-1 1000
(b) (c)
1
1 dt t2 1
(d) (e)
6x + 5 dx 3x2 + 5x + 1 0 -2 2 dx x -3
1
3. Find the area of the region bounded above by y = x3 - 3 and below by y = -4 - 2x from x = 0 to x = 2. Give an exact value. 4. Suppose that a certain company's marginal revenue (in dollars per item) for producing q items is given q2 . by the formula R (q) = ln q + 120000 (a) Find a decimal approximation for R (100), rounded to the nearest cent.
200
(b) Explain in words what the quantity
150 q2
R (q) dq represents.
(c) Suppose that R(q1 ) = 13, 456 and R(q2 ) = 19, 221 for some specific real numbers q1 and q2 . Determine the exact value of
q1
R (q) dq.
5. A company's total cost for producing q widgets is given by C(q) = 0.01q 3 - 0.1q 2 + 5q + 2000. (a) Find an expression representing the fixed cost for producing q widgets. (b) Find an expression representing the variable cost for producing q widgets. (c) Find the average cost for producing 100 widgets. (d) Is the average cost of producing 100 widgets greater than or less than the marginal cost at q = 100? (e) Is the graph of y = C(q) concave up for all q 0? Explain.
100
(f) Find the exact value of
0
C (q) dq.
6. the Suppose power consumption in Seattle on the morning of Friday, February 27th, 2009, is given by the function P (t) where t is measured in the number of hours after midnight, and P (t) is measured in megawatts. The table below gives measurements of P (t) for various values of t. For example, note that P (2.5) = 399: that means at 2:30 AM, 399 megawatts are being used. Time (hr) 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 Power (mW) 475 445 407 403 400 399 398 401 409 423 450 466 493
6
(a) Which of the following are the proper units for
0
P (t) dt: (you may have to just use a process
of elimination here) i. ii. iii. iv. v. megawatts/hour megawatts hours/megawatt miles/hour megawatt-hours
6
(b) Find a good estimate for What is t? What is n?
0
P (t) dt, the total energy used in Seattle from midnight to 6 AM.
2
7. Look at the graph for problem 4 on page 109 of your textbook. If that is the graph of y = f (x), do
3 b
you think
-3
f (x) dx is positive or negative? Explain. Find values for a and b for which
a
f (x) dx
will definitely be negative.
8. Suppose the instantaneous rate of change of Sayo, a small but growing girl, in inches per month, is given by f (t), where t is the number of months elapsed since she turned 10 years old. When she turned 10, she was 49 inches tall.
12
(a) What does
0
f (t) dt represent? Use appropriate units.
12
(b) Is it true that Sayo's height at t = 12 months is given by 49 +
0
f (t) dt?
(c) Suppose that f (t) = 0.05e0.16t. Find Sayo's height at t = 12. (Round to the nearest...