I have math assignment ( calculus)
sajdah90
1730-s16-final_review.pdf
Math 1730 Final Exam Review
Chapter 1
1. Given the above function find:
a. lim x3
f (x) _______
b. lim x3
f (x) _______
c. lim x3
f (x) _______
d. f (3) _______
e. _______)(lim 2
xf x
f. _______)(lim 2
xf x
g. _______)(lim 2
xf x
h. _______)2( f
i. _______)(lim 6
xf x
j. _______)(lim 6
xf x
k. _______)(lim 6
xf x
l. _______)6( f
m. _______)(lim 8
xf x
n. _______)(lim 8
xf x
o. _______)(lim 8
xf x
p. _______)(lim
xf x
2. At what x‐values is the graph above discontinuous? 3. For the graph above, at what values of x does the derivative not exist? 4. Find each of the given limits:
a.
lim x2
3x3 7x 2 3 b. lim x3
x 2 9 x 3
c. lim x1
x 2 3x 4 x 2 1
5. Are each of the functions continuous at the given point or over the given interval?
a. f (x) 3x 5 for x 2 x 1 for x 2
at the point x = 2
b.
3for2
3for54 )(
2
x
xxx xf at the point x = ‐3
c. g(x) x 2 4x 2 over the interval (‐7, 7)
6. For f (x) 2 x 2 3x 1, a. find f '(x) by determining lim h0
f (x h) f (x) h
b. find f '(2) and f '(3) 7. Find the derivative of each of the following:
a. f (x) x3 4x2 7x 5 b. f (x) 4x2 3x1 c. 3
645 )(
23
x
xxx xf
d. f (x) 4x1/2 7x3/2 e. f (x) x 23 f. f (x) 5x 0.7 3x 0.2
8. Find an equation of the tangent line to the graph of f (x) x3 2x 2 5x at the point (2, 10). 9. The average price in dollars of a share of a stock can be estimated by p(x) 9 0.2x 0.1x2
where x is the number of years after the year 2000. Find: a. the rate of change of the share price with respect to the year, . b. The average share price in 2015. c. The rate of change of the average share price in 2015. 10. Use the product rule to find the derivative of f (x) (x 2 3x)(4x 7).
11. Use the quotient rule to find the derivative of f (x) x3 2 x 5 x2
12. Find the derivative of each of the following:
a. 42 )125(3)( xxxf b. xxxf 764)( 2 13. Find the second derivative of each of the following:
a. 927)( 24 xxxf b. 32 745)( xxf 14. Find )()4( xf given that xexf 2)(
Chapter 2 15. Find the relative extrema (maximums and minimums) of 10243)( 23 xxxxf .
16. Given 2 4 3 find:
a. The critical values b. The x values where any relative extrema occur c. The x values where any points of inflection occur d. Interval(s) in which the function is increasing e. Interval (s) in which the function is decreasing f. Interval (s) in which the function is concave up g. Interval (s) in which the function is concave down
17. Find the horizontal asymptotes (use limits) and vertical asymptotes for each of the following functions:
a. 9
65 )(
2
2
x
xx xf b.
1
2 )(
2
x
x xf
18. Find the following limits:
a. 2
2
4
153 lim
x
xx x
b. 5
42 lim
2
x
xx x
19. Find the absolute maximum and absolute minimum for the function
323912)( xxxxf over [0, 3]
20. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. (assume R(x) and C(x) are in dollars).
306.001.)(
2)( 2
xxxC
xxR
21. At a ticket price of $20, there will be 40,000 people attending a football game. For every $2 the ticket price is dropped, 10,000 more people will attend the game. What price per ticket should be charged in order to maximize revenue? How many people will attend at that price? 22. A company finds that its monthly revenue, in dollars, from the sale of x leather bags is
xxxxR 1304.0006.)( 23 and the cost is 10040)( xxC . Currently, the company is selling 32 bags per month.
a. Find the marginal profit function. b. Find the marginal profit when 32 bags are sold.
Chapter 3 23. Find the derivatives of the following functions: a. 11 b. xexf 57)( c. xexxf 33)( d. xxexf 2
3
2)(
e. 3ln f. )65ln()( 2 xxxf g. )3ln()( 2 xxxf h. 4
4
)( x
e xf
x
24. An initial deposit is made of $12,000 in an account paying 4% interest compounded continuously. a. How much will the account be worth in 6 years? b. How long will it take the account to double? 25. Find the present value of a payment of $2300 that will be received 3 years for now, assuming that interest is 2.7% compounded continuously. 26. Find the derivatives of the following functions:
a. xxf 4)( b. xxf 9log)(
c. 752 2
7)( xxxf d. )7(log5)( 43 xxxf .
e. xxf x 2log3)( f. )5log()( 23 xxxxf
Chapter 4 27. Find the following indefinite integrals:
a. 15 5 7 b.
c. 7 7 d. √ √ 28. Find f such that 6 8 4 and 1 10. 29. Evaluate the following definite integral: 4 14
30. A technology firm finds that the marginal profit, in dollars, from sale of x hard drives is given by
3 . A customer orders 50 hard drives and later increases the order to 60. Find the extra profit resulting from the increase in order size. 31. Find the area under the curve over the interval given: a. 81 ; [‐9, 9] b. ; [0, 3] (round answer to 3 decimal places) 32. Find the area of the regions bounded by the graphs of
2, , 1,and 2 33. Given the graphs of 2 and 3 6. a. At what x value(s) do the graphs intersect? b. Find the area of the region bounded by the two functions. 34. Find the average value of the function 2 10 over the interval [‐2, 6]. 35. Use substitution to find the following integrals:
. b. c. 1
Chapter 5
36. The price, in dollars per unit, that consumers are willing to pay for x units of an item, is given by 5 . The price in dollars per unit, that producers are willing to accept for x units of an
item is given by 2 13. Find: a. The coordinates of the equilibrium point. b. The Consumer Surplus. c. The Producer Surplus.
37. A man receives a continuous income stream of $520,000 per year. If the income stream is invested at 6% compounded continuously for 25 years find (round answers to the nearest dollar): a. The accumulated future value of the income stream. b. The accumulated present value of the income stream. 38. Determine whether each improper integral is convergent or divergent, and calculate its value if it is convergent. a. b. c. 3 39. Find the accumulated present value of an investment for which there is a perpetual continuous money flow of $3900 per year, assuming continuously compounded interest at a rate of 8%.
Solutions
1.
a. 2
b. ‐7
c. DNE
d. ‐4
e. ‐2
f. ‐2
g. ‐2
h. undefined
i. ‐9.5
j. ∞
k. DNE
l. ‐9.5
m. 1
n. 1
o. 1
p. 0
2. ‐3, 2, 6
3. ‐3, 0, 2, 4, 6
4. a. ‐1 b. 6 c. 5/2
5. a. no b. yes c. yes
6. 34 x , 11, ‐9
7. a. 783 2 xx b. 23 38 xx c. 3
1 6815 234 xxx or
3
16815 234
xxx
d. 2/12/1 2
21 2 xx e. 3/1
3
2 x or 33
2
x f. 8.03.0 6.05.3 xx
8. 89 xy
9. a. 0.2 0.2 b. $28.50 c. $2.80/year
10. )74)(32(4)3( 2 xxxx or 213812 2 xx
11. 22
322
)5(
)2)(2()23)(5(
x
xxxxx
or 22
24
)5(
1013
x
xx
12a. 12 5 12 2 5
12b. 2 6 7 12 7
13a. 84 4
13b. 1920 4 7 120 4 7
14. xe 216
15. Relative minimum at (2, ‐18); relative maximum at (‐4, 90)
16a. x = ‐2/3, x = 2
16b. Relative maximum at x = ‐2/3, relative minimum at x = 2
16c. point of inflection at x = 2/3
16d. ∞, , 2,∞
16e. ,2
16f. ,∞ 16g. ∞,
17a. horizontal: y = 1, vertical: x = 3
17b. horizontal: y = 0, vertical: x = 1, x = ‐1
18a. ‐3
18b. ∞
19. Absolute maximum of 17 an x = 1 Absolute minimum of – 15 at x = 3
20. x = 70, maximum profit is $19.00
21. $14; 70,000 people.
22a. 0.018 0.8 90
22b. $82.83
23a. 11
23b. 35
23c. 3 3
23d. 2 3 2
23e. 3
23f. 2 5 5 6
23g. 2 3
23h. 4 4
4 4
24a. $15,254.99
24b. approximately 17 years
25. $2121.05
26a. 4 ∙ 4
26b. 1 9
26c. 7 ∙ 7 4 5
26d. 5 4 7 3 7
26e. f x 3 ∙ ln3 ∙ 3 ∙ log x
26f. f x
27a. 5 5 2
7
27b. 4 7
27c. 7 2
7 3
27d. 3 5
2 3
28. 2 4 4 8
29. or 101
30. $222.41
31a. 972
31b. 201.214
32. 6
33a. ‐1,4
33b. 125/6
34. 70/3
35a. ln 2 5 C
35b.
35c. 6561/16
36a. (1, $16)
36b. $4.33
36c. $1.67
37a. $30,174,639
37b. $6,732,872
38a. converges ,1/8
38b. diverges
38c. converges, 1
39. $48,750
