Final Exam Review Fall19 PDF

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MATH 120 - FINAL EXAM REVIEW

FINAL EXAM: SATURDAY DEC 14th, 12-2 pm LT 161 and PG 153 Must bring a scantron ParSCORE 289L and a PHOTO ID This is a sample of questions to review for the final exam. I encourage you to review lecture notes for a complete understanding of the concepts learned. 1. Find the derivative of the following functions: 2 1 a. g (x )  3x  5x  9  x

c. h( x) 

e.

3x x 3

b. k( x)  4 x  1

9

d. r ( x )  x 4 e 5x

7

2

f ( x)  2ln(6 x  5)

2. At what point on the curve y  3x 2  2 x  6 is the slope equal to -22?

3 3. Find the equation of the tangent line to the function y  2 x  4 x  1 at x = 2.

4. Find the area under the curve y  x 2  5 from x  2 to x  3 . Draw a picture and shade the area you are finding.

5. Find all maximums and minimums (if any) for the following function. Use the first derivative test. f ( x)  2 x 3  9 x 2  60 x  10

6. Find the intervals where the function is increasing or decreasing: f ( x) 

8x x  7x 2

4 5x 7. Find the absolute extrema of f ( x)  x e on the interval [0, 20].

8. Integrate: a.

 x  5 dx

b.

 1 

  2x  dx

15

c.



x x 2  1  dx

d.

7x e 

e.

x 2  3x  1 1 x dx

f.

 (x

e

2

3 dx x

x dx 3  5)

9. A projectile is launched vertically upward from the ground. Its height s(t) in feet at the end of t seconds is given by the function s (t )  360t  16t² . a. Find the time at which it reaches its maximum height. b. What is the maximum height?

3

10. Find the derivative of the function: f ( x )  x  x

11. Use the properties of logs to rewrite the expression and then find the derivative. f ( x)  3ln 3

e3 x x 5

12. The total cost (in dollars) of producing x tennis rackets per day is C( x)  800  60 x  0.25x 2 for 0  x  120 . Find the marginal cost at a production level of 60 rackets, and interpret the results

13. The Park and recreational department is planning to build a picnic area for motorists along a quiet dirt road. It has 2000 yards of material to enclose a rectangular area with no fence along the road. What is the maximum area that can be enclosed?

14. Find the values of t for which the function k( t)  3t 4  2t 3  12t 2  18t  15 is concave up and concave down. Use the second derivative test.

15. The cost of producing x widgets where 0  x  100 is C ( x)  5280  340 x  0.25 x² . The revenue from the sales is R( x)  500 x  20 x². Find the marginal profit when x = 40.

3

2

16. For the function F ( x)  x  6 x  9 x  1 Identify all the relative extrema and points of inflection.

17. Find the following limits given: x 2 1 if  f ( x)   x if x 2 1 if 

a. lim f (x )  x2

x 0 0 x  2 x 2

b. lim f (x )  x 2

c. lim f (x )  x2

18. Find any vertical asymptotes, horizontal asymptotes, and any holes for the graph of: 3x 2  2 x  1  f ( x) x2 1

x2  9 x  8 x1 x2 1

19. Find the following limit: lim

1

20. A manufacturer estimates marginal profit to be P ( x)  100x 2  0.4 x . Suppose the manufacturer’s profit is $520 when the level of production is 16 units. What is the manufacturer’s profit when the level of production is 25 units?

21.

When a theater owner charges $5 for admission, there is an average attendance of 180 people. For every $0.10 increase in admission, , there is a loss of one customer from the average number. What admission should be charged to maximize revenue?

22. What is the definition of the derivative and what does it represent?

ANSWERS: 1 x2 b) k( x)  36(4 x  1) 8 7 18 x  9 c) h( x)  7 2 ( x  3) x x d) r( x)  5x4 e5  4 x3 e5 24 x e) f ( x )  2 (6x  5)

1. a) g ( x)  6 x  5

2. 3. 4. 5. 6.

(4, -34) y  20 x  33 110/3 units² rel. min (5, -265) rel. max (-2, 78) Decreasing: ( ,7)  (7, ) 7. Abs min (0, 0) Abs Max (20, 160,000e 100 ) 1 2 8. a) x  5x  C 2 b) 12 ln x  C





16 1 2 x 1  C 32 1 d) e 7 x  3ln x  C 7 1 2 7 e) e  3e  2 2 1 f)  C 4( x 2 5) 2

c)

9. a) 11.25 sec. b) 2025 ft  12 3 10. f ( x)  x  2 x  12  1 11. f ( x )  3  x5 12. C( x)  60  0.5 x C(60)  30 the cost is increasing $30 for the 61st racket 13. 500,000 sq yards 2  14. Concave up:  ,    1,   Concave down: (-2/3, 1) 3   15. $1780 16. rel. max (1, 3) rel. min (3, -1) pt of inf. (2, 1) 17. a) 3 b) 2 c) DNE 18. vertical: x = 1 horizontal: y = 3 Hole at x = -1 19. -7/2 20. $646.20

21. $11.50 f x  h   f x  The slope of the tangent lines to a curve. 22. lim h 0 h...