Calculus problems

MAC 2311 Review for Test # 3

Name___________________________________

SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question.

Determine whether the function satisfies the hypotheses of the Mean Value Theorem for the given interval. 1) g(x) = x3/4, 0,3 Also, find all numbers c that satify the conclusion of the MVT. 1)

Identify the function's local and absolute extreme values, if any, saying where they occur. 2) f(x) = x3 + 3.5x2 + 2x - 1 2)

Find the extrema of the function on the given interval, and say where they occur. 3) sin x + cos x, 0 ≤ x ≤ 2π 3)

Find the absolute extreme values of the function on the interval.

4) F(x) = 3 x, -2 ≤ x ≤ 27 4)

5) g(x) = -x2 + 11x - 28, 4 ≤ x ≤ 7 5)

Find the absolute extreme values of the function on the interval.

6) f(x) = ln(x + 2) + 1 x , 1 ≤ x ≤ 10 6)

Determine all critical points for the function. 7) f(x) = x3 - 3x2 + 3 7)

Graph the equation. Include the coordinates of any local and absolute extreme points and inflection points. 8) y = 7x2 + 70x

x-10 -5 5 10

y

100

-100

x-10 -5 5 10

y

100

-100

8)

1

9) y = 8x x2 + 16

x

y

x

y

9)

Sketch the graph and show all local extrema and inflection points. 10) y = -x4 + 4x2 - 9

x-10 -5 5 10

y 10

5

-5

-10

x-10 -5 5 10

y 10

5

-5

-10

10)

Graph the rational function.

11) y = x + 4 x2 + 9x + 20

x-8 -4 4 8

y

8

4

-4

-8

x-8 -4 4 8

y

8

4

-4

-8

11)

2

12) y = x 2

x2 + 13

x-3 -2 -1 1 2 3

y

1.5

1

0.5

-0.5

x-3 -2 -1 1 2 3

y

1.5

1

0.5

-0.5

12)

Sketch the graph and show all local extrema and inflection points. 13) y = -x4 + 4x2 - 2

x-10 -5 5 10

y 10

5

-5

-10

x-10 -5 5 10

y 10

5

-5

-10

13)

Find dy. 14) y = ln(5 + x4) 14)

PLEASE MAKE SURE TO REVIEW THE PROBLEMS IN SECTION 3.11 ( UNIT 2) AND LOOK OVER THE GRAPHING PROBLEMS IS SECTION 4.4 PART 2 ( UNIT 3)

3

Answer Key Testname: MAC 2311 REVIEW FOR TEST # 3

1) Yes

2) local maximum at x = -2; local minimum at x = - 13

3) local maxima: 1 at x = 2π and 2 at x = π 4 ;

local minima: 1 at x = 0 and - 2 at x = 5π 4

4) absolute maximum is 3 at x = 27; absolute minimum is 0 at x =0

5) absolute maximum is 9 4

at x = 11 2 ; absolute minimum is 0 at 7 and 0 at x = 4

6) Minimum value is ln 4 + 1 2

at x = 2; maximum value is ln 12 + 1 10

at x = 10

7) x = 0 and x = 2 8) absolute minimum: (-5,-175) no inflection points

x-10 -5 5 10

y

100

-100

x-10 -5 5 10

y

100

-100

9) local minimum: (-4, -1) local maximum: (4, 1) inflection points: (0, 0), (-4 3, -2 3),

(4 3, 2 3)

x-4 -2 2 4

y 6

4

2

-2

-4

-6

x-4 -2 2 4

y 6

4

2

-2

-4

-6

4

Answer Key Testname: MAC 2311 REVIEW FOR TEST # 3

10) Absolute maxima: (- 2, -5), ( 2, -5) Local minimum: (0, -9)

Inflection points: - 23 , 5 9 ,

2 3 ,

5 9

x-10 -5 5 10

y 10

5

-5

-10

x-10 -5 5 10

y 10

5

-5

-10

11)

x-8 -4 4 8

y

8

4

-4

-8

x-8 -4 4 8

y

8

4

-4

-8

12)

x-3 -2 -1 1 2 3

y

1.5

1

0.5

-0.5

x-3 -2 -1 1 2 3

y

1.5

1

0.5

-0.5

5

Answer Key Testname: MAC 2311 REVIEW FOR TEST # 3

13) Absolute maxima: (- 2, 2), ( 2, 2) Local minimum: (0, -2)

Inflection points: - 23 , - 2

9 , 2 3 ,

- 2 9

x-10 -5 5 10

y 10

5

-5

-10

x-10 -5 5 10

y 10

5

-5

-10

14) 4x 3

x4 + 5 dx

6