Department

MathematicsCourse Code

MAT 145Professor

AllStudy Guide

MidtermThis

**preview**shows page 1. to view the full**5 pages of the document.**MAT 145: Test #3 (50 points)

Part 2: Calculator OK!

Name ________________________ Calculator Used ____________ Score ____________________

21. For

f(x)=8x3+81x2âˆ’42xâˆ’8

, defined for all real numbers, use calculus techniques to determine all

intervals on which f is increasing and on which f is decreasing. Show all calculations. Explain your

answer, using calculus, in a sentence. (2 pts)

Increasing on: _______________________

(intervals: exact values)

Decreasing on: _______________________

(intervals: exact values)

22. Use calculus techniques to determine the location of any local maxima and any local minima of

f(x)=8x3+81x2âˆ’42xâˆ’8

, where f is defined for all real numbers. Show all calculations. Explain your

answer, using calculus, in a sentence. (2 pts)

Local Maxima at: ____________________

(exact x values)

Local Minima at: _____________________

(exact x values)

23. For

g(x)=x3âˆ’3x2+xâˆ’2

, defined for all real numbers, use calculus techniques to determine all

intervals on which g is concave up and on which g is concave down. Show all calculations. Explain your

answer, using calculus, in a sentence. (2 pts)

Concave Up on: ______________________

(intervals: exact values)

Concave Down on: ___________________

(intervals: exact values)

24. Use calculus techniques to determine ordered pairs for any points of inflection of

g(x)=x3âˆ’3x2+xâˆ’2

,

where g is defined for all real numbers. Show all calculations. Explain your answer, using calculus, in a

sentence. (2 pts)

Points of Inflection: ___________________

(ordered pairs: exact values)

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25. Use the information here to sketch the graph of a function f that meets ALL the following requirements.

Label your graph to help me identify these requirements. (12 pts)

â€¢

lim

xâ†’âˆ’âˆž f(x)=2

â€¢

lim

xâ†’âˆž f(x)=0

â€¢

lim

xâ†’2+f(x)=âˆž

â€¢

lim

xâ†’2âˆ’f(x)=âˆž

â€¢ There is a local minimum at x = â€“2.

â€¢ There is an absolute minimum at x = 5.

â€¢

!

f(x)>0

on

âˆ’2,0

()

âˆª0,2

()

âˆª5,âˆž

()

â€¢

!

f(x)<0

on

âˆ’âˆž,âˆ’2

()

âˆª2,5

()

â€¢

!!

f(x)>0

on

âˆ’4,âˆ’1

()

âˆª0,2

()

âˆª(2,6)

â€¢

!!

f(x)<0

on

âˆ’âˆž,âˆ’4

()

âˆª âˆ’1,0

()

âˆª6,âˆž

()

â€¢ The y-intercept is (0, 2).

â€¢ Exactly two x-intercepts: (â€“2, 0) and (3, 0)

â€¢ A point on the graph is (â€“4, 1).

â€¢ A point on the graph is (5, â€“2).

5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

â€“0.5

â€“ 1

â€“1.5

â€“ 2

â€“2.5

â€“ 3

â€“ 6 â€“ 5 â€“ 4 â€“ 3 â€“ 2 â€“ 1 1 2 3 4 5 6 7

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