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Midterm

# MATH 145 Illinois State Test03part02S15Exam

Department
Mathematics
Course Code
MAT 145
Professor
All
Study Guide
Midterm

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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€¢
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