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Math Exam 106: September 1, 2006, Exams of Advanced Calculus

Christian Brothers University (CBU)Advanced Calculus

A math exam for a university-level course, math 106, from september 1, 2006. The exam covers various topics including finding average and percentage rates of change, interpreting functions and their derivatives, and estimating instantaneous rates of change. Questions involve calculating slopes, finding derivatives of functions, and analyzing graphs.

Typology: Exams

Pre 2010

Uploaded on 08/18/2009

koofers-user-79
koofers-user-79 🇺🇸

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EXAM 2
Math 106
September 1, 2006
Name
You must show all your work. Partial credit will be given.
1. Find the average rate of change and the percentage rate of change for the exponential function
y= 3(0.4x) from x= 1 to x= 3. (6 pts)
2. Imagine that C(f) is the number of bushels of corn produced on a tract of farmland where fis
the number of pounds of fertilizer used. What are the units of C0(f)? Interpret C0(13) = 112
(that is translate it into English). Is it possible for C0(f) to ever be negative? Why or why
not? (8 pts)
3. On the following graph estimate the slope of the line tangent to the curve at the point Band
the average rate of change for the curve from Ato C. (8 pts)
1
2
3
4
5
10 20 30 40 50 60 70
A
B
C
pf3
pf4
pf5

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EXAM 2

Math 106 September 1, 2006

Name

You must show all your work. Partial credit will be given.

  1. Find the average rate of change and the percentage rate of change for the exponential function y = 3(0. 4 x) from x = 1 to x = 3. (6 pts)
  2. Imagine that C(f ) is the number of bushels of corn produced on a tract of farmland where f is the number of pounds of fertilizer used. What are the units of C ′(f )? Interpret C′(13) = 112 (that is translate it into English). Is it possible for C ′(f ) to ever be negative? Why or why not? (8 pts)
  3. On the following graph estimate the slope of the line tangent to the curve at the point B and the average rate of change for the curve from A to C. (8 pts)

1

2

3

4

5

10 20 30 40 50 60 70

A

B

C

  1. Numerically estimate the instantaneous rate of change for

f (x) =

4 x + 15 when x ≤ 3 , 8(1. 5 x) when x > 3.

at x = 2. Is this function continuous at 3? Does it have a derivative at 3? (10 pts)

  1. Values of the cumulative capital investment in the cellular phone industry, beginning in 1985 are shown in the table below. year Cumulative Capital Investment (millions of dollars) 1985 0. 1987 2. 1989 4. 1991 8. 1993 14. 1995 24. 1997 46. 1998 60. Estimate the instantaneous rate of change of capital investment in the year 1991 (6 pts).
  1. For each of the following formulas find the derivative formula. (5 pts each)

(a) g(t) = − 3 t^2 + t − 2 t−^3.

(b) y =

2 x^2 − 2 x^2

(c) f (x) = 7x^2 + 13 ln x.

(d) f (t) = 5e^2 t^ + t + 1.

(e) y = (2x^2 + 15x − x)−^4.

(f) g(x) = ex

(^2) − 2

  • ln (x^3 + 2x + 1).

(g) f (x) =

(2x^2 − x)^5

(h) y = (ln (x))e^3 x.

(i) f (t) =

2(et) 6 − 3 ln t

(j) y = 12x + x 3

3 x + 2.