# MBA 6303 CSU Stakeholder Engagement Monitoring and Control Plan Discussion

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Math 31
Final Exam
May 22nd, 2021
NAME & ID:
DIRECTIONS:
• Do not open Exam until instructed to do so.
• Do each of the problems and show all work.
• NO WORK MEANS NO POINTS!
• Box or circle and LABEL your final solution.
• You will have 135 minutes to complete this Exam.
• No part of this document can be shared without the expressed written consent of Dr. Dashiell
Fryer
SCORES:
1.
/5
2.
/10
3.
/10
4.
/10
5.
/15
6.
/10
7.
/20
8.
/10
Total:
/90
1. 5 points Find derivatives of the following functions with respect to x.
Z xp
(a) 3 points
1 + t3 dt
0
Z
(b) 2 points
3×2
p
1 + t3 dt
0
1
2
2. 10 points Write the following integral as the limit of Riemann sums using either left or right
endpoints.
Z
5
(7 + x3 )1/3 dx
2
You do not need to find the limit.
3
4
3. 10 points Compute the following integrals.
Z
(a) (ln x)2 dx
Z
(b)
3x + 1
dx
x3 − x2
5
6
4. 10 points Let f (x) =

x and g(x) = x2 .
(a) Find the area of the region entirely bounded by f and g.
(b) Rotate the region about the y-axis and find the solid’s volume.
7
8
5. 15 points Let f (x) = x2
(a) 3 points Set up the integral for the arc length of f from x = 0 to x = 1.
(b) 2 points Set up the integral for the surface area of the surface obtained by rotating
(c) 10 points Find the arc length and surface areas exactly.
9
10
6. 10 points Does the following integral converge or diverge?
Z
0
−∞
x4
11
x
dx.
+ 16
12
7. 20 points The hyperbolic cosine and hyperbolic sine functions are defined as
cosh x =
(a) 2 points Show that
ex + e−x
ex − e−x
, sinh x =
2
2
d
d
cosh x = sinh x, and
sinh x = cosh x.
dx
dx
(b) 2 points What are cosh 0 and sinh 0?
(c) 10 points Derive the Taylor series for f (x) = cosh x centered at c = 0, using only the
derivatives of f (x). Do not use a Taylor Series for ex .
(d) 6 points What is the interval of convergence for the Taylor series for f (x) = cosh x
centered at x0 = 0.
13
14
8. 10 points
Approximate
1
Z
2
e−x dx
0
accurately to two decimal places.
You must describe the method used, eg Taylor Series, Simpson’s Rule, etc, in full detail.
Clearly explain how you know your approximation is correct to the specified accuracy.
15
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stakeholders

Monitoring and Control Plan

soft skills

Economic Studies

historical project documents

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