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Mathematics 1220 Midterm 1 Solutions: Problems and Their Resolutions - Prof. Michael Wills, Exams of Mathematics

The solutions to mathematics 1220 midterm 1 exam problems. It covers topics such as calculus, trigonometry, and logarithms. Students can use these solutions to check their work, understand the concepts better, or prepare for exams. The document also includes explanations for each problem's solution.

Typology: Exams

Pre 2010

Uploaded on 07/23/2009

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MATHEMATICS 1220 MIDTERM 1 SOLUTIONS
Overview
The exam went well. The average was 23 but that is due to there
being an outlier. The median is a better measure of central tendency
in this case. The median is 26.5. Here is the score distribution.
Score Number of Students
28 4
25 1
24 1
23 1
3 1
I do not curve exams. I do sometimes curve classes if I think it is
appropriate. It would not be appropriate to curve the class if the class
scores were along these lines.
The preset scale outlined on the syllabus tells us that the grade scale
when scores are given out of 30 is as follows.
Score Grade
27-32 A, A-
23-26 B+, B, B-
20-22 C+, C
18, 19 C-
14-17 D+, D, D-
0-13 E
Thus, 7 out of 8 passed with 4 people earning an ‘A’. Well done!
Here are the solutions.
1. Problem 1
Let f(x) = xxfor x > 0. Compute f0(e).
Solution 1.1.Write f(x) = exlnx. By the chain and product rules,
(1.1) f0(x) = exln x³(ln x+x
x´=xx(ln x+ 1).
Hence,
(1.2) f0(e) = ee(ln e+ 1) = 2ee.
1
pf3
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MATHEMATICS 1220 MIDTERM 1 SOLUTIONS

Overview The exam went well. The average was 23 but that is due to there being an outlier. The median is a better measure of central tendency in this case. The median is 26.5. Here is the score distribution. Score Number of Students 28 4 25 1 24 1 23 1 3 1 I do not curve exams. I do sometimes curve classes if I think it is appropriate. It would not be appropriate to curve the class if the class scores were along these lines. The preset scale outlined on the syllabus tells us that the grade scale when scores are given out of 30 is as follows.

Score Grade 27-32 A, A- 23-26 B+, B, B- 20-22 C+, C 18, 19 C- 14-17 D+, D, D- 0-13 E

Thus, 7 out of 8 passed with 4 people earning an ‘A’. Well done! Here are the solutions.

  1. Problem 1 Let f (x) = xx^ for x > 0. Compute f ′(e).

Solution 1.1. Write f (x) = ex^ ln^ x. By the chain and product rules,

(1.1) f ′(x) = ex^ ln^ x

(ln x +

x x

= xx(ln x + 1).

Hence,

(1.2) f ′(e) = ee(ln e + 1) = 2ee. 1

  1. Problem 2 Compute

ln xdx.

Solution 2.1. We integrate by parts. Let u = ln x and dv = 1. Then du = (^) x^1 and v = x. Hence,

ln xdx = x ln x −

xdx x

= x ln x −

1 dx = x ln x − x + C

where C is an arbitrary constant.

  1. Problem 3 Show that for every x, y ∈ R

(3.1) cosh(x + y) = cosh x cosh y + sinh x sinh y.

Solution 3.1. From the definition of cosh and sinh we have:

cosh x cosh y + sinh x sinh y

=

ex^ + e−x 2

ey^ + e−y 2

ex^ − e−x 2

ey^ − e−y 2

=

exey^ + exe−y^ + e−xey^ + e−xe−y^ + exey^ − exe−y^ − e−xey^ + e−xe−y 4

=

2 ex+y^ + 2e−x−y 4

=

ex+y^ + e−x−y 2 = cosh(x + y).

  1. Problem 4 Compute

0

(2x^ − x^3 )dx.

Solution 4.1. We compute:

0

(2x^ − x^3 )dx =

2 x ln 2

x^4 4

4 0

ln 2

ln 2

ln 2

for some constant c. At x = 1,

(7.4) ln(y) = ln(1 · y) = g(1) = ln 1 + c = c.

Hence,

(7.5) g(x) = ln x + ln y

as required. Since this equation holds for any y > 0, our claim has been shown.

  1. Problem B It is well known that

2 is an irrational number. What sort of psychoanalysis should be given to straighten the poor number out?

Solution 8.1. Answers will vary.