MATH 100—Midterm—Solutions
version 1
Date: October 27, 2017 Time: 90 minutes
Section EA1 EB1 EC1 ED1 EE1 EG1 EH1
Instructor Yaskin Leonard Aldabbas Shen Yaskin Wiersma Aldabbas
X
Instructions
1. Place your U of A Student ID card on your table or desk.
2. The exam has 11 pages including this cover page. Please ensure that you have all
pages.
3. The exam will be marked out of 60 points. There are 7 questions. The points for each
question are indicated beside the question number.
4. Answers for questions 1 through 6 must be accompanied by adequate justification.
Please note that this exam will be marked electronically. Anything written
on the back of the pages will not be marked.
5. For multiple-choice questions you are not required to show your work.
6. This is a closed book exam. No books, notes, calculators, cell phones or other
electronic aids are allowed!
, 1. [7 pts] Use logarithmic differentiation to find the derivative of the function
x cos x
4 3 x2 e +x
y = (x + 1) e .
arcsin(2x)
Do not simplify your answer.
Solution. Applying the natural logarithm to both sides of the equation and simplifying
produces
ln y = 3 ln(x4 + 1) + x2 + cos(x) ln(ex + x) − ln(arcsin(2x) .
Differentiating with respect to x, we get
1 dy 3
(4x3 ) + 2x − sin(x) ln(ex + x) − ln(arcsin(2x)
= 4
y dx x +1
1 x 1 2
+ cos(x) x (e + 1) − p .
e +x arcsin(2x) 1 − (2x)2
Thus
cos x
ex + x 12x3
dy 2
= (x4 + 1)3 ex + 2x − sin(x) ln(ex + x) − ln(arcsin(2x)
· 4
dx arcsin(2x) x +1
x
e +1 2
+ cos(x) x − p .
e + x arcsin(2x) 1 − (2x)2
version 1
Date: October 27, 2017 Time: 90 minutes
Section EA1 EB1 EC1 ED1 EE1 EG1 EH1
Instructor Yaskin Leonard Aldabbas Shen Yaskin Wiersma Aldabbas
X
Instructions
1. Place your U of A Student ID card on your table or desk.
2. The exam has 11 pages including this cover page. Please ensure that you have all
pages.
3. The exam will be marked out of 60 points. There are 7 questions. The points for each
question are indicated beside the question number.
4. Answers for questions 1 through 6 must be accompanied by adequate justification.
Please note that this exam will be marked electronically. Anything written
on the back of the pages will not be marked.
5. For multiple-choice questions you are not required to show your work.
6. This is a closed book exam. No books, notes, calculators, cell phones or other
electronic aids are allowed!
, 1. [7 pts] Use logarithmic differentiation to find the derivative of the function
x cos x
4 3 x2 e +x
y = (x + 1) e .
arcsin(2x)
Do not simplify your answer.
Solution. Applying the natural logarithm to both sides of the equation and simplifying
produces
ln y = 3 ln(x4 + 1) + x2 + cos(x) ln(ex + x) − ln(arcsin(2x) .
Differentiating with respect to x, we get
1 dy 3
(4x3 ) + 2x − sin(x) ln(ex + x) − ln(arcsin(2x)
= 4
y dx x +1
1 x 1 2
+ cos(x) x (e + 1) − p .
e +x arcsin(2x) 1 − (2x)2
Thus
cos x
ex + x 12x3
dy 2
= (x4 + 1)3 ex + 2x − sin(x) ln(ex + x) − ln(arcsin(2x)
· 4
dx arcsin(2x) x +1
x
e +1 2
+ cos(x) x − p .
e + x arcsin(2x) 1 − (2x)2
