Math exam

MATHEMATICAL ANALYSIS III, MATH 183

Exam 2

1. Use the product rule for differentiation to find derivatives in each of the following:

(a) (b)

(c)

2. Use the quotient rule for differentiation to find derivatives in each of the following:

(a) (b) (c)

3. Use the chain rule to find derivatives in each of the following:

(a) (b) (c)
4. Find derivatives of the following functions

(a) (b) (c)

5. Find the intervals on which the given function is increasing or decreasing

(a) (b)

6. Find all critical numbers for the given function, and then use the First or Second Derivative Tests to determine whether the critical number implies a local extrema

(a) (b)

7. Sketch complete graphs of the following; indicate where the function is increasing/decreasing, has local extrema, where the graph is concave up or concave down, asymptotes and coordinate intercepts

(a) (c)

(
)
(
)
1
x
2
x
3
x
2
x
1
x
f
2
+
÷
ø
ö
ç
è
æ
+
+
=
(
)
3
x
3
x
x
f
2
2
–
+
=
(
)
x
2
4
x
x
f
–
=
(
)
1
x
x
x
x
f
2
2
3
+
+
=
(
)
(
)
2
3
2
x
4
x
f
+
=
(
)
(
)
3
2
/
1
3
/
1
x
3
x
2
x
f
–
+
=
(
)
2
1
2
x
3
x
x
2
x
f
÷
÷
ø
ö
ç
ç
è
æ
+
=
(
)
(
)
x
3
2
e
1
x
x
f
×
–
=
(
)
1
x
ln
x
y
3
2
–
=
x
e
e
y
x
2
x
2
–
=
–
(
)
25
x
x
f
2
–
=
x
2
x
4
x
8
y
3
–
=
(
)
x
8
x
2
x
f
3
–
=
(
)
1
x
3
x
x
f
–
+
=
(
)
x
2
x
4
x
2
x
f
2
3
+
–
=
(
)
2
x
1
x
x
f
–
–
=
(
)
(
)
(
)
x
3
x
4
x
2
x
3
x
x
f
2
3
2
+
+
+
+
=
(
)
(
)
(
)
3
x
2
3
x
x
f
2
–
+
=