Math 140 Final Exam Review
- For
the function
a) Find the x-coordinates of all critical points and identify the interval(s) where y=f(x) is increasing and decreasing.
b) Find the x-coordinates of all inflection points and identify the interval(s) on which y=f(x) is concave up and concave down.
- Find all critical numbers for
the function (a)
, (b) 
- Find
the x –coordinate of the inflection points of the function
on the
interval 
. - Find
the absolute minimum value of
on the interval
[0,5].
- a) State the definition of the derivative.
b) Use the definition of the derivative to find the derivative of
(i) 
(ii) 
- Find the following limits:
a) 
b) 
c) 
d) 
e) 
f) 
(Consider a similar
problem by replacing 
with 
.)
- What is the derivative of
a) 
b) 
c)
?
- If the
tangent line to y=f(x) at (8,9)
passes through the point (5,6) find
. - If
, find 
. - Find
the most general antiderivative of the function
- Evaluate the integral
a) 
b) 
c) 
- Use
implicit differentiation to find
at the point (1,2), for 
. - Evaluate
the integral
by interpreting
it in terms of areas. - If a
snowball melts so that its surface area decreases at a rate of 1
, find the rate (in cm/min) at which the diameter
decreases when the diameter is 43 cm.
(Hint: Surface area of a sphere = 
) - A farmer with 700 feet of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens?
- Find
the volume of the solid obtained by rotating the region in the first
quadrant bounded by
and y = 5 and the
y-axis. - Set
up, but DO NOT evaluate the integral for the volume of the solid obtained
by rotating about the y-axis the region bounded by the curves
and 
and 
. - Find
the area of the region enclosed by the graphs of
and 
. - A
region in the xy-plane is
bounded by and
a) Graph the region
b)
Using the Washer Method, set up the integral to determine the volume of the solid generated
by rotating the region about the y-axis.
c)
Using Cylindrical Shells set up the integral to determine the volume of the solid generated
by rotating the region about the y-axis.
- Consider
the function
where 
and 
. Find, domain,
intercepts, symmetry, asymptotes, critical numbers, increasing and
decreasing intervals, local maximum and minimum, concavity intervals,
inflection points, and then at last… graph f(x).
Note the following should also be reviewed
- Continuity
- Intermediate Value Theorem, Squeeze Theorem, Extreme Value Theorem, Rolles Theorem, Mean Value Theorem, Fundamental Theorem of Calculus Parts I and II
- Possibly other topics as they come up in class review or review of your notes.
Answers:
1. (a) x=-4, x=0, Inc. 
and 
, Dec. 
(b) x = 2 is a point of inflection, Concave Down 
and 
, Concave Up 
2. (a) x = 5, (b) 
, where n is an integer
3. x = 
4.
y= -15
5. (a)
(b) (i) 
(ii) 
6. a) 4 b) -3 c) 3/2 d) 13/4 e) 
f) 0 , (0)
7. a) 
b) 
c) 
8.
1 9. 
10. 
11.
a) 
b) 
c) 
12.
2 13. 
14. 
or 
15.
16. 
17. 
18. 
19. a)
b) 
c)
20. Domain:
, Intercepts:
(0,0), Symmetry: None
Asymptotes:
No horizontal asymptotes, Vertical asymptote: x = -8, Slant asymptote: y = x-8
Critical numbers: x = 0 and x = -16. Note: x = -8 is not in the domain.
Increasing: 
, Decreasing: 
Local maximum: (-16,-32), Local minimum (0,0)
Concave
down: 
,
Concave up: 
Inflection points: None
Graph f(x).