Review Guide for MAT220 Final Exam Part I.
Part 1 is worth 50% of your Final Exam grade. Syllabus approved calculators can be used on this part of the
exam but are not necessary. All work will be done on the test itself; you may NOT use any scratch paper.
Partial credit WILL be awarded for partially correct work so be sure to show ALL of your steps. Correct
answers without the correct corresponding work are worth nothing. 14 Questions…some with parts. Since
you are about to finish up this calculus class (hopefully with a passing grade) you should be able to calculate
how much time that gives you per problem. Some problems will take MUCH less time than this number
whereas some problems may take slightly longer.
Things you should make sure that you can do! Note: Section numbers have been provided by each topic so
that you can go back through your NOTES, HOMEWORK and OLD TESTS to find problems to practice. You can
also go back to the class HELP page and view some of the relevant supplemental readings and videos. I have
provided a few examples for particular problems for you to practice (you should still find others of those types
to practice on your own!). For those that I did not provide examples for you should have no problem finding
examples in your notes, HW and on old tests!
BE SURE THAT YOU HAVE LOOKED AT, THOUGHT ABOUT AND TRIED THE SUGGESTED PROBLEMS ON THIS REVIEW
GUIDE PRIOR TO LOOKING AT THESE COMMENTS!!!
1. Be able to find the absolute max and absolute min of a given continuous function on a closed interval.
Extreme Value Theorem (section 2.7). Like Test 4 problem # 1.
2. Be able to find critical numbers (values that make the derivative zero or undefined) and be able to
determine open intervals on which a function is increasing and decreasing by creating the appropriate table.
Be able to use the first derivative test to determine x values where a function has a relative maximum and a
relative minimum (section 2.9). Like Test 4 problems # 3, 4
3. Be able to find possible points of inflection (the x-values anyway) for a function and be able to determine
open intervals on which a function is concave up and concave down by creating the appropriate table. Also be
able to tell which possible points of inflection are actually points of inflection (section 2.10). Like Test 4
problem # 6
Note: If after practicing # 1 – 3 if you have any questions be sure to ask in class!!!
4. Know when L’Hopital’s Rule applies and how to use it to evaluate limits where direct substitution yields the
appropriate indeterminate form (section 2.11). Like Test 4 problem # 8 BUT be sure to also go back in your
notes and HW and practice some non-polynomial fraction type examples!
For example….. Lim
x 0
x2
sin x
or
Lim
x 1
x 1
3Lnx
x2
02
0
direct substitution yeilds
 thus L'Hopital's rule applies.
x  0 sin x
sin 0 0
2
x
2x
20 0
Lim
 Lim

 0
x  0 sin x
x  0 cos x
cos 0 1
Lim
x 1
1 1
0 0
direct substitution yeilds

= thus L'Hopital's rule applies
x 1 3Lnx
3Ln1 3  0 0
x 1
1
x 1
Lim
 Lim
 Lim 
x 1 3Lnx
x 1
1 x 1 3 3
3
x
Lim
Remember if after applying L’Hopital’s rule you STILL get 0/0 then DO IT AGAIN!
5. In section 2.12 we did a summary of curve sketching where we put together all of the material we had
learned from the previous few sections and applied it to graphing a function. Try the following problem in
addition to reviewing what we did in class for section 2.12 and what you did on HW!
Sketch the graph of y  f  x  on the axes below given the following information!!!!
The Domain of
y  f  x  is
0, d 
y  f  x  is
The Range of
 2, 2
 ab
f
  1 f  b   2 f  c   1 f  d   2
 2 
The critical numbers (places where f   x  = 0 or f   x  = undefined ) are x  a, b, c
f  0  2
f a  0
The "possible" points of inflection (places where f   x  = 0 or f   x  = undefined ) occur when x  a,
TEST #
a
2
 ab
 a,

2 

ab
a
2
2
Sign of f   x 

+
+


Sign of f   x 
+

+
+

 0, a 
 ab 
,b

 2

ab
b
2
2
 b, c 
 c, d 
bc
2
cd
2
ab
,c
2
2
1
1
a
b
c
d
2
This problem may look long and complicated BUT it isn’t. Just plot the given points,
 ab
f  0  2 f  a   0 f 
  1 f  b   2 f  c   1 f  d   2
 2 
and then use the information in the table to sketch the graph using the appropriate increasing/decreasing
information along with sketching the appropriate curvature of the segments! Ask in class if you are unsure!
6. Be able to evaluate an indefinite integral (section 3.1). Like Test 5 problems # 1 – 4 (which were ALL
problems that YOU did in class!)
7. During the week of 4/14 there were a couple of days where you were given the graph of a velocity vs time
function and asked various questions about it (Try Now problems). This test question will ask you four
questions similar to some of those. See below for a third problem that you could use as practice (although
YOUR test problem will only have 4 parts).
The graph given below represents the velocity of an object moving right and left along straight line for 16 seconds.
Velocity
(meters / second)
2 m/s
1 m/s
-1 m/s
1
2
3
4
5
9
12
14
16
time (seconds)
-2 m/s
3
A) (5 pts) Find
 v(t )dt
0
Just find the “net signed area” on the interval from 0 to 3! 2x3 = 6
B) (5 pts) What does your answer to A) represent?
As we discussed in class, the net signed area represents the displacement of the object. Thus your answer of 6 in part A) means the
object is 6 meters to the right of wherever it stated 3 seconds into the trip.
C) (5pts) What direction is the object moving from 3 seconds to 5 seconds? Explain HOW you know!
The object is moving right during this time interval. We know this because the velocity is positive (graph is above the time axis)
D) (5 pts) What direction is the object moving from 9 seconds to 11 seconds? Explain HOW you know!
The object is moving left during this time interval. We know this because the velocity is negative (graph is below the time axis)
E) (5 pts) Where is the object located at the end of its 16 second trip relative to its starting position?
To answer this just calculate the “net signed area” over the whole trip. 6 + 2 -6 +0 +2 = 4. Thus after 16 seconds this object is located
4 meters to the right of wherever it started at.
F) (5 pts) What is the total distance travelled by the object during its 16 second trip?
To figure this out we have to calculate the “actual area” =
6  2  6  0  2  16 . Thus the object travels a total distance of 16
meters during its 16 second trip.
G) (5pts) What is the object doing during the time interval
11,14  ?
The object is standing still during this time interval (velocity is zero).
H) (5 pts) During what time interval(s) is the object moving but not accelerating?
Since acceleration is the derivative of velocity the slope of the tangent line indicates the acceleration. In order
to NOT be accelerating the object must have zero acceleration (or a horizontal tangent line). This occurs in
two places;
0,3
and
11,14.
The object is not moving during the later time interval so our answer to this
question is from 0 seconds to 3 seconds.
8. Be able to evaluate a definite integral by using the limit definition (sections 3.2, 3.3). You should be able to
find MANY problems in your notes and HW to practice! Also Test 5 problem #8 is an example (although on
your final exam you will NOT be given the limit definition so be sure that you KNOW it). Here is another
problem that you could use as practice…
1
 x
Find
2
 2 x  3 dx using the limit definition of the definite integral. Use a regular partition and choose
1
to be the right endpoint.
1
 x
2
 2 x  3 dx
1
b  a 1  1 2


n
n
n
x 
xk*  a  k x  1  k
2
2 2k

1
n n
4k
 2k 
 2k 
f  x      1  2   1  3   2  4
n
 n

 n

2
*
k
n
n
 4k 2
2
1 n 
 8 n
 2 x  3 dx  Lim  f  xk*  x  Lim    2  4   Lim   3  k 2   8 
n 
n 
n
n k 1 
k 1
k 1 
 n n  n k 1
1
 8 n  n  1 (2n  1) 1

8
16
 8 4 4

 Lim   3 
  8n   Lim     2  8     0  0  8 
n 
n

6
n
3
3
 3 n 3n

 n

1
 x
2
Be sure to practice several problems and be sure to be familiar with ALL of the summation formulas that we worked with
n
n
 k3
k2
k 1
k 1
n
k
k 1
n
c .
k 1
9. Be able to evaluate a definite integral using the FTC (see YOUR notes and HW from section 3.4).
1
Try
 x
2
 2 x  3 dx and see if you get the same answer as you did when you practiced the limit definition of
1
derivative!
1
 x
1
2
1
2
3
2
 1 3
  1

 2 x  3 dx   x 3  x 2  3x ]11    1  1  3 1      1   1  3  1 
3
 3
  3

1
1
2
16
  1 3  1 3    6 
3
3
3
3
See I got the same answer here as when I used the limit definition!
10. Be able to find the average value of a function on a given interval. Also be able to find the x value(s) on
that given interval that generate the average value (section 3.4). See homework problems # 18, 19 and 20 in
section 3.4. You could also practice the following…
Find the average value of  x  2 x  3 on
2
1,1 .
Note: This is the function and interval from question 9.
1
1
1 16 8
Average Value =
 x 2  2 x  3 dx   


1  1 1
2 3 3
Also be able to find the answer to
*
What is the value of x on
Solve  x 2  2 x  3 
1,1 that gives you the average value?
8
  3x 2  6 x  9  8  3x 2  6 x  1  0 this one doesn't factor so solve
3
3  2 3
3  2 3
only
is in the given interval.
3
3
Note: YOUR actual final exam questions #8 – 10 have been created to be much quicker and easier than these
samples!
using the quadratic formula and obtain x 
11. Be sure to review the second part of the Fundamental Theorem of Calculus (section 3.4). For example,
can you do the following problem?
x

d 
2
Find
   t  2t  1 dt  . Write your answer as a polynomial in standard form with terms in descending order.
dx  4

3
x

d 
2
   t  2t  1 dt  
dx  4

3
 x   2  x   1  3x   x  2x 1  3x  3x  6x  3x
3 2
3
2
6
3
2
8
5
2
12 – 14. Be able to evaluate a variety of integrals (both indefinite and definite). Some may require
“substitution”, some may lead to natural logarithms and some may lead to inverse trigonometric function
(sections 4.1, 4.2, 4.3). See ALL problems on Test #6 as examples!
It is unlikely that you will finish this test in the given amount of time unless you are EXCEPTIONALLY well
prepared. You have only 65 minutes to complete as much as you can. This test may prove to be very
challenging unless you have taken the necessary steps throughout the semester to learn all of the material we
covered. If you haven’t figured it out yet, there is no rule that says these problems must be done in the order
that they appear on the test. A wise student would have prepared for this test so well by utilizing this study
guide that they know exactly what questions are going to be easiest for them and complete those problems
first!