Review Guide for MAT220 Final Exam Part II.
Part 2 is worth 50% of your Final Exam grade. NO CALCULATORS are allowed on this portion of the Final Exam.
You may NOT use your own scratch paper for this Multiple Choice Exam. Instead, your instructor will give you two
sheets of scratch paper that you can use to do whatever work you feel that you need to in order to obtain the answer to
each question. This portion of the Final Exam consists of five pages. Each page has a blank side that can also be used for
scratch work if needed. You will turn in BOTH sheets of scratch paper with your Final Exam (even if they have nothing
written on them). NO PARTIAL CREDIT will be given on this portion of the Final Exam. You will have 110 minutes to
complete this portion of the Final Exam (assuming that you show up on time). There are 50 questions on this part of
your final exam. MANY of these questions will be very quick and require little to no written work!
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. Some problems are not necessarily
specific to a section, rather will test how well you understand an idea that is covered over possibly multiple sections.
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 - 2. Make sure you understand a few basic properties of antiderivatives. For example, is an antiderivative of a
difference of two functions, f  g , the same as an antiderivative of f minus an antiderivative of g ?(section 3.1)
True.
  f  x   g  x  dx   f  x  dx   g  x  dx . Does this property hold for addition and multiplication?
3 - 4. Be sure that you understand the Fundamental Theorem of Calculus (part I) and what MUST be true about f  x  ,
in order for
b
b
a
a
 f  x  dx to exist. If  f  x  dx does exist, what does the answer “look like”? What does the answer to
 f  x  dx look like? (sections 3.1, 3.4)
One of these is a “family of functions” and the other is a “number”. Which is which?
5 - 6 If f  x   g  x  , does f   x   g   x  (True)? What about the converse of that statement(False)? Ask yourself
this same question (both ways) about indefinite and definite integrals. i.e If f  x   g  x  does
b

b
f  x  dx   g  x  dx ? (True)
a
If
a
b
b
a
a
 f  x  dx   g  x  dx does f  x   g  x  ?(False) Now answer both questions if the integral was indefinite rather
than definite (True) (True).
Try to gain some insight into what each of these statements are saying!
7. If the derivative of f  x  at x  c is zero what does that mean happens at x  c in the graph of f  x  ? (sections 2.9,
2.12). Think carefully!
It means that the graph has a horizontal tangent line at x = c (does this mean that the graph HAS to have a relative max
or min at x = c?).
8. If the second derivative of f  x  at x  p is zero what does that mean happens at x  p in the graph of f  x  ?
(sections 2.10, 2.12). Think carefully!
It means that there is a POSSIBLE point of inflection with an x coordinate of p (i.e. it is POSSIBLE that the graph changes
concavity at this location).
9 - 10. Be sure that you understand the second part of the Fundamental Theorem of Calculus (section 3.4). For example
can you find F   x  if
x
x2
3
3
F  x    cos t dt. What if F  x    cos t dt ? Can you find F   x  for both of these
examples?
x
F  x    cos t dt  F   x  
3
x2
x
d
d
cos t dt  cos x  F   x   cos x   sin x

dx 3
dx
x2


d
F  x    cos t dt  F   x  
cos t dt  cos  x 2   2 x  2 x cos x 2

dx 3
3
F   x  


d
2 x cos x 2  2 x    sin x 2   2 x  cos x 2 1  2  2 x 2 sin x 2  cos x 2 
dx
11. What is the maximum number of vertical asymptotes that a function could have? What does a horizontal
asymptotes represent in Calculus (how many of those could a function have)? (section 2.11, 2.12)
A function could have unlimited vertical asymptotes (y=tanx). A horizontal asymptote for
y  f  x  is a limit at infinity..... i.e. y  Lim f  x  and
x 
y  Lim f  x 
x
12. When doing related rates problems be sure to understand how to write down the “given” rate using derivative
notation. For example if the radius of a circle is getting smaller by 2cm each second then the given rate would be
dr
cm
 2
. (section 2.5)
dt
s
13. Be able to find an indefinite integral (section 3.1). Example: Find
  3x  4 
2
dx
u  3x  4
du  3dx 
1 2
1 1
1
3
u du   u 3  c   3x  4   c

3
3 3
9
1
du  dx
3
  3x  4 
2
dx
14. If an object undergoing rectilinear motion is moving right and slowing down, what would the graph of its position
function look like (i.e increasing concave up, decreasing concave down etc.)? (section 2.9)
Moving right means the velocity is positive (thus the slope of the tangent line to the position graph would be positive
and consequently the position graph must be increasing). If the object is ALSO slowing down then the acceleration must
have the opposite sign of velocity so in THIS example that would mean negative. If the acceleration is negative the
graph of the position function must be concave DOWN as acceleration is the second derivative of position. So in THIS
example the graph of the position function would be increasing concave down. Can you figure out what the graph
would look like if the object was moving left and slowing down, moving left and speeding up and moving right and
speeding up?
15. How would you find the second derivative of a function? How would you find the “second antiderivative” of a
function?
To find the second derivative of a function just take the derivative of the derivative. To find the second antiderivative
just take the antiderivative of the antiderivative (or the integral of the integral).
b
16. Be sure that you understand when the
 f  x  dx is positive, when it is negative and when it is zero (section 3.4)
a
If f(x) >0 and a<b then the integral is positive. What if f(x)>0 and a>b? What if f(x)<0 and a<b? What if f(x)>0 and a>b?
17. A simple definite integral problem for you. (section 3.4)
18. Given the graph of some function f . If g  x  
x
 f  t  dt where “a” is some number (like 0 or 1 for example). Be
a
able to look at the graph of f and find things like g  2 and
g   2 for example (remember you are given the graph
of f NOT the graph of g ) .
g(2) would be the net signed area under the graph of f(t) between “a” and 2. g   2 would be the “y” value of the graph
of f(t) at 2 (recall the second part of the FTC). Think about this!
19. Another definite integral problem for you BUT this one is not as simple as #17. This time the antiderivative will be
an inverse trig. function (so be sure that you can evaluate things like section 4.3 HW # 5 and 6).
20. Be able to solve simple related rates problems (section 2.5)
21 – 22. Be able to evaluate definite integrals using substitution (section 4.1). Note: Be sure to practice a VARIETY of
problems involving different integrands (some involving trig functions (like HW #7 section 4.1 and some involving “e”
0
like
 xe
x2
dx )
1
0
x
 xe dx
2
1
u  x2
du  2 xdx 
1
du  xdx
2
1 u
1
1
1
1
1
1 1
1 e
e du  eu ]10  e0  e1  1   e   e 

21
2
2
2
2
2
2 2
2
0
23. Given a function know how to tell things like….where it is continuous, where it has a max or min, where it is
increasing or decreasing, where the derivative exists, where the second derivative is positive and negative. (sections 2.9,
2.10, 2.12)
24. If you are given the derivative of a function, be able to tell where the original function has relative max(s) and min(s)
(section 2.9). See Test #4 question #4 for an example.
25. Given the acceleration function of some particle along with two initial conditions (one for velocity and one for
position), be able to find the position function. (section 3.1)
Remember that in order to find the velocity function just integrate the acceleration function (use the initial condition
given for velocity to HELP you find the constant of integration that you get in your velocity function). THEN integrate
your velocity function to obtain your position function AND now use the initial condition given for position to find the
constant of integration for the position function.
26. Be able to find out where a given function is decreasing (section 2.9).
f  x  0
27. Make sure that you understand L’Hopital’s Rule and WHEN it applies! (section 2.11)
If you obtain
0
0
or

when you do direct substitution then it applies so long as the function meets the criteria

spelled out in L’Hopital’s rule (go read it)
28. Be sure that you understand the various properties of the definite integral that we covered (section 3.4)
29. Be able to evaluate definite integrals for basic sine and cosine functions (section 3.4)
30. Be able to evaluate a definite integral for a piece-wise defined function (section 3.4)
Remember that IF the piece-wise defined function is defined differently on one part of the interval that you are
integrating over than an another then you must split the integral up using properties of the definite integral!
31. Be able to solve a related rates problem (section 2.5). Be sure to practice a variety of different types, including ones
where angles change (obviously the one I select for your test must be easily doable without a calculator). In addition to
your notes and HW you can try the following…
(Note: “Usually” when an angle is changing your “equation” will involve a trig function)
The top of a 25 foot ladder is sliding down a vertical wall at a constant rate of 3 feet per minute. When the top of the
ladder is 7 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the
wall?
A. 
7
ft / min
8
B. 
7
ft / min
24
dy
ft
dx
 3
Find
when
dt
min
dt
Equation : x 2  y 2  252
2x
C.
7
ft / min
24
y  7 ft.
dx
dy
dx
y dy
7 ft 
ft  7 ft
 2y
0 


 3

dt
dt
dt
x dt
24 ft  min  8 min
Note: x was found to be 24 WHEN y=7 by using the Pythagorean theorem.
D.
7
ft / min
8
E.
21
ft / min
25
32. Be able to find points of inflection for a given function (section 2.10). Remember just because an x-value is a
“possible” point of inflection, it doesn’t mean that it actually is.
Example: What is the Point of Inflection for f  x    x  1  5 x  2 ?
5
A.
 3,1
B.
 1,3
C.
f  x    x  1  5 x  2 
 0,0
D.
1,3
f   x   5  x  1 1  5 
5
4
E.
3,1
f   x   20  x  1
3
only "possible" p.o.i is at x =  1
Set up a table and check to see there is a sign change in f   x  as you move across x =  1.
f  1   1  1  5  1  2  0  5  2  3
5
Answer   1,3
33. Be able to find limits at infinity! (section 2.11) Here are a couple of problems for you to practice BUT be sure to
review your notes and HW!!!
3x 4  6 x  1
x  2 x 3  5 x 2  5
A.
3
2
B. 0
C.
9 x 2  4 x  20
x  3  4 x  5 x 2
A. 
B. 0
C. 
Find lim
Find lim
4
3
D. 
9
5
1
5
D. 3
E. 
E.
9
5
Be sure to review your notes on limits at infinity. There are QUICK and easy ways to do these type of problems!
34. Be able to apply the extreme value theorem to find an absolute max of absolute min value of a function on a given
interval. (section 2.7)
Example: What is the maximum value of the function f  x   2 x3  3x2 12 x  1 on
A.  3
B. 0
C.
f  x   2 x 3  3x 2  12 x  1 
2
D. 6
2,3 ?
E. 8
f   x   6 x 2  6 x  12  6  x 2  x  2   6  x  2  x  1
critical numbers are 2, 1 Note that BOTH of these are in our interval (that is not always the case)
f  2   3
f  1  8
f  2   19
f  3  8
So the absolute maximum of the function on the given interval is 8
35. Be able to determine intervals on which a function is increasing or decreasing (section 2.9).
Example: On what interval(s) is f  x   2 x3  3x2  36 x decreasing?
A.
 , 3
B.
 2,3
C.
 3, 2
Take the derivative and find the critical numbers 2
D.
and
 2,3
E.
 2, 
3 . Break up the domain of f(x), your table will have 3
intervals (since you have two critical numbers). The sign of the derivative is only negative on the interval from -3 to 2 so
that is the only interval on which the original function is decreasing.
36 and 37. Review over the rectilinear motion problems from section 2.9
Example: A particle moving along a horizontal line has position function s  t   2t 4  4t 3  2t 2  8 . Find it’s
acceleration function. When does the object speed up and slow down? When does the object change direction?
Note: The one on your final exam will work out MUCH nicer (and quicker) than this one BUT this still gives you one to
practice that you haven’t seen (you can find others in your notes and HW etc.). Critical numbers will be 0, ½, and 1
(places where the velocity is zero….i.e. the only places where the object MIGHT change direction). Your PPOI will be
3 3
 .789 and
6
3 3
 .211 (your acceleration function is a quadratic that doesn’t factor so use the quadratic
6
formula…obviously you won’t have THIS issue on the Final Exam as no calculators are allowed on this part).
a  t   4  6t 2  6t  1
Speeds up on  0,.211 ; .5,.789  ; 1,  
Slows down on .211,.5  ; .789,1
Changes direction at
1
and 1 second.
2
38. Be able to find the value of “c” guaranteed by the Mean Value Theorem for derivatives (section 2.8)
f c 
f b  f  a 
ba
c
39. If c is a positive real number, find an expression for
 3x
in terms of c. What if c were a negative real number?
0
Draw a picture of f  x   3x . If c is a positive number then f  c   3c but if c is a negative number then
f  c   3c  3 c . Use your knowledge of this type of “net signed area accumulator” function to derive expressions for
the “net signed” are under your graph.
Note: If c is negative you should come up with
HOWEVER since c is positive isn’t
3
3
c c whereas when c is positive you should come up with c 2
2
2
3 2 3
c = c c anyway?
2
2
40. Be able to evaluate a definite integral involving an absolute value function. (like section 3.4 HW # 8)
41 – 45. Be sure that you understand Newton’s Method (section 2.6) , the first derivative test (section 2.9) , the second
derivative test (section 2.100, the mean value theorem for derivatives (section 2.8) and the mean value theorem for
integrals (section 3.4). In particular make sure that you know exactly what each of these is used for in Calculus!
46 – 50. Several indefinite integral problems. Most of these are VERY quick and require little if any work at all! Be sure
to know ALL of the integration formulas and rules that we have covered in this course!
THIS TEST WILL FAVOR THE EXTRAORDINARILY PREPARED STUDENT!