Review Guide for MAT220 Midterm Exam Part II. Fall 2015
Part 2 is worth 50% of your Midterm Exam grade. NO CALCULATORS are allowed on this portion of the Midterm Exam.
You may NOT use your own scratch paper for this Multiple Choice Exam. Instead, your instructor will give you a sheet 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 Midterm Exam consists of two pages. Each page has a blank side that can also be used for
scratch work if needed. You will turn in your sheet of scratch paper with your Midterm Exam (even if they have nothing
written on it). NO PARTIAL CREDIT will be given on this portion of the Midterm Exam. You will have 65 minutes to
complete this portion of the Midterm Exam (assuming that you show up on time).
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.
A. Be sure that you understand the relationship between the derivative of a function and the derivative of its inverse
function (section 2.4).
If the slope of the tangent line at 1, 4  of f  x  is 8 , give a point that lies on f 1  x  AND the slope of the tangent line
there.
B. Be able to find derivatives for inverse trigonometric functions (know the formulas!) (section 2.4). Don’t forget to
apply the chain rule when appropriate.
h  x   arctan x 4
f  x   arc sec  3x 2 
g  x   arccos
 x
C. Be able to calculate derivatives of various functions (sections 1.6 and 2.1 – 2.3). Be sure that you know when (and
how) to apply the chain rule, product rule and quotient rule. Be sure that you know when you can use the power rule
and when you need to use logarithmic differentiation (section 2.3). There will be several problems on the Midterm
Exam that will allow you to demonstrate your ability to calculate derivatives.
f  x 
x3
cos x
y  x x 1
 
y  ln e x
2
D. Given a rational function know when a particular function value exists and when it doesn’t (vertical asymptote, hole).
Also know when the limit exists at a particular x value and when it doesn’t (section 1.4).
h  x 
 x  2  x  1
 x  3 x  2 
List any x value for which the function value does not exist. List any x value for which the limit
of the function does not exist.
E. Be sure that you understand the relationship between continuity and differentiability (section 1.5).
Can you give an example of a function that is differentiable but not continuous?
Can you give an example of a function that is continuous but not differentiable?
F. Be sure that you understand left hand and right hand limits (section 1.3).
What must be true about the left hand and right hand limit at x=2 in order for the limit at x = 2 to exist?
G. Make sure that if you are given the graph of the derivative of some original function that you can answer questions
about the original function by looking at the graph of the derivative and interpreting what it is telling you about the
original function (section 1.5).
If this is the graph of f   x  THE DERIVATIVE OF f  x  then answer the following questions about f  x  .
1. What is the slope of the tangent line to the graph of f  x  when x = 3?
2. List any x- coordinates where f  x  would have a relative maximum?
3. List any x- coordinates where f  x  would have a relative minimum?
4. On what interval(s) would f  x  be increasing? decreasing?
H. Given a function f  x  be able to find f  16  for example! (various sections but you will NOT be asked to use the
limit definition of derivative here so you can apply the relevant derivative rules to f  x  )
f  x  5 x
find
f  16 
I. Given a rational function equation, be able to find the coordinates (x,y) of a removable discontinuity. (section 1.3 is
where we discussed discontinuities)
f  x 
x2  x  6
x 2  7 x  10
J. Be able to find the limit of a rational function at a specific “c” value where direct substitution yields 0/0. (section 1.2)
Lim
x 2
x 4  16
x3  5 x 2  6 x
K. Be able to find the second derivative of a relation at a given point. For example…
Find the second derivative of x3 y 2  400 at
 2,5 .
This can best be done using implicit differentiation (section
2.3). First find y and be sure to simplify it. Then use the quotient rule on y to find y  . Substitute in (2,5) to evaluate
y  . This is one of two “longer” problems that should probably be done towards the end of your testing time. By the
way, the answer to THIS sample problem is
75
.
16
L. Know how to tell by working with the equation of a piecewise defined function whether it is continuous and whether
it is differentiable. See your class notes for some examples. Also can you solve the following?
Find all values of “a” so that the following piecewise defined function is differentiable everywhere.
sin x  2 x x  0
f  x  
2
x0
3ax  x
M. Make sure that you can do a few related rates problems. (section 2.5)
This was the most recent section that we covered in class so it should be most fresh in your mind. If you have paid
attention to the examples that we did in class and did your homework then you will have no trouble doing the
problems of this type on the test!
NOTE: The sample problems provided here are NOT intended to be the only problems that you look at while preparing
for this Midterm exam. Be sure to READ the information provided after each alphabetic letter above and spend time
searching your old tests, notes and homework to review the mentioned topics. Make sure that you understand the
mathematics relevant to each topic NOT just how to do a set of particular problems!
THIS TEST WILL FAVOR THE EXTRAORDINARILY PREPARED STUDENT!