MATH 110
Ron Buckmire
Fall 2007
Study Guide for Exam II
The exam will cover material that appears on homeworks 13 through 24, Classes 13 through
24 and Quiz 4 through 7. Sections of the textbook included are all of Chapter 3 and Chapter 4
except for Section 4.4. I try to make the study guides complete, but there is no guarantee.
You should review homeworks, homework solutions, quizzes, handouts, labs, and class notes.
This might change, but my current plan is not to allow calculators.
Topics
 Derivatives
o know all the different notations for derivative covered on the homework
assignments
o compute a derivative using the limit definition(s)
o Evaluate a limit by interpreting it as a particular derivative, i.e. f  (a) as a limit of
a difference quotient
o use the limit definition of derivative to prove differentiation formulas
(e.g., ( f ( x)  g ( x))  f ( x)  g ( x) ). I will assume that you know the values of
sin x
cos x  1
 1 and lim
 0 . In fact I might give you information
the limits lim
x 0
x 0
x
x
about certain limits like lim
x 0
o
o
o
o
o
o
o
o
o
o
o
o
o
x
 1
 1 
 x
 e , and have you use these to prove the
derivative of an elementary function
Know the derivatives of elementary functions on Derivatives To Remember
compute using all the derivative formulas on the Rules of Differentiation handout
Ability to evaluate or estimate numbers such as f (3)
Give graph of f , sketch the graph of f  ; given a graph of f  sketch the graph of f
estimate derivatives with left estimates, right estimates, symmetric difference
quotients, and tables of values (i.e., successive approximations).
Equation of the line tangent to a function at (a,f(a)) is y= f (a)+ f  (a)(x-a)
Local linear approximation to estimate the value of a function near a value you
already know such as sin(3) (i.e. near π) or 3 7 (near 3 8 =2).
d3y
compute higher-order derivatives such as
dx 3
units
the units on a derivative are always function
variableunits
Physical interpretation of derivative as the rate of change of the output variable
relative to the input
When the function is increasing(decreasing), the derivative is positive(negative)
Differentiability implies continuity, a function not continuous at a point implies
the derivative does not exist at that point
Difference between average rate of change and instantaneous rate of change and
their graphical interpretation (secant line versus tangent line)
Review Exercises
Let f ( x)  x . Use the definition of derivative to compute f (x) .
Use the definition of derivative to prove that [Cf ( x)  D]  Cf ( x) .
If f ( x)  ln[(cot x)( 47) x ] , find f (x) .
Determine whether y  t sin t is the solution to the differential equation y   2 cos t  y .
ln( 47  t )  ln 47
(5) Find lim
. [Hint: recognize the limit as a derivative, and use
t 0
t
differentiation rules.]
(6) In this problem, you will prove the quotient rule in two different ways. [The two
purposes of this problem are (1) to see that the quotient rule is true, and (2) to get
differentiation practice.]
(a) Use the product and chain rules to continue the computation


 f ( x) 

  f ( x)g ( x)1   . After computing the derivative, use algebra to make your
 g ( x) 
answer look like the quotient rule.
f ( x)
(b) Let h( x) 
, so that g ( x)h( x)  f ( x) . Take derivatives of both sides and
g ( x)
f ( x)
solve for h(x) . While simplifying your answer, you will use again that h( x) 
.
g ( x)
(7) Several times, we’ve used the fact that differentiable implies continuous. (For example,
when sketching f  from the graph of f , you draw open circles in the graph of f  wherever
the graph of f is discontinuous.) Your task in this problem is to prove this fact. You’ll need
the “limits are nice” theorem, the definition of continuous, the version of the definition of
derivative that you developed in problem (13) on homework 6, and two applications of the
“introduce a canceling pair” algebra trick! Assume that f is differentiable at x  a , and fill
in the blanks to show that f is continuous at x  a .
lim f ( x)  lim  f ( x)  f (a )  ___ 
(1)
(2)
(3)
(4)


xa
xa
 f ( x)  f (a )

 lim 
 (___  ___)  same as first line' s blank 
xa
xa


 ___  ___  ___  ___
limits
are
nice
(8) Fill in the blanks, then find the value of k that makes f continuous.
x 2 x

 x 2 if x  __
f ( x)  
lim kx k if x  __

 x1 2 x 2
(9) Suppose f (x) is the number of snarfs living in Snarfville, x years from now. What are
the units on f  and f  ?
(10) Let f ( x)  1x . Find an equation for the tangent line at x  2 , and use it to estimate 11.7 .
(11) Suppose at a price $p, a quantity q of a commodity (IPods!) is sold, and q=f(p). Explain
in economic terms the meaning of the statements f(100)=240,000 and f  (100)=-29,000?
2