Definitions You Should Know:
1) A function f is increasing on an interval [ a, b ] if f ( x
1
) < f ( x
2
) whenever x
1
< x
2
2) A function f is decreasing on an interval [ a, b ] if f ( x
1
) > f ( x
2
) whenever x
1
< x
2
3) A function f is continuous at a if lim x → a f ( x ) = f ( a )
4) The limit definition of the derivative of f at the point a lim x → a f ( x ) − f ( a ) x − a
5) Another limit definition of the derivative of f at the point a lim h → 0 f ( a + h ) − f ( a ) h
6) The Fundamental Theorem of Calculus:
Suppose f is continuous on [ a, b ]. Then
Z x a) If g ( x ) = f ( t ) dt , then g
0
( x ) = f ( x ) a
Z b b) f ( x ) dx = F ( b ) − F ( a ), where F is any antiderivative of f , that is, F
0 a
= f .
7) An Antiderivative of a function f is a function F ( x ) such that F 0 ( x ) = f ( x ) for all x in the domain of f .
8) A number c is a Critical Point of a function f if either f
0
( c ) = 0 or f
0
( c ) is undefined.
9) We say that f ( c ) is an absolute maximum of f if f ( c ) ≥ f ( x ) for all x in the domain of f .
10) We say that f ( c ) is an absolute minimum of f if f ( c ) ≤ f ( x ) for all x in the domain of f .
11) We say that f ( c ) is a local maximum of f if f ( c ) ≥ f ( x ) when x is near c .
12) We say that f ( c ) is a local minimum of f if f ( c ) ≤ f ( x ) when x is near c .
13) The Mean Value Theorem for Integrals : If f is continuous on [ a, b ], then there exists a number c between a and b such that f ( c ) = f ave
=
1 b − a
Z b a f ( x ) dx

Functions:
ˆ
Know the different types of function classes that we talked about in Section 1.2.
ˆ
Know all of the different operations (shifting, stretching, reflecting) from Section 1.3 for getting new functions from old ones.
ˆ
Know how to work with the absolute value function, including its definition.
ˆ
Know how to do Exponential growth/decay problems.
ˆ Know how to algebraically solve for the inverse of a one-to-one function.
ˆ Know properties of logarithms and exponential functions, and how they relate to one another (including cancellation equations)
ˆ
Know how to find the average rate of change of a function on an interval. In general, if you want the average rate of change of a function f ( x ) on the interval [ a, b ], the equation is f ( b ) − f ( a ) b − a
Think about our example finding average velocity from Section 2.1.
ˆ
Know Point-Slope Form equation for finding the equation of a line between two given points.
ˆ
Know how to determine left and right-sided limits and two-sided limits from the graph of a function.
ˆ
Know how to determine where a function is increasing, decreasing, and continuous from its graph.
ˆ
Know the Limit Laws from Section 2.3 and how to apply them to compute limits algebraically.
ˆ Know how to find the Domain and Range for all of the kinds of functions we have discussed, and use these to determine where a function is continuous.
ˆ
Know how to take limits of functions to ∞ and −∞ .
ˆ
Know how to determine if the one-sided limit of a function is + ∞ or −∞ at a vertical asymptote.
ˆ
Know how to find Horizontal and Vertical asymptotes of functions.
Differentiation:
ˆ
Know how differentiability and continuity are related (see Section 2.7).
ˆ
Know how to determine properties of f from the graphs of f
0 and f
00
.
ˆ
Know how to determine properties of f from its graph (i.e. increasing, decreasing, concavity, etc.)
ˆ
Know how to draw graphs of functions with certain derivative and second derivative properties (as in Section 2.8).
ˆ Know all differentiation rules from Section 3.1.
ˆ
Know Product, Quotient, and Chain Rules and how to compute derivatives using them.
ˆ
Know derivatives of all basic functions.
ˆ
Know derivatives of Trig functions (from section 3.3)
ˆ
Know how to apply differentiation techniques to solve real world type problems (see
Section 3.8)
ˆ
Know how to find the absolute maximum and minimum of a function on a closed interval.
ˆ
Know how to find critical points of a function
ˆ
Know first and second derivative tests
ˆ
Understand how to formulate an optimization problem, and solve for the maximum and minimum asked for. (Section 4.6) This includes showing that your answer is indeed the maximum or minimum.
Integration:
ˆ Know indefinite integrals of basic functions (or equivalently, the general antiderivatives)
ˆ
Know how to solve for a function given its first or second derivative and information about the function itself.
ˆ
Know how to calculate definite integrals as the area under a curve

ˆ
Know how to calculate definite integrals using the Evaluation Theorem (i.e. using the antiderivative)
ˆ
Know Comparison Properties of Integrals (at the end of Section 5.2)
ˆ
Know integrals of all basic functions and trig functions
ˆ
Know how to use the Fundamental Theorem of Calculus to differentiate integrals. Also know how to do this when the limits are functions of x rather than x itself.
ˆ Know how to use the Substitution Method to evaluate integrals.
ˆ Know how to find the area of the region between two functions by integration.
ˆ Know how to compute the average value of a function over an interval (by integration) as well as how to determine when the average value occurs.
General Comments:
Since much of the final will be word problems, there are a couple of things to keep in mind.
1) If you are given the rate of change of a function (say for example, you are given the rate of change of the Population), and you are asked to find how much the population increased from time a to time b , then you should evaluate the definite integral of the rate of change from a to b . This is by the Fundamental Theorem of Calculus (or the Evaluation
Theorem) since
Z b
P
0
( t ) dt = P ( b ) − P ( a ) a
2) If you are given a function and you are asked to find the instantaneous rate of change , then you are being asked to compute the derivative.
3) Recall that the average rate of change of a function between a and b is just the difference of the function values divided by the difference b − a . That is, the average rate of change of the function f between a and b is f ( b ) − f ( a ) b − a
4) It would be prudent to look back at all of the graphical sections of your old exams.
Some similar things will be tested in the graphical section for the final.

Example Problems:
Problem 1) What is lim x →∞ e
− x
Problem 2) What is lim x →∞ e x
Problem 3) What is d dx e cx
Problem 4) What is
Z e cx dx
Problem 5) A shuttle is being launched from Cape Canaveral today. In order to enter orbit around the Earth, it must achieve a velocity of 11.2 km/s by the time it reaches a height of 700 km above the Earth. Assuming that the shuttle accelerates at a constant rate of 1 km/ s 2 , will the shuttle achieve orbit? You may assume that the initial velocity is 0 m/s and that the initial height prior to launch is also 0 m.
a) Write the equation for the acceleration a ( t ).
b) Find the equation for velocity, v ( t ).
c) Find the equation for position (in this case height), s ( t ).
d) Will the shuttle achieve orbit? (You must justify your answer).

Problem 6) a) Verify that the functions y = ( c − x 2 )
− 1
2
, where c is any constant, are solutions to the differential equation y
0
= xy
3 b) Find the particular solution to the differential equation from part (a) that satisfies the condition y (0) = 2.
Problem 7) a) Find the average value of the function g ( x ) =
√
3 x on the interval [1 , 8].
b) Find c such that g ( c ) = g ave
.

Problem 8)
Suppose the amount of capital that a company has at time t is C ( t ). The rate of change of the capital is called the net investment flow . Suppose that the net investment flow is t million dollars per year (where t is measured in years). Find the increase in capital from the fourth year to the eighth year.
Problem 9)
Find the area enclosed by the curves y = 2 x
2 and y = 4 + x
2
.