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Kathryn
Bollinger, February 4, 2014
Concepts to Know #1
Math 142
A.8, 1.0, 1.1 topics, 1.2 topics, 1.3, 1.5, 3.1-3.3
• Prerequisite Knowledge
Functions: each element x corresponds to exactly
one element y (passes vert. line test)
Domain: all possible x values
(independent variable) know how to find this
Range: all possible y values
(dependent variable)
Function Notation
f (a) represents the function value (y) when
x=a
Interval Notation
(a, b) = exclude endpoints a and b
[a, b] = include endpoints a and b
f (x + h) − f (x)
Difference Quotient:
h
Basic/Parent Functions : know basic graphs
Identity Function (y = x)
Square Function (y = x2 )
Cube Function (y = x3 )
√
Sq. Root Function (y = x)
√
Cube Root Function (y = 3 x)
Absolute Value Function (y = |x|)
Know properties of lines
Be able to find the equation of a line
Be able to interpret slope and
y-int.
Know applications (linear cost, linear supply
and demand)
Linear Supply and Demand
S(x) = p = mx + b
D(x) = p = mx + b
**All points on supply and demand
curves are of the form
(x, p) = (quantity, price)!!**
Know properties of quadratic functions
Be able to determine if a parabola opens up
or down
Know how to find the vertex
Know how to find the zeros
• A.8 - Some Special Functions and Graphing
Techniques
Be able to classify different functions based on
their equation and/or shape of their graph
Polynomials:
P (x) = an xn + an−1 xn−1 + · · · + a1 x + a0
(n is a positive integer)
Leading Coefficient = an
(determines direction of the ends of the
graph as x → ±∞)
Degree of the Poly = n (if an 6= 0)
Examples of Polynomials:
Linear, Quadratic, Cubic, Quartic
Rational Functions: a polynomial divided by a
polynomial
Power Functions: f (x) = xn
Special Case: Root Functions
Know the difference between even and odd
root functions
Graph Transformations
Know how each of the following will change
the graph of f (x) based on the value of c:
f (x) + c, f (x) − c (vert. shift)
f (x + c), f (x − c) (horiz. shift)
cf (x) (vert stretch or shrink)
−f (x) (reflection about x-axis)
Be able to shift a given graph
Be able to formulate a function’s equation
by knowing how a basic function is being
transformed
Be able to choose a basic function and list
the transformations (in the correct order)
performed on that function which will produce a function with a given equation
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Kathryn
Bollinger, February 4, 2014
• 1.1 Topics - Increasing/Decreasing,
Concavity,
Continuity,
and PiecewiseDefined Functions
Piecewise-Defined Functions
Be able to graph
Be able to formulate from a word problem
Write the absolute value of a function as its
equivalent piecewise-defined function
Be able to find the domain
From a graph, know where a function is increasing
or decreasing, and concave up or down
Be able to identify where a graph is
continuous
• 1.2 Topics - Break-Even Analysis and
Market Equilibrium
Cost, Revenue and Profit
Linear Cost: C(x) = cx + F where c is the
cost to produce each unit and F is the
fixed costs
Linear Revenue: R(x) = sx where s is the
selling price of each unit
(a constant price)
Revenue when price changes:
R(x) = px where p is demand function
Profit: P (x) = R(x) − C(x)
Break-Even Point
R(x) = C(x) (or P (x) = 0 to find quantity)
x = break-even quantity
y = break-even revenue
Be able to find the quantity to be sold where
quadratic revenue or profit is maximized
Be able to find the price where quadratic revenue
or profit is maximized
Be able to find the maximum revenue or profit
Equilibrium Point
Supply = Demand
x = equilibrium quantity
y = equilibrium price
Linear Depreciation: V (t) = mt + b
|m| = rate of depreciation
b = value of asset at time zero
Scrap Value = lowest value asset attains
• 1.3 - Exponential Functions
Basic Function: f (x) = a ∗ bx (b > 0, b 6= 1)
Exponential Growth when a > 0; b > 1
Exponential Decay when a > 0;
0<b<1
Domain: (−∞, ∞)
Range: (0, ∞)
HA: y = 0 in one direction
Be familiar with the graphs of basic exponential
functions
Be able to find the domain of any exponential function
Know exponential properties in order to simplify
expressions and solve equations
Simple Interest
I = P rt where I = interest earned, P =
principal (money originally deposited), r
= interest rate (in decimal format), and t
= time of investment (in same time units
as r)
A = P + I = P (1 + rt) where A = the accumulated amount in the account after time
t
Compound Interest
m = the number of times your money is
compounded per year (annually=1, semiannually=2, quarterly=4, monthly=12,
weekly=52, daily=365)
Know how to use TVMSolver on the calculator
N = mt total # of times money is compounded
I= interest rate per year (in % form)
P V = P (amount initially deposited into
an acct)
P M T = $0 in this class
F V = A (the amount of money in the
account after t years)
P/Y = number of payments made per
year (m in this class)
C/Y = m = number of times your account is compounded per year
PMT: should be set to END
Continuously Compounded Interest:
A = P ert
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Kathryn
Bollinger, February 4, 2014
1.3 continued
Effective Rate of Interest:
The simple interest rate that would produce
the same accumulated amt in one year as
the given rate compounded m times per
year
EF F (r, m) on the calculator when compounded m times a year
r
e − 1 when compounded continuously
Used to compare different bank accounts
• 1.5 - Logarithms
One-to-One Functions
Each y-value of a function corresponds to exactly one x-value
Passes both vert. and horiz. line tests
Have inverses
Inverse Functions
Know that domain and range of a function’s
inverse are the range and domain of the
original function, respectively
Logarithmic Functions
Basic Function: f (x) = loga x
(a > 0, a 6= 1)
Inverses of Exponentials
Domain: (0, ∞)
Range: (−∞, ∞)
Base e is “natural” log: ln x (on calc.)
Base 10 is “common” log: log x (on calc.)
Be familiar with the graph of a basic
logarithmic function
Be able to find the domain of any logarithmic
function
Know properties of logarithms in order to
simplify expressions and solve equations
Change of Base Formula:
ln a
log a
=
logb a =
log b
ln b
• 3.1 - Limits
One-Sided Limits
Be able to find graphically
Be able to find numerically from a table of
values
lim f (x) = L ⇔ lim f (x) = lim f (x) = L
x→a
x→a+
x→a−
Know the difference in a limit value and a function
value
Algebraically Calculating Limits
Know the properties of limits
Know when to use Direct Substitution to find
the value of a limit
If an indeterminate form is found, algebraically simplify (factor and cancel, multiply by the conjugate, get common denominators,...) to determine the limit
value, if it exists
Know how to find limits of piecewise-defined
functions
Be able to draw conclusions about the graph
of a function from limit values
Vertical Asymptotes
Know limit definition (as x → a number, the
function → ±∞)
Know how to determine if a limit “=” ±∞
Definition of Continuity at x = c
f (c) is defined
lim f (x) exists
x→c
lim f (x) = f (c)
x→c
Be able to find where a function is continuous or
discontinuous
Piecewise-Defined Functions
Be able to find discontinuities
Be able to find variable values to make a
piecewise-defined function continuous everywhere
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Kathryn
Bollinger, February 4, 2014
• 3.2/3.3 - Rates of Change and The Derivative
Know how to find the average rate of change
∆y
(AROC) between two points: ∆x
(slope of a secant line)
Be able to estimate the instantaneous rate of
change (IROC) from a set of data by finding
the average of two AROC
Understand that finding the IROC of f (x) at
x = a is equivalent to finding the slope of the
tangent line at x = a and is found by
f ′ (a) = lim
h→0
f (a + h) − f (a)
h
If f (t) represents the position of an object, then
f ′ (a) represents the velocity of the object at
t=a
Be able to find the equation of the line tangent to
a curve at a specified point
The Derivative as a Function
Know how to find f ′ (x) by using
the limit definition:
f ′ (x) = lim
h→0
f (x + h) − f (x)
h
Be familiar with the different notations for
the derivative:
f ′ (x) = y ′ =
df
d
dy
=
=
f (x)
dx
dx
dx
Be able to sketch the graph of f ′ (x), from the
graph of f (x)
Values of Slopes of Tangent Lines
→ y-values on f ′ (x)
Points where f (x) has horizontal tangent
lines → x-intercepts of f ′ (x)
Know
where
f (x)
is
increasing/decreasing
→ f ′ (x) above or below x-axis
Know when a function is nondifferentiable
(the derivative DNE):
At sharp turns or corners
At a vertical tangent
Where the function is not continuous