MATH 150 Review
Lines
Limit Theorems
y = mx + b
(slope-intercept equation of a line)
Suppose that
lim f ( x)
and
x a
y = b is the equation of a horizontal line
lim g ( x )
both exist. Then we have the
x a
following theorems.
Limit of a Polynomial Function
Let p(x) be a polynomial function and a any number. Then
lim p( x)  p(a)
xa
Limit of a Rational Function
Let r(x) = p(x)/q(x) be a rational function where p(x) and q(x) are polynomials.
Let a be a number such that q(a) does not equal zero. Then,
y
m
x
The slope of a nonvertical line is given by the formula
lim r ( x)  r ( a )
xa
Derivative via Limits
The basic rules of differentiation are obtained from the limit definition of the
derivative. There are three main steps for calculating the derivative of a function
f(x) at x = a.
1.
2.
3.
Write the difference quotient f (a  h)  f (a) .
h
Simplify the difference quotient.
Find the limit as h approaches zero.
Continuity and Differentiability
We say that a function is continuous at x = a if its graph has no breaks or gaps as
it passes through the point (a, f(a)). If a function f(x) is continuous at x = a, it
should be possible to sketch its graph without lifting the pencil from the paper at
the point (a, f(a)).
Secant Lines
To calculate f’(x) using a secant line approximation of the tangent line
1.
Calculate the difference quotient f ( x  h)  f ( x) .
2.
Let h approach zero.
h
The quantity f ( x  h)  f ( x) will approach f’(x).
3.
h
f’(a) =
lim
h 0
f ( a  h)  f ( a )
h
If f(x) is differentiable at x = a, then f(x) is continuous at x = a.
A function f(x) is continuous at x = a provided the following limit relation
holds:
lim f ( x)  f (a )
xa
In order for this to hold, three conditions must be fulfilled.
1.
f(x) must be defined at x = a
2.
lim f ( x ) must exist
x a
3.
The limit
lim f ( x ) must have the value f(a)
x a
Basic Rules for Differentiation
If we take the derivative of the velocity function v(t), we get what is called the
acceleration function
a(t) = v’(t) = s’’(t)
Approximating the Change in a Function
Consider the function f(x) near x = a. We know that
f ( a  h)  f ( a )
 f ' (a)
h
When h is small, hf’(a) is a good approximation to the change in f(x). In
applications, hf’(a) is calculated and used to estimate f(a+h) – f(a).
Marginal Cost
If we differentiate the function f’(x), we obtain what is called the second
derivative of f(x), denoted f’’(x). So, we have
Economists often use the adjective marginal to denote a derivative.
If C(x) is a cost function, then the value of the derivative C’(a) is called the
marginal cost at production level a. The number C’(a) gives the rate at which
costs are increasing with respect to the level of production when the production is
currently at level a.
d
f ' ( x)  f ' ' ( x)
dx
Graphing
A relative maximum point is a point at which the graph changes from increasing
to decreasing.
A relative minimum point is a point at which the graph changes from decreasing
to increasing.
The maximum value of a function is the largest value that the function assumes on
its domain.
The minimum value of a function is the smallest value that the function assumes
on its domain.
Note: Functions might or might not have maximum and/or minimum values.
We say that a function is concave up at x = a if there is an open interval on the xaxis containing a throughout which the graph of f(x)lies above its tangent line.
We say that a function is concave down at x = a if there is an open interval on the
x-axis containing a throughout which the graph of f(x)lies below its tangent line.
Rates of Change
The average rate of change of f(x) over this interval is the change in f(x)
divided by the length of the interval.
An inflection point is a point on the graph of a function at which the function is
continuous and the concavity of the graph changes, i.e., goes from concave up to
concave down, or concave down to concave up.
average rate of change of f ( x) y f (b)  f (a)
over the int erval a  x  b
  x 
ba


The x-intercept is a point at which a graph intersects the x-axis. (x,0)
The y-intercept is a point at which the graph intersects the y-axis. (0,y)
In the special case where b is a+h, the value of b – a is (a + h) – a or h, and the
average rate of change of the function over the interval is the difference
quotient
Graphs sometimes straighten out and approach some straight line as x increases
(or decreases). Theses straight lines are called asymptotes.
f ( a  h)  f ( a )
h
Asymptotes of a graph may be horizontal, vertical or diagonal.
First Derivative Rule
The derivative f’(a) measures the (instantaneous) rate of change of f(x) at x =
a.
If f’(a) > 0 then f(x) is increasing at x = a.
Motion, Velocity and Acceleration
If f’(a) < 0 then f(x) is decreasing at x = a.
The average velocity from t = 2 to t = 2 + h is
Second Derivative Rule
dis tan ce traveled s(2  h)  s(2)

time elapsed
h
If f’’(a) > 0, then f(x) is concave up at x = a.
If f(‘’(a) < 0, then f(x) is concave down at x = a.
If s(t) denotes the position function of an object moving in a straight line, then
the velocity v(t) of an object at time t is given by
v(t) = s’(t)
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Cost, Revenue and Profit
C(x) = cost of producing x units of a product
R(x) = revenue generated by selling x units of a product
P(x) = R(x) – C(x) = the profit (or loss) generated by producing and selling x
units of the product.
R(x) = xp
The demand equation p = f(x) determines the total revenue function. If a firm
wants to sell x units, the highest price it can set is f(x) dollars per unit. The
revenue function becomes
R(x) = xp = xf(x)
More Rules for Differentiation
Product Rule
d
 f ( x) g ( x)  f ( x) g ' ( x)  g ( x) f ' ( x)
dx
Look for possible relative extreme points of f(x) by setting f’(x) = 0 and
solving for x.
Is the point a relative maximum point or a relative minimum point? How can
we tell?


Quotient Rule
d  f ( x)  g ( x) f ' ( x)  f ( x) g ' ( x)

dx  g ( x) 
g ( x)2
Check concavity at relative extreme point using second derivative.
Examine slope of nearby points on either side using the first
derivative.
Chain Rule
Look for possible points of inflection by setting f’’(x) = 0 and solving for x.
d
f ( g ( x))  f ' ( g ( x)) g ' ( x)
dx
1.
2.
When applying the chain rule, begin by identifying f(x) and g(x).
To differentiate f(g(x)), first differentiate the outside function f(x) and
substitute g(x) for x in the result. Then, multiply by the derivative of
the inside function g(x).
Alternate Notation for the Chain Rule
Given the function y = f(g(x)), set u = g(x) so that y = f(u). With this notation we
have y = f(u) and we note that
du
 g ' ( x)
dx
and
dy
 f ' (u )  f ' ( g ( x))
du
Using this notation, the chain rule states that
dy dy du


dx du dx
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Curves of the form y = ekx, where k is negative, have several properties in
common:
Related Rates
There are some applications where x and y are related by an equation, and
both variables are functions of a third variable t. Often the formulas for x and
y as functions of t are not known.
When we differentiate such an equation with respect to t, we derive a
relationship between rates of change
dy
dt
and
dx .
dt
We say that theses
Properties of Natural Logarithms
derivatives are related rates.
1.
The equation relating the rates may be used to find one of the rates when the
other is known.
2.
3.
4.
5.
Laws of exponents
The point (1,0) is on the graph of y = ln x [because (0,1) is on the
graph of y = ex]. So,
ln 1 = 0.
ln x is defined only for positive values of x.
ln x is negative for x between 0 and 1.
ln x is positive for x greater than 1.
ln x is an increasing function and concave down.
Differentiating Natural Logarithmic Functions
d
1
(ln x ) 
dx
x
Differentiating Exponential Functions
More Properties of Logarithms
Differential Equations
Suppose that y = f(x) satisfies the differential equation
The y is an exponential function of the form
More on Differential Equations
The function y = Cekt satisfies the differential equation
y’ = ky
Properties of Exponential Functions
Conversely, if y = f(t) satisfies the differential equation, then
Curves of the form y = ekx, where k is positive, have several properties in
common:
y = Cekt
for some constant C.
It is important to note that if f(t) = Cekt, then by setting t = 0, we have
f(0) = Ce0 = C
So, C is the value of f(t) at t = 0.
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Functions for compound interest
Rules for Antidifferentiation
When interest is compounded continuously, the compound amount A(t) is an
exponential function of the number of years t that interest is earned
A(t) = Pert
and A(t) satisfies the differential equation
A’(t) = r A(t)
Relative Rates of Change
Recall that the logarithmic derivative of a function f(t) is defined by the
equation
Constructing the (Riemann Sum) Rectangles
d
f ' (t )
ln f (t ) 
dt
f (t )
Given a continuous nonnegative function f(x) on the interval
The quantity on either side of this equation is often called the relative rate of
change of f(t) per unit change of t, since it compares the rate of change of f(t)
with itself.
1.
The percentage rate of change is the relative rate of change of f(t) expressed as
a percentage.
2.
In each subinterval, select a point (any point in the subinterval will
do). Let x1 be the point in the first interval, x2 be the point in the
second, etc. These points are used to form the rectangles that
approximate the area under the graph of f(x).
3.
Construct the first rectangle with height f(x1) and the first subinterval
as the base. The top of the rectangle touches the graph directly above
x1. Then,
The elasticity of demand E(p) at price p for the demand function q = f(p) is
defined to be
pf ' ( p)
f ( p)
Economists say that demand is elastic at price p0 if E(p0) > 1.
Economists say that demand is inelastic at price p0 if E(p0) < 1.
[area of first rectangle] = [height][width] = f(x1) Δx
The second rectangle rests on the second interval and has height f(x2).
Its area is
When demand is elastic at some price p0, E(p0) > 1 and 1 – E(p0) is negative.
So, R’(p0) is negative and R(p0) is decreasing. That is, an increase in price
will result in a decrease in revenue, and a decrease in price will result in
an increase in revenue.
[area of second rectangle] = [height][width] = f(x2) Δx
4.
When demand is inelastic at some price p0, E(p0) < 1 and 1 – E(p0) is positive.
So, R’(p0) is positive and R(p0) is increasing. That is, an increase in price
will result in an increase in revenue, and a decrease in price will result in a
decrease in revenue.
or
[area estimate] = [f(x1) + f(x2) +…+ f(xn)] Δx
Given a derivative F’(x), we must find the function F(x). The process of
determining F(x) from F’(x) is called antidifferentiation.
This sum is called a Riemann sum. It provides an approximation to
the area under the graph of f(x) when f(x) is nonnegative and
continuous
If F1(x) and F2(x) are two antiderivatives of the same function f(x), then F1(x)
and F2(x) differ by a constant. In other words, there is a constant C such that
As the number of subintervals increases indefinitely, the Riemann sums approach
a limiting value…the area under the graph.
F2(x) = F1(x) + C
If F’(x) = 0 for all x, then F(x) = C for some constant.
It can be shown that the Riemann sums still approach a limiting value as Δx
approaches zero. This number is called the definite integral of f(x) from a to b
and is denoted by
Suppose that f(x) is a function whose antiderivatives are F(x) + C. The
standard notation to express this fact is

 f ( x)dx  F ( x)  C

If we continue in this fashion, our estimate of the area under the graph
will be given by summing the area of these n rectangles
[area estimate] = f(x1) Δx + f(x2) Δx +…+ f(xn) Δx
Antidifferentiation
The symbol
Divide the x-axis into n equal subintervals, where n is some positive
integer. This subdivision is called a partition of the interval from a to
b. The width of the entire interval is b – a, so the width of each
subinterval is (b – a)/n. We will denote this width as Δx.
Δx = (b – a)/n
Elasticity of Demand
E ( p)  
a  x  b,
b
a
is called an integral sign, and the entire notation
f ( x)dx
That is
 f ( x)dx is called an indefinite integral and stands for the
antidifferentiation of the function f(x).
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Area and the Definite Integral
Suppose that f(x) is continuous on the interval

b
a
Future Value of an Income Stream
a  x  b.
Then
The future value, of a continuous income stream, of K dollars per year for N years
at interest rate r compounded continuously is
f ( x)dx

N
0
Ke r ( N t ) dt
is equal to the area above the x-axis bounded by the graph of y = f(x) from
x = a to x = b minus the area below the x-axis.
Fundamental Theorem of Calculus
where C is any constant.
Suppose that f(x) is continuous on the interval
an antiderivative of f(x). Then

b
a
a  x  b , and let F(x) be
f ( x)dx  F (b)  F (a)
This theorem connects the two key concepts of calculus – the integral and the
derivative.
F(b) – F(a) is called the net change of F(x) from a to b. It is represented
symbolically by
Area Between Two Curves
If y = f(x) lies above y = g(x) from x = a to x = b, then the area of the region
between f(x) and g(x) from x = a to x = b is
The Average Value of a Function
Let f(x) be a continuous function on the interval a  x  b . The definite
integral may be used to define the average value of f(x) on this interval.
The average value of a continuous function f(x) over the interval
a  x  b is defined as the quantity
1 b
f ( x)dx
b  a a
Consumers’ Surplus
The consumers’ surplus for a commodity having demand curve p = f(x) is

A
0
[ f ( x)  B] dx
where the quantity demanded is A and the price is B = f(A).
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