Notesale: Turn your study into money

Document Preview

Extracts from the notes are below, to see the PDF you'll receive please use the links above

---

Math 1232

Section R
...
)
The set of first elements of a relation or function is its domain
...

The set of second elements of a relation or function is its range
...

Vertical line test: If no vertical line cuts a graph in more than one point, then the graph is
the graph of a function
...
If f(x) = 3x2  2x, find
a
...
f

FG 3 IJ
H 2K

c
...


f ( x  h)  f ( x)
h

1

2
...

b
...
Graph the function y  6  x  x 2 ; find the domain and range
...

5  2 x if x  1

if 1  x  5 , find the following function values:
4
...
f(4)

5
...
f(1)

c
...
f(5)

e
...
3

Finding Domain and Range

The set of first elements of a relation or function is its domain
...

The set of second elements of a relation or function is its range
...


DOMAIN OF A FUNCTION
1
...

2
...
To find the domain, set the radicand > 0
and solve for x
...

3
...
Set the denominator = 0 and solve for x
...

1
...
Find the domain and range of each function:
a
...
f ( x)  x 1

3

3
...
f ( x ) 
x 5

b
...
The percent of the U
...
population that were foreign-born can be modeled by the
function f ( x)  0
...
00397 x 2  0
...
581 , where x is the number of
years after 1900
...
Use the graph to estimate the domain and the
range
...
4

Slope and Linear Functions

Slope of a Line
The slope of a line is a number that measures the steepness of the line
...


A line with 0 slope represents
a constant function
...


A vertical line has undefined slope
and is not a function
...
The
slope of a horizontal line is 0
...
The
slope of a vertical line is undefined
...
(8, 4) and (3, 2)
2
...
Graph the line
...
3x  2 y  6

Write the equation of the line described
...

2
4
...
horizontal, passing through (5, 3)

6
...

Variable cost = (cost per unit)(number of units)
Fixed costs remain constant regardless of the number of units produced
...
If x units are produced and sold,
P(x)  R(x)  C(x)
Break-even occurs when Revenue = Cost, or when Profit = 0
...
A computer manufacturer has a fixed cost of $3300 and a variable cost of $85 for each
computer produced
...
If x is the number of computers
produced and sold,
(a) Write the equation that represents total cost
...


(c) Write the equation of the profit function
...


(e) Find the break-even value
...
5

Nonlinear Functions

A quadratic function has the form f ( x)  ax 2  bx  c
...
Its graph is a
parabola that opens downward (is concave down) if a is negative, and the vertex is a
maximum point
...
The y-coordinate of the
2a
vertex can be found by evaluating the function at the x-coordinate of the vertex
...

2a
A polynomial function of degree n is a function of the form
f ( x)  a0 x n  a1 x n1    an1 x  an

where a0 , a1 ,
...
[All exponents of the
variables are nonnegative integers
...
Find the vertex and graph the function

f ( x)   x 2  x  6

A rational function is the quotient of two polynomial functions
...
Example
g( x)
6x  5
f ( x)  2
x  3x

2
...

a
...
Find the domain and then graph the square-root function f ( x)  x

4
...
In the case where n is negative or rational, we
will need the following properties:
Rules of Exponents
m n
m n
Product Rule a  a  a
am
 a mn
Quotient Rule
Defn zero exponent: a 0  1, a  0
n
a

ea j  a
Power of a Product babg  a b
F aI a
Power of a Quotient G J 
H bK b
Power Rule

m n

m

m
Defn rational exponent: a n  n a

mn

m

Defn negative exponent: a 1 

m m

m

m

an 

m

Also,

if a x  a y , then x  y

1
, a0
a

1
, a0
an

( a  1)
9

5
...


3

x4

6
...

a
...


6

3
x

b
...
A supply equation expresses the relationship between the unit price and the
quantity supplied
...

The law of demand states that the quantity demanded will increase as the price
decreases
...

Market equilibrium occurs when the quantity of commodity demanded is equal to the
quantity supplied
...

6
...

Demand: q  5800  60 x Supply: q  600  40 x

7
...

5
x
Demand: q 
Supply: q 
x
4

10

Math 1232

Section 1
...
If the y-values (outputs) also get closer and closer to another number, then
that number is called the limit of f(x) as x approaches a
...
The limit L must be a unique real number
...
Consider the function f(x) = 2x – 5, and suppose we want x to get closer and closer to 3
...

As x gets closer to 3 from the left, y gets closer to 1
...

As x→ 3+, y → 1, or lim f ( x)  1

x 3

Since y → 1 as x→ 3 from both the left and the right
of 3, we say that lim f ( x)  1
x 3

11

y

4
2

-5

-4

-3

-2

-1

1

2

3

4

5

x

Note that it is also true that f(3) = 1
...


-2
-4
-6
-8
-10

x2  4

...
(Why?) Complete the table:

2
...

Write this mathematically:_________________
As x approaches 2, from the right of 2, y approaches ___________
(2, 4)

Write this mathematically:_________________

x2  4
Thus, the limit of f ( x) 
as x → 2 is _____________
...
(a) Given the graph of the function, evaluate lim f ( x) , if it exists
x 0

 x  3 if x  0
f ( x)  
3x if x  0
y

8
6
4
2

-5

-4

-3

-2

-1

1

2

3

4

-2

5

x

-4
-6
-8
-10
-12
-14

(b) Graph the function and evaluate lim f ( x) , if it exists
x1

2 x  3 if x  1

f ( x )  2
if x  1
  x 2  6 if x  1

y

12
10
8
6
4
2

-5

-4

-3

-2

-1

1

2

3

-2

4

5

x

-4
-6

Limits at Infinity

1

...


4
...
1
−0
...
001
−0
...
1
0
...
001
0
...
For the function f ( x) 
(a) lim f ( x) (b)

y

1
, Complete the table and find
x

lim f ( x)

x

x 
10
100
1000
10,000

f(x)

x  
-10
-100
-1000
-10,000

f(x)

y

5
4
3
2
1

6
...

x 3
x 
x 3

14

Math 1232

Section 1
...

The limit of a constant function is that constant
...

“plug and chug”
1
...

(a) lim 3
(b) lim 4 x

b g

b g

x 2

x2

c

hc

(c) lim x 2  1 x 2  4
x 2

h

(d)

FH

lim x 3  2 x

x 1

IK

Now look at the graph of 1(d) and compare your answer
...

1
...

2
...

3
...


This expression is undefined, and the limit does not exist
...
If you get a fraction with 0 in both numerator and denominator,

0
, we call this
0

indeterminate form
...
Then substitute c into the
reduced function and simplify to get the limit
...
Evaluate the limits, if they exist
...

y

f ( x) 

-4

18
16
14
12
10
8
6
4
2

x  3x  2
x 1
2

-3

-2

-1

1

-2
-4
-6
-8
-10
-12
-14

2

3

4

5

x

Confirm your answer by completing the table
...
5
0
...
99
0
...
9999

2
1
...
1
1
...
001
1
...
The following polynomial functions are continuous over the entire domain,  ,  
y

20
18

y

y

10

20

16
8

14

10

12

6

10
8

4

6
4

-5

2

-4

-3

-2

-1

1

2

3

4

2
-8

-7

-6

-5

-4

-3

-2

-1

-2
-4
-6

1

2

3

x

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

-10

x

-2
-4

-8

f ( x)  ( x  3)

2

-20

f(x) = −x + 5

f ( x)  x  4 x
3

Note that there are no “jumps” or “holes” in the graphs
...


17

5

x

4
...

y

14

y

12
10

20

8
6
15

4
2
-1

1

-2

2

3

4

10

5

x

-4
-6

5

-8
-10
-12

-5

-4

-3

-2

-1

1

2

3

-14

f ( x) 

1
x2

f ( x) 

4

5

x

x3  4x
x2

The function f is continuous at x = c if all of the following conditions are met:
1
...
lim f (x) exists
3
...

Note: 1
...

2
...

3
...
Is the function f ( x)  3x 2  2 x continuous at x = 2? Why or why not?

6
...
Is the function f ( x)  
continuous at x = 2? Why or why not?
4 x  7 if x  2

8
...
If it is not, identify where it is
discontinuous and which condition fails to hold
...
3

Average Rates of Change

The average rate of change of y with respect to x, as x changes from x1 to x2, is the ratio
y y
of the change in output to the change in input: 2 1
x2  x1

1
...

800
700

700

Number of units sold

600

600

500

480

400
300

300

200
100
0

1
0

1

2

3

4

5

6

Money spent (in thousands of dollars)

Find the average rate of change of N as x changes from 1 to 2; 2 to 3; 3 to 4; 4 to 5
...
For the function f ( x)  x3  3x , find the average rate of change as
a
...
x changes from 1 to 3

20

Difference Quotients as Average Rates of Change
y

y2 = f(x+h)

y1 = f(x)
x

x1 = x

x2 = x + h

If we let x2  x  h , then
y2  y1 f ( x2 )  f ( x1 ) f ( x  h)  f ( x) f ( x  h)  f ( x)



x2  x1
x2  x1
( x  h)  x
h
And the average rate of change is called a difference quotient
...

h
The difference quotient is also the slope of the secant line from
( x, f ( x)) to ( x  h, f ( x  h))
...
For f ( x)  3x  4 , find the difference quotient when
a
...
x = 2 and h = 0
...
find a simplified form of the difference quotient

21

4
...


5
...


22

Math 1232

Section 1
...


For a function y = f (x),
its derivative at x is the
function f  defined by
f ( x  h)  f ( x)
f ( x )  lim
h0
h
provided the limit exists
...


EXAMPLE:
1
...

1
...
f ( x  h)  f ( x)

23

3
...
lim

h 0

f ( x  h)  f ( x )
h

Thus, f ( x) 

SLOPE OF TANGENT LINE
The derivative of a function gives a formula to determine the slope of a tangent to the graph
at any given point:
The slope of the tangent to the graph of y = f(x) at point A x, f  x  is





f ( x  h)  f ( x)
h0
h
( x)
...
That is, m  f
is found by evaluating the derivative at x1
...
Find the slope of the tangent line to y = f(x) = x2 at the point (-3, 9)
...
Given y  f ( x)  3x 2  2 x , find
(a) the derivative of f(x) at any point (x, f(x))

(b) the slope of the tangent line at (1, 5)

(c) the equation of the tangent to y  3x 2  2 x
at (1, 5)

2
find
x
(a) the derivative of f(x) at any point (x, f(x))

4
...
5)

(c) the equation of the tangent to f ( x) 

2
at
x

(4, 0
...


2)

A function f (x) is not differentiable at a point
x = a, if there is a vertical tangent at a
...

Example: g(x) is not continuous at –2, so g(x) is not differentiable at x = –2
...
5

Derivatives: The Power Rule and the Sum-Difference Rule

Power Rule: If f(x) = xn, where n is a real number, then f ( x)  nx n1
...
]
Note: If we take the derivative of a function with respect to x, we can write

d
dx

Note: To use the Power Rule, u n cannot be in a denominator or under a radical
...
]
Coefficient Rule: If f ( x)  c  u( x) , where c is a constant and u(x) is a differentiable function
of x, then
f ( x )  c  u ( x )
...


EXAMPLE:
1
...
Give answers in the form of the original
problem
...

[The derivative of a sum (or difference) of functions is the sum (or difference) of the
derivatives of each function
...
Find the derivatives of the functions
...
Find the slope of the tangent to the curve f(x) = x3 at x = 2
...


28

1
2

4
...


5
...
Verify by
graphing the curve and the tangent line
...
Find all points on the graph of f ( x)   x3  6 x 2 for which the tangent line has slope −15
...
Find all points on the graph of f ( x)  3x3  6 x2  45x  4 where the tangent line is
horizontal
...


8
...

a
...


b
...

dt

c
...
What does this mean?

30

Math 1232

Section 1
...

Product Rule:

EXAMPLES
1
...
Find dy/dx if y  3x 7  4 8x 6  6x 4  9







3
...


u(x)
, u(x) is the numerator and v(x)is the denominator, and both
v(x)
are differentiable functions of x, then
(denom)  (numer)  (numer)  (denom)
f
(denom)2
This says that the derivative of a quotient is the denominator times the derivative of the
numerator minus the numerator times the derivative of the denominator, all divided by the
square of the denominator
...
If f (x) 

5
...

(x)
x 5

2 x2  5x
x2  1

, find f 
...
Find an equation of the tangent line to the graph of the function f ( x ) 

3x
at the
x 2
2

point (2, 3)
...
Naomi’s Greenhouse finds that the cost, in dollars, to grow x hundred petunias is
C ( x)  750  20 x , and the revenue for selling x hundred petunias is R( x)  95 x
...
Find the average cost, average revenue, and average profit when x hundred petunias are
grown and sold
...
Find the rate at which the average profit is changing when 1500 petunias are grown and
sold
...
7

The Chain Rule

If y  u n , where u is a differentiable function of x, then
y  n(u n1 )  u

Note: To use the Chain Rule, u n cannot be in a denominator or under a radical
...
Find y  if y = (x2 + 1)4

2
...
Differentiate the function y  3x3  4 x  1

4
...



...
Find the derivative of g ( x)  1  x2 1  2 x2

6
...
Use the Product Rule, then the Chain Rule


...


35

7
...
2 x , in thousands of dollars, from
the sale of x items
...


36

Math 1232

Section 1
...

EXAMPLES
1
...
Find the first four derivatives of f ( x)  5x 3  3x 2  9 x  6

3
...
Find the first and second derivative of y  

 4x  5 

37

5
...
Suppose a particle travels
according to the equation s  100t  16t 2  200 where s is the distance in feet and t is the
time in seconds
...
So acceleration is dv/dt
...

a
...


b
...


(0  t  9) gives the consumer price index
6
...
2t 3  3t 2  100
(CPI) of an economy, where t = 0 corresponds to the beginning of 2000
...
It is possible for C (t ) to be
positive and C (t ) to be negative at t = c
...
“Inflation is slowing
...
Find the inflation rate at the beginning of
C(t) CPI
2006 (t = 6)
...
Show that inflation was moderating at
that time
...
1

Using First Derivatives to Find Maximum and Minimum Values and Sketch Graphs
AN ALGEBRA INTRODUCTION
Consider the function f (x)  x 2  6
...
The domain of the function is

...

The vertex is actually called the relative maximum
...

A relative minimum is a point on the graph of a function where it changes from decreasing to increasing
...
The
critical values divide the domain of the function into intervals, and only at these values will the function
change from increasing to decreasing or vice versa
...


Determining the Intervals Where a Function is Increasing or Decreasing
1
...

2
...


f (c) is positive, then f is increasing on that interval
...
If f (c) is negative, then f is decreasing on that interval
...
If

Relative Extrema (Maximum or Minimum) will occur only at critical values from the first derivative
...
Find the first derivative of the function
...
Set the derivative = 0 and solve for x
...
Values that make
undefined are also critical values
...
Determine the sign of f ( x) to the left and right of each critical value
...

 If f ( x) changes sign from negative to positive as we move across a critical value c,
then the point (c, f(c)) is a relative minimum
...



39

EXAMPLES
1
...
Find the relative
maxima and minima of the function, if they exist
...


1

1

f ( x )  x 4  x 3  3x 2  8
4
3
4
3

40

3
...


f ( x) 

x
x4

f ( x)  3 x  2

5
...
01x 2 1000, where x is the number of units
produced and sold
...


41

Math 1232

Section 2
...
A function can increase at an increasing
rate or a decreasing rate
...

A point of inflection is a point where the graph changes concavity
...

y

10

The function f ( x)  x3 is concave down on the

8

interval  , 0  and concave up on the interval

6
4

 0,  
...


-4
-6
-8
-10

The function is increasing at a decreasing rate on
the interval  , 0  , and increasing at an
increasing rate on the interval  0,  
...
Find the second derivative of the function
...
Solve f  equal to 0 and solve for x
...

3
...

4
...

5
...

6
...
Use the original
function to get the y-coordinate of the inflection point
...
Find the relative extrema, points of inflection, where the curve is concave up or concave down
...
Find the points of inflection and where the curve is concave up or concave down
...
Suppose that the daily sales (in thousands of dollars) of a product is given by

S 

x3 3 x 2

 1 where x is
6
2

thousands of dollars spent on advertising
...


Note: the function is increasing on the interval (0, 6)

Math 1232

44

Math 1232

Section 2
...

Below is the graph of f ( x) 
y

2 x2 1
x2  4
Note that as x approaches 2 from the right, the
function values get larger and larger,
approaching ∞
...


10
8
6
4
2

-5

-4

-3

-2

-1

1

2

-2

3

4

5

x

So lim f ( x)   and lim f ( x)  
...

Similarly the line x = −2 is also a vertical
asymptote
...
Also, as x approaches −∞, the
function values get closer and closer to 2
...

The line x = a is a vertical asymptote if any of the following are true:

lim f ( x)  , lim f ( x)   , lim f ( x)  , lim f ( x)  




x a

x a

x a

x a

The graph of a function never crosses a vertical asymptote
...

Note: If a rational function is simplified, then if x = a makes the denominator equal to 0, the line x = a is
a vertical asymptote
...

x 

x 

The graph of a rational function may or may not cross a horizontal asymptote
...

1
...

2
...

3
...
However, if the degree of the numerator is exactly one more than the denominator
degree, the there is a slant asymptote
...
Determine the vertical asymptotes for f ( x) 

2x 1

...
Determine the vertical asymptotes for f ( x)  2

...
Determine the horizontal asymptote for f ( x) 

4x 1

...
Determine the horizontal asymptote for f ( x) 

x 1

...
Sketch the graph of the function
...

4
f ( x) 
x 3

6
...
Indicate the intercepts, asymptotes, where it is increasing or
decreasing, where any relative extrema occur, where it is concave up or concave down, and
where any points of inflection occur
...
Sketch the graph of the function
...

2x 1
f ( x) 
x3

Math 1232

48

Math 1232
Values

Section 2
...

An absolute minimum occurs at a point on the graph of a function on a closed interval
[a, b] where the y-value is the smallest
...

Finding the absolute extrema of f on a closed interval [a, b]
...
Find the critical values of f that lie in (a, b) from f 
...
Compute the value of f at each critical value and f(a) and f(b)
...
The absolute minimum corresponds to the smallest value of f in Step 2
...

EXAMPLES
1
...


Math 1232

49

2
...


3
...
5, 10]
...
Find the absolute maximum and absolute minimum, if they exist, for f ( x)  x 2  3x  2 on
the entire domain,  ,  
...
An apartment complex has 100 two-bedroom units
...
To maximize profit,
how many units should be rented out? What is the monthly profit realizable?

6
...
The monthly revenue function is R( x)  0
...
00002 x2
...
To maximize its profits, how many CD’s
should be produced each month?

Math 1232

51

Math 1232
Section 2
...
After making a list of
what quantities are given in the problem, decide what quantity is to be maximized or minimized
...
Then use the derivative to find the
maximum or minimum
...
The owner of a ranch has 3000 yards of fencing to enclose a rectangular piece of grazing land
along the straight portion of a river
...
No fencing is required along the river
...
Suppose a travel agency charges $300 for a certain tour when the tour group has 25 people,
but will reduce the price by $10 per person for each additional person above the 25
...
The price per unit for Pilot V5 pens is p  50  0
...

The cost for producing x pens is C ( x)  500  0
...
How many pens must be sold to maximize
profit? What is the maximum profit? What is the price per pen needed to maximize profit?

Math 1232

53

Math 1232 Section 2
...

If C ( x) is the total cost for the production of x units, then the marginal cost at x, denoted C ( x) ,
st
is the approximate cost of the (x + 1) item
...

If P( x) is the profit from the production and sale of x units, then the marginal profit at x,
st
denoted P( x) , is the approximate profit from the (x + 1) item
...
Suppose C ( x)  55x 2  37,500 and R( x)  x3  10 x2  30 x  20
...
The total profit P(x)

b
...
The marginal cost, marginal revenue, and marginal profit when 100 units are produces
...
Suppose that the monthly cost, in dollars, of producing x carpets is
C ( x)  0
...
06 x2  16 x  800 , and that currently 50 carpets are produced monthly/
a
...
What would be the additional cost of increasing production to 55 carpets?

c
...
Use the marginal cost to estimate the difference in cost between 50 and 55 carpets
...
Use the answer from part (d) to predict C(55)
...
The average cost for a company to produce x units of a product is given by
Use A( x) to estimate the change in average cost as production goes from 80 to 81 units
...


4
...
Find the marginal revenue
...
1

Exponential Functions

Rules of Exponents
Product Rule
Quotient Rule

a a  a
m

n

am
a

n

m n

 a mn

Defn zero exponent: a 0  1,

ea j  a
Power of a Product babg  a b
F aI a
Power of a Quotient G J 
H bK b
Power Rule

m n

m

m
Defn rational exponent: a n  n a

mn

m

Defn negative exponent: a  n 

m m

m

a0

1
an

m

m

Also,

if a x  a y , then x  y

( a  1)

If a is a positive real number and a  1 , then the function f ( x)  a x is an exponential function
...
Graph y  3x
What is the domain of the function?

2
...
718, are important in many applications
...
Graph y  e x

4
...
That is,
Find the derivative of each function:
1
...


f ( x) 

Math 1232

x
ex

d x
e  ex
dx

2
...
f ( x)  4 x 2e x

58

The derivative of e to some power is the product of e to that power and the derivative of the
d f ( x)
power
...
f ( x)  e4 x

5
...


f ( x)  4  e 3x

e

e

j

j

8
...
Determine the intervals where the function f ( x)  e x is increasing and where it is
decreasing
...

2

10
...
Find the inflection point(s) of f ( x)  e x

Math 1232

2

2

60

12
...
09t , where t is the number of years
...
A company’s total cost, in millions of dollars, is given by C (t )  160  80et , where tis the
time in years since the start-up date
...
2

Logarithmic Functions

A logarithm is defined as follows: log a x  y means a y  x
1
...


23  8

b
...


log5

1
64

2
...


log10 100  2

1
 3
125

3
...


IMPORTANT LOGARITHMS
The logarithm with base 10 is the common logarithm and is written log
...
The logarithm with base e is the natural logarithm and is written ln
...
In calculus, the main focus is the natural logarithm
...
Graph the function y = ln x
...
loga 1  0
[because a 0  1 ]
2
...
loga a k  k
[inverse property]
4
...
If log a x  log a y , then x = y
Product Property 6
...
)
Quotient Property

Power Property

x

7
...
)
8
...
)

5
...
log5
5

b
...
Write as the sum or difference of two logarithms containing no exponents
...


d

id

i

ln x  1 4 x  5

b
...


Use properties of logarithms to write each expression as a single logarithm
...
ln z  ln y
b
...
Solve the equation

1 3x
e
 0
...
y  4ln x

11
...
y  x 2 ln x

ln x
x3

The derivative of the natural logarithm of a function is one divided by the function, times the
derivative of the function
...


y = ln 5x



14
...


f ( x)  ln x 3



15
...


y  ln 




x4 


 3x  1 



17
...
Determine the intervals where the function is increasing and where it is decreasing
...
A model to consumers’ response to advertising is given by N ( x)  1000  200ln x, x  1 ,
where N(x) is the number of units sold and x is the amount spent on advertising in thousands of
dollars
...
How many units were sold after spending $5000 on advertising?

b
...
Find the maximum and minimum values of N, if they exist
...
What is lim N ( x) ? Does it make sense to spend more and more dollars on advertising?
x 

Math 1232

66

Math 1232

Section 3
...
Its derivative is f ( x)  3e2 x  2  f ( x)  2 , which is 2 times
the original function
...

dy
That is,
 ky if and only if y  cekx for some constant c
dx
dA
1
...
So
P(t )  cekt or P(t )  P0ekt

2
...
That is, the balance grows at a rate of

dP
 0
...
Find the function that satisfies the equation
...
06
...
Suppose that $200 is invested
...
In what period of time will the investment of $200 double itself?

Math 1232

67

Doubling Time
The growth rate k and the doubling time T are related by kT = ln 2
3
...
04 billion at the beginning of 2000
...
016, or 1
...
per year
...
016P , where t is the time in years after 2000
...
Find the function that satisfies the equation, with P0 = 6
...
016

b
...


c
...

4
...
A function that models the growth of sales of iPods is
The logistic equation P(t ) 

f ( x) 

53
...
6664 x

where x is the number of years after 2000
...
Graph the function
...
Find the number of iPods in the years 2003, 2005, 2010
...
Find the rate of change of f(x)
...
4
Applications: Decay
dP
The equation
 kP, where k  0 , shows P to be a decreasing function of time, and the
dt
solution P(t )  P0e kt shows it to be decreasing exponentially
...
The
amount present at time t is P0
...
A company finds that its daily sales begin to fall after the end of an advertising campaign,
dS
and that the rate of change is
 200S , where t is the number of days after the campaign
dt
ends
...
Let S0 represent the amount of sales when t = 0
...


b
...

k

Math 1232

69

2
...
The decay
rate of this isotope is 0
...
What is its half-life?

3
...
Interest is
compounded continuously at 5%
...
6 Elasticity of Demand

Retailers and manufacturers often need to know how a small change in price will affect the
demand for a product
...
To measure the sensitivity of demand to a small percent
increase in price, economists calculate the elasticity of demand
...
(The response to price change is considerable)
The demand is said to be unitary if E ( x)  1
...
(Price changes cause relatively small changes in
demand for a product
...
Find the elasticity of the demand function q  D( x )  20  x when
5
a
...
the price is $60
c
...


2
...
Find the elasticity
...
At what price is the elasticity of demand equal to 1?

c
...
At what prices is the elasticity of demand inelastic?

e
...
For the demand equation q  D( p ) 

600
, find
( p  2)2

a
...
The elasticity at p = 6, stating whether the demand is elastic, inelastic, or has unit
elasticity

c
...
1

Antidifferentiation

A function F(x) is called an antiderivative of a function f(x) if, for every x in the domain of f,
F ( x)  f ( x)
...
But the function could also be f ( x)  x 3  4 because
f ( x)  x 3 , because
dx
d 3
x  4  3x 2
...

n 1
The process of finding the antiderivative is called integration
...
We can denote the
indefinite integral of f(x) by f ( x) dx
...

1
...


ze

5
...


j

8  x 2 / 3 dx

4
...
Find the integral

  2e

x



 x3 dx

7
...
Find the integral   2x  dx
x


Initial Conditions
The constant of integration is often of interest in applications
...
This point is called an initial
condition
...
Find a function f such that f ( x)  2 x  5 and f (2)  1
...
A company determines that the marginal revenue R , in dollars, from selling the xth unit of a
product is given by R( x)  x 2  5
...


11
...
Find the demand function if it is known that 1000 units of the
product are demanded by consumers when the price is $15 per unit
...
3

Area and Definite Integrals

Without knowledge of calculus, to find the area under a curve involves drawing nonoverlapping
rectangles under the curve, finding their area, and adding these areas
...

1

0
...
6

0
...
2

0

0
...
4

0
...
8

1

Note that we would get a better approximation of the area under the curve if we take more
rectangles
...
The area is actually defined as the limit as
n   of the sum of n rectangles under the curve
...
Find the area under the graph of f ( x)  4 x  x2 on the interval [0, 4]
...


Find the area under the graph of y  e2 x on the interval [0, 5]
...

We write F (b)  F (a) by F(x)| b
a

z FH

x 3  4 dx

1

3

4

3
...
Evaluate

 3x dx
a

z FGH
5

5
...
Evaluate

z

7
...
The management of an office equipment company has determined that the daily marginal cost
function associated with producing battery-operated pencil sharpeners is given by
C( x)  0
...
006x  4 where C'(x) is measured in dollars per unit and x denotes the
number of units produced
...


9
...

A( x)  100e01x where x is time in hours and 0  x  10
...


Math 1232

78

Math 1232

Section 4
...


x

2
...


x

2



2

 1  2 x dx

2 x  3  2 dx

2

 4x



3

 (2 x  4) dx

If du is “incomplete,” (missing a constant), you can “introduce a constant” by multiplying inside
the integral and dividing outside the integral by that constant
...


z ex  4j  x dx
2

Math 1232

4

79

5
...


2 x dx
 1  x2

2

3

7
...


x
 2 x e dx
2

9
...
Find the demand function given that

D = 13,000 when x = $3 per unit

---