APPLICATIONS OF DIFFERENTIATION
In this chapter we will use derivatives to graph functions while finding maximums and minimums,
increasing and decreasing intervals, concavity intervals, and points of inflections. We will then look at
optimization applications where we can find the price to charge to maximize our profit. Also, we will
look at maximizing area given a certain perimeter.
(2.1) Using First Derivatives to find Max and Min Values
In this section we will use derivatives to find intervals of increasing and decreasing for a function which
will then lead us into finding relative extrema (maximums and minimums).
Let us begin by defining critical values:
Critical Values are values for x where f ( x) 0 or where f (x) is undefined.
The critical values are boundaries values that will define where a function is increasing and decreasing.
Critical values can also determine the relative extrema.
Relative Extrema: the relative maximum and minimum values of a function. Graphically:
To find extremas:
1) Find f (x)
2) Find the critical values, where f ( x) 0 or where f (x) is undefined
3) Using the critical values as boundaries, test each interval using f (x) to determine if the
function is increasing or decreasing and thus determine the relative extrema.
c1 c2 c3
4) Find the ordered pair (x, y) using f(x)
Examples: Find the intervals where the function is increasing or decreasing and any relative extrema.
1) f ( x) 5 3x x²
2) f ( x)
3) f ( x) x 4 4 x ³ 20 x ² 12
(2.2) Using Second Derivatives to find Max and Min
In this section we will see that the second derivative, f ( x) ,also provides useful information about the
graph of f ( x) . We will look at the concavity of the graph.
CONCAVE UP: CONCAVE DOWN:
f is increasing as x increases f is decreasing as x increases
the slope of the tangent lines is increasing the slope of the tangent lines is decreasing
Test for Concavity:
1) If f ( x) 0 for all x in the interval, then f(x) is concave up
2) If f ( x) 0 for all x in the interval, then f(x) is concave down
Inflection Points: points where the graph changes from concave up to concave down. These will occur
where f ( x) 0 or is undefined. Graphically:
To Find Concavity Intervals and Points of Inflection:
1) Find where f ( x) 0 or is undefined
2) Using the values as boundaries, test each interval using f (x) to determine if the function is
concave up or concave down
* if f ( x) 0 then f(x) is concave up
* if f ( x) 0 then f(x) is concave down
3) Find any Points of Inflection (x, y) using f(x)
Examples: Find where the functions are concave up or down, and any points of inflection.
1) f ( x) x 3 3x 2 5 x 1
2) f ( x) x 4 20 x 3
Examples: Graph the following functions using the 1st and 2nd derivatives
1) f ( x) x³ 3x² 9 x 7
2) f ( x) x 4 8 x ³ 18 x ² 8
3) f ( x) (2 x 4) 5
1st DERIVATIVE 2nd DERIVATIVE
increasing and decreasing intervals concavity intervals
relative extremas relative extrema
critical points points of inflection
(2.3) Curve Sketching: Asymptotes and Rational Functions
We have looked at sketching continuous functions using the tools of calculus.. We now will look at
some discontinuous functions, most of which are rational functions.
Rational Functions are in the form: f ( x) where P(x) and Q(x) are polynomials.
The graphs of Rational Functions typically involve Asymptotes. The following are definitions when
The line x = a is a vertical asymptote if any of the following limit statements are true:
lim f ( x) lim f ( x) lim f ( x) lim f ( x)
x a x a x a x a
The graph will never cross a vertical asymptote. When the rational function is completely simplified,
vertical asymptotes will occur at values that make the denominator zero.
1) Determine the vertical asymptotes of the function: f ( x)
2) Determine the vertical asymptotes of the function: f ( x)
x 6x 8
The line y = b is a horizontal asymptote if either or both the following limit statements is true:
lim f ( x) b lim f ( x) b
The graph of a rational function may or may not cross a horizontal asymptote. Horizontal asymptotes
occur when the degree of the numerator is less than or equal to the degree of the denominator.
1) Determine the horizontal asymptote of the function: f ( x)
3x x 2
4 x3 3x 2
2) Determine the horizontal asymptote of the function: f ( x) 3
x 2x 4
1) When the degree of the numerator is the same as the degree of the denominator, the line y = a/b is a
horizontal asymptote. Where a is the leading coefficient of the numerator and b is the leading
coefficient of the denominator.
2) When the degree of the numerator is less than the degree of the denominator, the x-axis, or the line
y = 0, is the horizontal asymptote.
SKETCHING RATIONAL FUNCTIONS:
a) Find the x and y intercepts of the graph. Recall that the x-intercept is the points where
y f ( x) 0 and the y-intercepts is the points where x = 0.
b) Find all asymptotes (vertical and horizontal)
c) Use the first derivative to find relative extrema, increasing and decreasing intervals
d) Use the second derivative to find any points of inflections, concave up and concave down
e) Sketch the graph finding extra points if needed.
1) Sketch the graph of the function: f ( x)
2) Sketch the graph of the function: f ( x)
3) Sketch the graph of the function: f ( x)
x 2 16
(2.4) Using Derivatives to find Absolute Extrema
Absolute Max: the largest value of a function in its domain
Absolute Min: the smallest value of a function in its domain
The Absolute Extrema exists on a closed interval [a, b] either at the endpoints of the domain or the
critical values. Graphically:
Finding Absolute Extrema:
1) Find all critical values, where f ( x) 0 or is undefined
2) Evaluate f at the critical values and the endpoints to find the functional value
3) Absolute Max exists at the largest f value
Absolute Min exists at the smallest f value
Examples: Find any Absolute Extrema
1) f ( x) x 2 4 x 5 on the interval [-1, 5]
2) f ( x) x 4 8 x 2 3 on the interval [-3, 0]
(2.5) Business and Economic Applications
In this section we will look at optimization problems. Optimizing is finding the maximum or minimum
value where we can maximize profits, maximize revenue, or minimize costs.
Formulas to Use:
Profit = Revenue – Costs P( x ) R ( x ) C ( x )
Revenue = quantity x price R( x) x p( x)
R( x) C ( x) at maximum profit
1. A company wants to build a parking lot along the side of one of its buildings using 800 feet of
fence. If the side along the building needs no fence, what are the dimensions of the largest
rectangle possible to maximize the area? What is the maximum area?
2. A farmer wants to make 2 identical adjoining rectangular enclosures along a straight river. If he
has 600 yards of fence, and if the sides along the river need no fence, what should be the
dimensions of each enclosure if the total area is to be maximized?
3. Find the maximum profit and the number of units that must be produced and sold in order to
yield the maximum profit given the following revenue, R(x) and cost, C(x) functions.
R( x) 5 x C ( x) 0.001 x 2 1.2 x 60
4. Riverside Appliance is marketing a new refrigerator. It determines that in order to sell x
refrigerators, the price per refrigerator must be p( x) 280 0.4 x . It also determines that the
total cost of producing x refrigerators is given by C ( x) 5000 0.6 x 2 .
a) Find the total revenue R(x).
b) Find the total profit P(x).
c) How many refrigerators must the company produce and sell in order to maximize profit?
d) What is the maximum profit?
e) What price per refrigerator must be charged in order to maximize profit?
5. A baseball team plays in a stadium that holds 55,000 spectators. With ticket prices at $10, the
average attendance had been 27,000. When ticket prices were lowered to $8, the average
attendance rose to 33,000. How should ticket prices be set to maximize revenue?
6. During the summer months Terry makes and sells necklaces on the beach. Last summer he sold
the necklaces for $10 each and his sales averaged 20 per day. When he increased the price by $1,
he found that he lost two sales per day. If the material for each necklace costs Terry $6, what
should the selling price be to maximize his profit?
(2.6) Marginals and Differentials
R(x): Revenue Function – the income for selling x-units R(x) = price x quantity
C(x): Cost Function - the cost for producing x-units
P(x): Profit Function - the profit for selling x-units P(x) = R(x) – C(x)
When we take the derivative of the R(x), C(x), and the P(x) we get the rate of change in revenue, cost,
and profit. This is called the Marginal Revenue, Marginal Cost, and Marginal Profit.
MR(x): Marginal Revenue - the approximate revenue from the (x+1)st item.
MC(x): Marginal Cost – the approximate cost of the (x+1)st item.
MP(x): Marginal Profit – the approximate profit from the (x+1)st item.
1) Suppose that the daily cost, in dollars, of producing x radios is:
C ( x) 0.002 x3 0.1x 2 42 x 300
and currently 40 radios are produced daily.
a) What is the current daily cost?
b) What would be the additional daily cost of increasing production to 41 radios daily?
c) What is the marginal cost when x = 40?
d) Use marginal cost to estimate the daily cost of increasing production to 42 radios daily.
(2.7) Implicit Differentiation and Related Rates
Implicit differentiation is a techniques used to find when the function is not and can not be put in
the form y f (x) . That is, the y can not be isolated in the equation. If we need to find it is much
easier if the y is isolated.
Consider the function xy 1 to find
Consider the function x² y ² 1 to find
Consider the function x² y ³ y 0 to find
1. Use implicit differentiation to find for: x² y ² 25
2) Use implicit differentiation to find for: p 2 p 2 x 40
3) Find and then find the slope of the curve at the given point: 2 x3 4 y 2 12 at (-2, -1)
4) Find and then find the slope of the curve at the given point: 2 x3 y 2 18 at (2, -1)
5) Evaluate at (6, 2) for the function xy y ² 12
6) Find the rate of change of profit with respect to time given the function:
R( x) 50 x 0.5 x 2 and C ( x) 10 x 3
when x = 10 and dx/dt = 20 units per day.
7. The number of traffic accidents per year in a city of population p is predicted to be A 0.002 p 2 . If
the population is growing by 500 people a year, find the rate at which traffic accidents will be rising
when the population is p=40,000.
8. A rocket fired straight up is being tracked by a radar station 3 miles from the launching pad. If the
rocket is traveling at 2 miles per second, how fast is the distance between the rocket and the tracking
station changing at the moment when the rocket is 4 miles up?