CHAPTER 3APPLICATIONS OF THE DERIVATIVE

3.1 Maxima and Minima

Extreme Values

1. Does 𝑓(𝑥) have a maximum or minimum value on 𝑆?

2. If it does have a maximum or a minimum, where are they attained?

3. If they exist, what are the maximum danminimum values?

The Existence of Extreme Values

Where Do Extreme Values Occur?

Steps for Finding Extreme Values of a Function 𝑓 on an Interval 𝐼1. Find the critical points of 𝑓 on 𝐼

2. Evaluate f at each of these critical points. The largest of these values is the maximum value; the smallest is the minimum value.

Examples

1. Find the maximum and minimum values of 𝑓 𝑥 = −2𝑥3 + 3𝑥2 on [−

1

2, 2].

2. Find the maximum and minimum values of 𝑓 𝑥 = 𝑥 + 2 cos 𝑥 on [−𝜋, 2𝜋].

3. Find the maximum and minimum values of 𝑓 𝑥 = 𝑥2/3 on [−1,2].

3.2 Monotonicity and Concavity

Increasing and Decreasing

Examples

Find where the given function is decreasing and where it is increasing.

1. ℎ 𝑧 =𝑧4

4−

4𝑧3

6

2. 𝑓 𝑥 =𝑥−1

𝑥2

Concavity

Concavity Theorem

Examples.

Find where the given function is concave up and where it is concave down.

1. ℎ 𝑧 =𝑧4

4−

4𝑧3

6

2. 𝑓 𝑥 =𝑥−1

𝑥2

More Examples

1. A news agency reported in September 2016 that the unemployment in eastern Asia was continuing to increase at an increasing rate. On the other hand, the price of food was increasing, but at a slower rate than before. Interpret these statements in terms of monotonicity and concavity.

2. Coffee is poured into the cup shown in the figure at the rate of 4 ml per second. The top radius is 3.5 cm, the bottom radius is 3 cm, and the height of the cup is 10 cm. This cup holds about 0.17 l. Determine the height ℎas a function of time 𝑡, and sketch the graph of ℎ(t) from time 𝑡=0 until the time that the cup is full.

Inflection Points

Let 𝑓 be continuous at 𝑐. We call (c, 𝑓 𝑐 ) an inflection point of the graph of 𝑓 if 𝑓 is concave up on one side of 𝑐 and concave down on the other side.

Points where 𝑓′′ 𝑐 = 0 or where 𝑓′′ 𝑐 does not exist are the candidates for points of inflection.

Examples

Find the points of inflection of the given function.

1. ℎ 𝑧 =𝑧4

4−

4𝑧3

6

2. 𝑓 𝑥 =𝑥−1

𝑥2

Show that the following statements are true.

1. A quadratic function has no point of infection.

2. A cubic function has exactly one point of inflection.

3.3 Local Extrema and Extrema on Open Intervals

Global Extrema vs Local Extrema

Where Do Local Extreme Values Occur?

The Second Derivative Test

Example.

Find the local extreme values of 𝑓 𝑥 = sin 𝑥 2/3 on −𝜋

6,2𝜋

3.

Extrema of Open Interval

Examples.

Find (if any exist) the maximum and minimum values of the given functions.

1. 𝐺 𝑝 =1

𝑝(1−𝑝)on (0,1).

2. ℎ 𝜃 = sin 𝜃 on 0,2𝜋 .

3.5 Graphing Function using Calculus

Graphing Function

Examples1. Sketch the graph of the given function.

a) 𝐹 𝑠 = 𝑠4 − 2𝑠2 − 3

b) 𝑤 𝑧 =𝑧2+1

𝑧

2. Sketch the graph of a function 𝑓 that has the following properties:a) 𝑓 is everywhere continuous;b) 𝑓 −3 = 1;c) 𝑓′ 𝑥 < 0 for 𝑥 < −3, 𝑓′ 𝑥 > 0 for 𝑥 > −3, and 𝑓" 𝑥 < 0 for 𝑥 ≠ −3.

3. Let 𝑓 be a continuous function with 𝑓 −3 = 𝑓 0 = 2. If the graph of 𝑦 = 𝑓′ 𝑥 is as shown in the figure, sketch a possible graph for 𝑦 = 𝑓(𝑥).

3.4 Practical Problems

Step-by-step Method for Practical Optimization Problem

Examples

1. For what number does the principal square root exceed eight times the number by the largest amount?

2. Find the points on the parabola 𝑦 = 𝑥2 that are closest to the point 0,5 .

3. A farmer wishes to fence off two identical adjoining rectangular pens, each with 900 square meters of area, as shown in the figure bellow. What are 𝑥 and 𝑦 so that the least amount of fence is required.

More Examples

4. A right circular cone is to be inscribed in another right circular cone of given volume, with the same axis and with the vertex of the inner cone touching the base of the outer cone. What must be the ratio of their altitudes for the inscribed cone to have maximum volume?

5. At 7:00 am one ship was 60 kilometers due east from a second ship. If the first ship sailed west at 20 kilometers per hour and the second ship sailed southeast at 30 kilometers per hour, when were they closest together?

6. A wire of length 100 centimeters is cut into two pieces: one is bent to form a square, and the other is bent to form an equilateral triangle. Where should the cut be made if (a) the sum of the two areas is to be a minimum; (b) a maximum? (Allow the possibility of no cut).

3.6 The Mean Value Theorem for Derivatives

MVT for Derivative

Illustrations

Decide whether the MVT for Derivative applies to the given function on the given interval. If it does, find all possible values of 𝑐; if not, state the reason.

1. 𝑓 𝑥 = 𝑥 ; 1,2

2. 𝑔 𝑥 = 𝑥 ; −2,2

3. 𝑆 𝑡 = sin 𝑡 ; −𝜋, 𝜋

Using MVT for Derivative

Example.1. Use the MVT to prove that

lim𝑥→∞

𝑥 + 2 − 𝑥 = 0

2. John traveled 112 km in 2 hours and claimed that he never exceeded 55 km per hour. Use the MVT to disprove John’s claim.

3.8 Antiderivatives

Antiderivative

addition vs subtractionmultiplication vs divisionexponentiation vs root takingthe second operation undoes the first, and vice versa

Example.Find an antiderivative of the function 𝑓 𝑥 = 4𝑥3 on −∞,∞ .

Notation and Rules for Antiderivative

𝐴𝑥 𝑥2 =1

3𝑥3 + 𝐶 or න𝑥2𝑑𝑥 =

1

3𝑥3 + 𝐶

More Rules

Examples

1. Evaluate the indicated indefinite integrals.

a. 𝑠 𝑠+1 2

𝑠𝑑𝑠.

b. 𝑡2 − 2cos 𝑡 𝑑𝑡 .

c. 3𝑦

2𝑦2+5𝑑𝑦 .

d. sin 𝑥 cos 𝑥 1 + sin 𝑥 2𝑑𝑥 .

2. Find 𝑓"(𝑥) 𝑑𝑥 if 𝑓 𝑥 = 𝑥 𝑥3 + 1.

3.8 Introduction to Differential Equation

What is a Differential Equation?

Find the 𝑥𝑦-equation of the curve that passes through −1,2and whose slope at any point on the curve is equal to twice the 𝑥-coordinate of that point.

Any equation in which the unknown is a function and that involves derivatives (or differentials) of this unknown function is called a differential equation.

A function that, when substituted in the differential equation yields an equality, is called a solution of the differential equation.

Examples.

Show that 𝑦 = 𝐶1 sin 𝑥 + 𝐶2 cos 𝑥 is a solution of 𝑑2𝑦

𝑑𝑥2+ 𝑦 = 0.

First-Order Separable Differential Equation

A first-order separable differential equation is an equation involving just the first derivative of the unknown function and is such that the variables can be separated, one on each side of the equation.

Examples.

1. Find the particular solution that satisfies the indicated condition.

1.𝑑𝑦

𝑑𝑥=

𝑥

𝑦; 𝑦 = 4 𝑎𝑡 𝑥 = 1.

2.𝑑𝑢

𝑑𝑡= 𝑢3 𝑡3 − 𝑡 ; 𝑢 = 4 𝑎𝑡 𝑡 = 0.

More Examples

1. A ball is thrown upward from the surface of the earth with an initial velocity of 96 feet per second. What is the maximum height that it reaches?

2. Starting at station A, a commuter train accelerates at 3 meters per second per second for 8 seconds, then travels at constant speed 𝑣𝑚for 100 seconds, and finally brakes to a stop at station B at 4 meters per second per second. Find (a) 𝑣𝑚 and (b) the distance between A and B.