Application of Derivatives

APPLICATIONS OF DIFFERENTIATION

APPLICATIONS OF DIFFERENTIATION

Here, we learn how derivatives affect the shape of a graph of a function and, in particular, how they help us locate maximum and minimum values of functions.

Increasing and Decreasing Function

Rate of change of quantities

APPLICATIONS OF DIFFERENTIATION

Many practical problems require us to minimize a cost or maximize an area or somehow find the best possible outcome of a situation.

In particular, we will be able to investigate the optimal shape of a can and to explain the location of rainbows in the sky.

Maximum and Minimum Values

In this section, we will learn:How to find the maximum and minimum values of a function.APPLICATIONS OF DIFFERENTIATION

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EXAMPLES

Here are some examples of such problems that we will solve in this chapter.

What is the shape of a can that minimizes manufacturing costs?

What is the maximum acceleration of a space shuttle? (This is an important question to the astronauts who have to withstand the effects of acceleration.)

EXAMPLES

Here are some more examples.

What is the radius of a contracted windpipe that expels air most rapidly during a cough?

At what angle should blood vessels branch so as to minimize the energy expended by the heart in pumping blood?

OPTIMIZATION PROBLEMS

These problems can be reduced to find the maximum or minimum values of a function.

Lets first explain exactly what we mean by maximum and minimum values.

MAXIMUM & MINIMUM VALUESA function f has an absolute maximum (or global maximum) at c if f(c) f(x) for all x in D, where D is the domain of f.

The number f(c) is called the maximum value of f on D.Definition 1

MAXIMUM & MINIMUM VALUESThe figure shows the graph of a function f with absolute maximum at d and absolute minimum at a.

Note that (d, f(d)) is the highest point on the graph and (a, f(a)) is the lowest point.

LOCAL MAXIMUM VALUE

If we consider only values of x near bfor instance, if we restrict our attention to the interval (a, c)then f(b) is the largest of those values of f(x)

It is called a local maximum value of f.

LOCAL MINIMUM VALUE

Likewise, f(c) is called a local minimum value of f because f(c) f(x) for x near cfor instance, in the interval (b, d).

The function f also has a local minimum at e.

MAXIMUM & MINIMUM VALUES

This corresponds to the fact that the origin is the lowest point on the parabola y = x2.

However, there is no highest point on the parabola.So, this function has no maximum value.Example 2

MAXIMUM & MINIMUM VALUES

You can see that f(1) = 5 is a local maximum, whereas the absolute maximum is f(-1) = 37.

This absolute maximum is not a local maximum because it occurs at an endpoint.

EXTREME VALUE THEOREMThe theorem is illustrated in the figures.

Note that an extreme value can be taken on more than once.

EXTREME VALUE THEOREMThis does not contradict the theorem because f is not continuous.

Nonetheless, a discontinuous function could have maximum and minimum values.

LOCAL EXTREME VALUES

The figure shows the graph of a function f with a local maximum at c and a local minimum at d.

LOCAL EXTREME VALUESIt appears that, at the maximum and minimum points, the tangent lines are horizontal and therefore each has slope 0.

LOCAL EXTREME VALUESIt appears that, at the maximum and minimum points,tangent lines are horizontal and therefore each has slope 0.

LOCAL EXTREME VALUES

We know that the derivative is the slope of the tangent line.So, it appears that f (c) = 0 and f (d) = 0.

Critical point and Stationary PointA function is said to be monotonic if its either increasing or decreasing.

The points for which f (x) =0 or does not exist are called critical points.

The stationary points are the points where f(x) =0 in the domain.

FERMATS THEOREMThe theorem does suggest that we should at least start looking for extreme values of f at the numbers c where either:

f (c) = 0

f (c) does not exist

FERMATS THEOREMSuch numbers are given a special namecritical numbers.

If radius of circle increasing at uniform rate of 2 cm/s, find rate of increasing of area of circle, at instant when radius is 20 cm.

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How Derivatives Affect the Shape of a GraphAPPLICATIONSOF DIFFERENTIATION

In this section, we will learn:How the derivative of a function gives us the direction in which the curve proceeds at each point.

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WHAT DOES f SAY ABOUT f ?

Here, tangents to these curves have been drawn at several points.

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CONCAVE UPWARDIn the first figure, the curve lies above the tangents and f is called concave upward on (a, b).

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CONCAVE DOWNWARDIn the second figure, the curve lies below the tangents and g is called concave downward on (a, b).

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CONCAVITYDEFINITION If the graph of f lies above all of its tangents on an interval I, it is called concave upward on I.

If the graph of f lies below all of its tangents on I, it is called concave downward on I.

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CONCAVITY

The figure shows the graph of a function that is concave upward (CU) on the intervals (b, c), (d, e), and (e, p) and concave downward (CD) on the intervals (a, b), (c, d), and (p, q).

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CONCAVITYLets see how the second derivative helps determine the intervals of concavity.

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Increasing and Decreasing Function

Tangents to a curve

Increasing and Decreasing FunctionA function is said to be monotonic if its either increasing or decreasing.

The points for which f (x) =0 or does not exist are called critical points.

The stationary points are the points where f(x) =0 in the domain.

If radius of circle increasing at uniform rate of 2 cm/s, find rate of increasing of area of circle, at instant when radius is 20 cm.

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APPLICATIONAt the Sri Sri Gaura Nitai temple, they want to make the rectangular temple room such that the perimeter is 1000 feet. Find the maximum length and width so that everyone can have the fortune to the Sri Sri Gaura Nitai. In addition, find the area.

OPTIMIZATIONOptima- what???Optimization problems is a fancy way of saying maximum minimum problems. The best way to start out is to draw a picture to help yourself visualize the problem.

THEN..Since we have two variables, and w, we have to solve everything in terms of one variable. We know that A= w and Prect = 2 +2w.So we can solve for in either equation. Most preferably use the P equation.

And finallySubstite the expression for l into the area formula.

According to the problemP = 1000 ft so

But to find maximum w, the derivative will have to equal 0.Then solve for w.

ThenPlug w in for the expression for length.

Coincidentially, the width and length are the same. Therefore, the temple room will be square so that everyone could see the Lord Sri Sri Gaura Nitai Jaya!

FIND THE AREASimple.. The length is 250 ft. The width is 250 ft. Multiply them together..A= 625,000 sq ft.

Nice big temple aint it??

Extreme Values of a FunctionUnderstand the following terms:Critical PointsLocal Extreme ValuesLocal maximumLocal minimumEnd Point Extreme ValuesEnd Point maximumEnd Point minimum

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Global Extreme Values

Understand the following terms:

Global Extreme Values

Global maximumGlobal minimum

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