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Mathematics QS 026Topic4: Applications of Differentiation Lesson Plan

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TOPIC : 4.0 APPLICATIONS OF DIFFERENTIATION

SUBTOPIC : 4.1 Tangents and Normals.

LEARNINGOUTCOMES :

a) To find the equations of tangent and normal to a curve.

1.dx

dyxf )(' is the gradient to the function of )(xfy

2. At point ( 11, yx ) on the curve of )(xfy , the equation of tangent is

)( 111 xxmyy

dx

dym 1 is the gradient of the tangent to the curve at the point ),( 11 yx

11, yx

Example 1:

Find the tangent and normal equation to the curve 322

xy at 2x

Example 2:

Find the tangent and normal equation to the curve 223 xyyx at P(1,1)

Curve )(xfy

normaltangent

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Exercise: Find the tangent and normal equation to the curve 8xy at P(4,2)

Answer;

x

y

dx

dy

Tangent equation 82 xy

Normal equation 62 xy

Example 3:

Find the equation of normal to the curve given parametrically by

tx

2 and 13 2 ty at point where t= 1

Example 4:

Given parametric equation1

3

t

x and2)1(

2

ty

where t 1

Find the tangent equation which is parallel to line 3y = x

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SUBTOPIC : 4.2 Extremum Problem

LEARNING

OUTCOMES : a) To find the critical points.

b) To discuss and use the first derivative test.c) To determine the intervals where a function increases or decreases.

Ask students to observed the movement of the coaster and then identify its maximum

or minimum position.

Figure 2: Application of extremum values

From the simulation above, student could predict the maximum value and minimum

value.

As a conclusion, the lecturer verifies the answer and shows them a few more different

situations.

4.2.a) EXTREMUM PROBLEMS

1. A critical point for a functionf(x) is any value of x in the domain offat

which

f(x) = 0 orf(x) is not defined.

2. The critical point wheref(x) = 0 are called stationary point and the value of

the function at that point a stationary value.

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Example 1

Find the stationary points and stationary values of the function 4x3 + 15x218x + 7.

Example 2

Find the critical numbers off(x) = )4(53

xx .

a) forf(x) = 0

cc

c c

(b) forf(x) is undefined

c = critical value

c

c c

c

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4.2 b) & 4.2 c) Function Increases and Decreases

In sketching the graph of a function it is very useful to know where it rises and where

it falls. The graph shown in Figure 4.21 rises from A to B, fall from B to C, and rises

again from C to D. The function f is said to be increasing on the interval [a, b],decreasing on [a, b], and increasing again on [c, d]. Notice that ifx1 and x2 are any

two numbers between a and b with x1 < x2 then f(x1) < f(x2). We use this as the

defining property of an increasing function.

Figure 4.21

Definition

A function f is called increasing on an interval I iff(x1)

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Exercises:

Answer:

i) x = 0 &x =2

1; (,0) and (

2

1, ) are increasing, (0,

2

1) is decreasing

ii) x = 0 ; (,0) and (4, ) are increasing

2

2

3xi) f(x) =x3 + 1

Find the critical value for the following and hence, determine the interval which

the function are increasing or decreasing for the following function.

1,4

10,1

0,1

)(

2

2

xx

x

xx

xfii)

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SUBTOPIC : 4.2 Extremum Problems

LEARNING

OUTCOMES :

d).To find the relative extremum using the First Derivative Test.e).To Determine the concavity and the point of inflection.

f).To apply the second derivative test.

4.2 d) The First Derivative Test

Suppose that c is a critical number of a function f that is continuous on (a, b).

a. If f (x) > 0 for (a, c) and f (x) < 0 for (c, b), then f has a relative maximum at

c

b. If f (x) < 0 for (a, c) and f (x) > 0 for (c, b), then f has a relative minimum at cc. If f (x) does not change sign at c, then f has no relative extremum at c

It easy to remember the First Derivative Test by visualizing diagrams such as those in

Figure 3.22

a. Relative maximum b. Relative minimum

c. No extremum d. No extremum

f (x) > 0

f x

x x

f (x) > 0

c

f (x) < 0

f x

c

f (x) > 0

c

f (x) > 0 f (x) < 0

f x

f (x) < 0

f (x) < 0

f x

xx

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Example 1

Find the maximum or minimum values of the function y = 4x 3 + 15x 218x + 7.

Example 2

Find the relative extremum of f(x) = 32

3

5

185

9xx .

Example 3

Find the relative extremum for the function f(x) = ( 3x + 2)e -x

Exercise

1. The curve y = x 2 + ax + b has a stationary point at (1, 3 ). Find a and b2. For what value of a and b will the function f(x) = x 3 + ax 2 + bx + 2 have a

relative maximum when x = -3 and relative minimum when x = -1.

3. Find a cubic function f(x) = a x 3 + bx 2 + cx + d that has a relative maximumvalue of 3 at -2 and a relative minimum value of 0 at 1.

4.2 e) The Concavity and points of inflection

For point of inflection, then f(x) = 0, with a changes of sign in the value of f(x) forthe

interval before and after the expected point of interval

Example 1

Given f(x) = x36x 2 + 9x + 2 . Find the inflection point and determine the interval

where f(x) is concave upward and concave downward.

Example 2

Given a fuction f(x) = -x 3 + 3x 21. Determine the

a) point of inflectionb) intervals where f(x) is concave upward and where f(x) is concave downward

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4.2 f) The Second Derivative Test

Example 3

Find the relative extremum and points of inflection of f(x) = x3

3x2

and sketch thegraph

Example 4

Find the relative extremum and the inflexion points of f(x) = 3x 55x 3

Exercise

Sketch the graph of the following functions, showing all stationary points and

inflexion points ( if any )

a) f (x ) = 2x33x212xb) f (x ) = x33x29x + 7c) f (x ) = x22x4 x3d) f (x ) = x55x45x3 -1e) f (x ) = x3 -

2

3x2 + 1

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SUBTOPIC : 4.2 Extremun problem

LEARNING

OUTCOMES :

g) find the relative extremum using the Second DerivativeTest.

h) solve the optmization problems

4.2 g ) :The Second Derivative Test

Another application of the second derivative is in finding maximum and minimum

values of a function.

Suppose f (x) is continuous on an open interval that contains c

a. If f (x) = 0 and f (x) > 0, then f has a local minimum at cb. If f (x) = 0 and f (x) < 0, then f has a local maximum at c

c. If f (x) = 0 and f (x) = 0, the test fails, it is inconclusive. Thus we must use the

first derivative test.

Example 1

By using the Second Derivative Test, find the relative extremum of f(x) = x33x2

24x + 32.

Example 2

Using the Second derivative Test, find the relative extrema of the function

f(x) =x44x3

4.2 h. To solve Optimization problems

Steps for solving optimum problem

Step 1: identify the variable to be maximized or minimized, say variable y.

Step 2 : express y into a single variable function, say y=f(x).

Step 3 : Solve for f(x)=0

Step 4 : verify the solution of f(x)=0 by the second derivative f(x).

Step 5 : get the optimum value y=f(x) if it is required.

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Example 3

You have been asked to design a 1 liter oil can shaped like a right circular cylinder.

2r

h

What dimensions will use the least material?

Example 4

A rectangle is to be inscribed in a semi circle of radius 2. What is the largest area the

rectangle can have, and what are its dimensions?

Example 5

An open-top box is to be made by cutting small congruent squares from the corners of

a 12 cm by 12 cm sheet of tin and bending up the sides. How large should the squares

cut from the corners be to make the box hold as much as possible?

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SUBTOPIC : 4.3 Curve Sketching

LEARNING

OUTCOMES :

a) Sketch The Graph

For curve sketching, we can see the flow of the roller coaster movement from the

simulation.

figure3: Application of curve sketching

Sketch down the shape of the lane on a piece of paper and verify the answer.

In the last several section we saw how the first and second derivatives of

a function are used to reveal various properties of the function. These

techniques , combined with the knowledge of the properties of the function,

provide us with a powerful method for sketching the graph of the function.

The following procedure gives a systematic method of compiling some

useful information on the function to aid in sketching its graph.

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A Guide to curve sketching of Polynomials

1. Determine the domain of f2. Determine the intercepts (if exist)3. Find the relative extreme (turning point) using first derivative test or

second derivative test.

4. Find inflection point and determine the concavity of a function usingsecond derivative test

5. Sketch the graph.

Example 1

Sketch the graph of the function

f(x) = x36x2 + 9x + 2

Example 2

Sketch the graph of the function f(x) = 3x3 + 3x2 -3x + 2

Example 3

Sketch the graph of the function f(x) = 1 + 6x23x4

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SUBTOPIC : 4.3 Curve Sketching

LEARNING

OUTCOMES : a) Sketch The Graph

Curve Sketching of Rational Function

A Guide to curve

1. Determine the domain of f2. Determine the intercepts (if exist)3. Find all asymptotes.

i) Rewrite the equation so that it does not contain any algebraic fractions.

ii) Equate to zero the coefficient of the highest power of y to find theasymptote parallel to the y-axis.

iii) Equate to zero the coefficient of the highest power of x to find the

asymptote parallel to the x-axis.

4. Find the relative extreme using first derivative test or second derivativetest.

5. Find inflexion point and determine the concavity of a function usingsecond derivative test

6. Sketch the graph.

Example 4

Graph the function3

2)(

x

xxf .

Example 5

Sketch the graph of2)1(

)(

x

xxf

Exercise

Sketch the graph of the following functions , showing all stationary points and

inflexion points ( if any ).

1. f(x) = 2x3 +x2 - 20x + 5

2. f(x) =3

x3

- 2x2 + 3x

3. f(x) = x4 - 4x2

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SUBTOPIC : 4.4 Rates of Change

LEARNING

OUTCOMES : a) To solve related rates problems

The rate of change in velocity (the speed of an object in a certain direction) is known

as acceleration. Whether an object is speeding up, slowing down, or changing

direction, it is accelerating. Most amusement park rides involve acceleration. On a

downhill slope or a sharp curve, a ride will probably increase in velocity or accelerate.

While moving uphill or in a straight line, it may decrease in velocity or decelerate.

The force of gravity pulling a roller coaster down hill causes the roller coaster to go

faster and faster, it is accelerating. The force of gravity causes a roller coaster to go

slower and slower when it climbs a hill, the roller coaster is decelerating or going

slower. The acceleration of a roller coaster depends on its mass and how strong is the

force that is pushing or pulling it.

Ify = f (x) thandx

dyis the rate of change ofy with respect tox.

They are often, but not always, rates of change with respect to time. The rate of change of quality Q

dt

dQmeans: the rate of change of Q

with respect to time t.

Rate of Increase ispositive. Rate of Decrease is negative. Rate of change ofy and rate of change ofx can be shone as:

dt

dx

dx

dy

dt

dy

Strategies for solving related rate problems

1. Draw a picture and name the variables and constants. Use t for timeand assume all variables are differentiation function oft.

2.

Write down the numerical information (in term of the symbols youhave chosen)

3. Write down what you are asked to find (usually a rate, expressed as aderivative)

4. Write an equation that relates the variables you may have to combinetwo or more equations to get a single equation that relates the variable

whose rate you want to the variable whose rate you know.

5. Differentiation with respect to tto express the rate you want in terms ofthe rate and variables whose values you know.

6. Evaluate/calculate.

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Example 1:

Gas is escaping from a spherical balloon at the rate of 2 3m /min. How fast is

the surface area shrinking when the radius is 12 m?

Example 2:

Water is running out of a conical funnel at the rate of .sec/13

cm If the radius of thebase of the funnel is 4 cm and the height is 8cm, fine the rate at which the water level

is dropping when it is 2 cm from the top.

Example 3:

Given

fvu

111 withfas a constant. Iff= 10 cm and u decreases with the rate of

2 cm/sec , find the rate of change of v when u = 40 cm

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SUBTOPIC : 4.4 Rates of Change

LEARNING

OUTCOMES : a) To solve related rates problems

Example 4:

A 8 meter ladder rest against a vertical wall(see figure below). If the bottom of the

ladder is sliding away from the base of the wall at the rate of 3 m/sec, how fast is the

top of the ladder moving down the wall when the bottom of the ladder is 2 meter from

the base?

Example 5 :

A person 6 meter tall is walking away from a streetlight 20 meter high at the rate of

7m/s. At what rate is the length of the persons shadow increasing?

Example 6

At 1:00 P.M, ship A is 25 miles due south of ship B. If ship A is sailing west at rate of

16mi/hr and ship B is sailing south at rate of 25mi/hr, find the that at which the

distance between the ships is changing at 1:30P.M.

25

y

x

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Exercise

1. The radius r of circle is increasing at the rate of 2cm/min. Find the rate ofchange of area when 6r cm and 24r cm.

2. Suppose that the cost ( in Ringgit ) of manufacturing x items is modeledby 205.04200)( xxxC . Find the daily rate of change in this cost

function at the time when the production level is 15x items and

changing at the rate of 2dt

dx(items per day ).

3. In certain electronic circuit the resistence is given as 23 tr (ohms) andthe voltage is 23te (volts). Find the rate of change of the current i (amperes) with the respect to the time t (seconds).

( Note :rei )

4. A cylinder is expanding in such a way that its height h cm and radius rcm are both increasing at the rate of 1% per minute. Find the rate the

volume is increasing per minute.

( Note : 2rhV )

Answers :

1) min/96,24 cm

2) 11RM

3)2)23(

)43(3

t

tt

4) 3%

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