grafik dan optimisasi

7/28/2019 Grafik Dan Optimisasi

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LN 6- 2

MBM059

Graphing and Optimization

First Derivative Test for Local Extremum Let c be a critical value of the f ( f (c) defined and either f ′(c) = 0 or f ′(c) not

defined ). Construct a sign chart for f ′( x) close to and on either side of c.

Sign Test Critical Value

f ′( x) − − − + + +

m c n x f ( x) decreasing increasing

Local minimum

If f ′( x) changes from negative to positive at c,then f (c) is a local minimum

f ′( x) + + + − − −

m c n x f ( x) increasing decreasing

Local maximum

If f ′( x) changes from positive to negative at c,then f (c) is a local minimum

f ′( x) − − − − − −

m c n x f ( x) decreasing decreasing

Not a local extremum

If f ′( x) does not change sign at c, then f (c) is a

neither a local maximum nor a local minimum.

f ′( x) + + + + + +

m c n x f ( x) increasing increasing

Not a local extremum

If f ′( x) does not change sign at c, then f (c) is a

neither a local maximum nor a local minimum.

f ′( c)

= 0: horizontal tangent f ( x) f ( x) f ( x) f ( x)

f (c) f (c) f (c) f (c)

0 c x 0 c x 0 c x 0 c x

f ′( x) − − 0 + + f ′( x) + + 0 − − f ′( x) + + 0 + + f ′( x) − − 0 − −

local minimum local maximum not local extremum not local extremum

f ′( c) not defined but f ( c) is defined

f ( x) f ( x) f ( x) f ( x)tangent line

tangent line f (c) f (c) f (c) f (c)

tangent line tangent line

0 c x 0 c x 0 c x 0 c x

f ′( x) − − N + + f ′( x) + + N − − f ′( x) + + N + + f ′( x) − − N − −

local minimum local maximum not local extremum not local extremum



LN 6- 3

MBM060


f ′( x) = 3 x2 − 3 = 3( x − 1)( x + 1)

f ′( x)

−1 1 x

f ( x)

x

f ( x) = x3 − 3 x

Example 6.1 Locating local extremum

Given the function f ( x) = x3

− 3 x. Find the critical value

of f , local extremum of f , and sketch the graph of func-tion f and f ′.

Solution f ′( x) = 3 x2 − 3 = 3( x − 1) ( x + 1)

Find all number x in the domain f where f ′( x) = 0.

f ′( x) = 3 x2 − 3 = 3( x − 1) ( x + 1) = 0.

x = −1 or x = 1.

The critical values of function f are x = −1 and x = 1.

Sign test for local extremum :

(−∞,−1) (−1,1) (1,∞) f ′( x) + + + + + 0 − − − − − 0 + + + + +

−1 1 x

f ( x) increasing decreasing increasing

local max local min

The sign chart indicates that f increase on (−∞,−1), has

a local maximum at x = −1, decrease on (−1,1), has a

local minimum at x = 1, and increase on (1,∞).

The graph of f and f ′ are shown on the left figure.

B ′(t )

10

8

64

2

0 t

B(t )

0 t

Example 6.2 Agricultural Export and Import Over the past several decades, the USA has exported more agricultural product than it has imported, mainta-ining a positive balance of trade in this area. However,the trade balance fluctuated considerably during the pe-riod. The graph on the left figure approximates the rateof change of the trade balance over a 15-year period,where B(t ) is the trade balance in billions of dollars and t is time in years. (a) Write a brief verbal description of

the graph of y = B(t ), including a discussion of any lo-cal extreme. (b) Sketch a possible graph of y = B(t ).

Solution B′(t ) > 0 on (0,4), B′(4) = 0, B′(t ) < 0 on

(4,12), B′(12) = 0, and B′(t ) > 0 on (12,15). Function

B has local maximum at 4 and local minimum at 12.The scale of vertical axes is depend on information of B(t ).

−1 0 1

y = f ′( x)

−3

−2

2

y = f ( x)

−1 0 1

4 12 15

4 12 15



LN 6- 5

MBM062


Student Work-sheet

I.N Name Signature

y

x

Problems

If the function y = f ( x) continuous on (−∞,∞), use the given

information to sketch the graph of f .

f ′( x) + + + + + 0 − − − − − 0 − − − − −

−1 1 x

Sketch the graph of f on left figure!

y

x

Problems

If the function y = f ( x) continuous on (−∞,∞), use the given

information to sketch the graph of f .

f ′( x) + + + N + + 0 − − − − 0 + + + +

−1 0 2 x

Sketch the graph of f on left figure!

f ( x) 1 3 2 1 −1

−2 −1 0 1 2

x

f ( x) −3 0 2 −1 0

−2 −1 0 2 3



LN 6- 6

MBM063


Student Work-sheet

I.N Name Signature

Problems Medicine A drug is injected into the bloodstream of a patient through the right arm.The concentration of the drug in the bloodstream of the left arm t hours after the injection is given

by 2

0.14

1( )

t

t C t

+= , 0 < t < 24. Find the critical values for C (t ), the interval where the concentration

of the drug is increasing, the interval where the concentration of the drug is decreasing, and thelocal extreme.

Solution



LN 6- 7

MBM064


Second Derivative, Concavity, and Optimization Second Derivative For y = f ( x), the second derivative of f , denoted by y″,

f ″( x), or 2

2

d y

d x, provided that it exist, is ( ) ( )

d

dx f x f x=¢¢ ¢ .

Concavity For the function y = f ( x) in the interval (a,b) we have

f ″( x) f ′( x) Graph of f Examples

Positive (+) Increasing Concave upward

Negative (−) Decreasing Concave downward

f ″( x) > 0 over (a,b), f concave upward f ″( x) < 0 over (a,b), f concave downward

f

a b

f ′( x) is negativeand increasing

graph f falling

f

a b

f ′( x) increasefrom neg. → pos.

graph f fall → rise

f

a b

f ′( x) is positiveand increasing

graph f rising

f

a b

f ′( x) is positiveand decreasing

graph f rising

f

a b

f ′( x) decreasefrom pos. → neg.

graph f rise→fall

f

a b

f ′( x) is negativeand decreasing

graph f falling

Inflection point An inflection point is a point on a graph f where the concavity changed.

For the concavity to change at a point, f ″( x) must be change sign at that point.

Theorem If y = f ( x) is continuous on (a,b) and has an inflection point at x = c,

then either f ″(c) = 0 or f ″(c) does not exist.

A partition number c of f ″, c at the domain f produced in inflection point for the

graph of f only if f ″( x) changes sign at c.

Optimization

The quantity f (c) is absolute maximum of f if f (c) ≥ f (c) for all x in domain of f .

The quantity f (c) is absolute minimum of f if f (c) ≤ f (c) for all x in domain of f .Theorem A continuous function f on a closed interval [a,b] has both an absolutemaximum value and an absolute minimum value on that interval.Theorem If f is continuous on an interval I and c is the only critical value of I ,

f ′(c) = 0 and f ″(c) < 0 (> 0), than f has an absolute maximum (minimum) on I .



LN 6-10

MBM067


Student Work-sheet

I.N Name Signature

Problems Given f ( x) = 2 x2 − x

4.

(a) Find the interval where f ( x) is increasing, decreasing, and the local extreme.(b) Find the interval where f ( x) concave upward, concave downward, and the point of inflection. (c) Sketch the graph of f .

Solution

y

x



LN 6-11

MBM068


Student Work-sheet

I.N Name Signature

Problems A cardboard box manufacturer wishes to make open boxes from pieces of cardboard 12 inches square by cutting equal squares from the four corners and turning up the sides. Find the length of the side of the square to be cut out in order to obtain a box of the largest possiblevolume and find the largest volume.

Solution

←⎯⎯⎯⎯⎯ 12 ⎯⎯⎯⎯⎯→

x x

x ? x

? ? 12

x ? x

x x



LN 6-12

MBM069


Exercise 6

1. If f ( x) = 32 x − x4, find the intervals where f ( x) is increasing, the intervals where

f ( x) is decreasing, and the local extreme.

2. If f is continuous on (−∞,∞) and satisfy the condition

(a) f (−1) = 2, f (0) = 0, f (1) = 2,

(b) f ′(−1) = 0, f ′(1) = 0, f ′(0) is not defined,

(c) f ′( x) > 0 on (−∞,1) and (0,1),

(d) f ′( x) < 0 on (−1,0) and (1,∞);

use these information to sketch the graph of f .

3. Find the critical value, the intervals where f ( x) is increasing, the intervals where

f ( x) is decreasing, and the local extreme of f ( x) = 1 21 x x- -

+ + .

4. Given f ( x) = x4 − 8 x

2 + 10.

(a) Find the intervals where f ( x) is increasing, decreasing, and the local extreme. (b) Find the interval where f ( x) concave upward, concave downward, and the

point of inflection.

5. Average cost A manufacturer incurs the following cost in producing x blenders

in one day for 0 < x < 250; fixed cost $450, unit production cost $30 per blender,

equipment maintenance and repair $0.08 x

2

. What is the average cost ( )C x per blender if x blenders are produced in one day? Find the intervals where the ( )C x

is decreasing, is increasing, and the local extreme.

6. Botany If it is known from past experiments that the height (in feet) of a given

plant after t months is given approximately by H (t ) = 4 t − 2t , 0 ≤ t ≤ 2. How

long the plant to reach its maximum height? What is the maximum height?

7. Politics In a new suburb, it is estimated that the number of registered voters

will grow according to N = 10 + 6t 2 − t

2, 0 ≤ t ≤ 2, when t is time in years and N

is in thousand. When will the rate of increase be maximum?

Math QuoteNothing is more important than to see the source of invention whichare, in my opinion more interesting then the inventions themselves.

A method of solution is perfect if we can foresee from the start, andeven prove, than following that method we shall attain our aim.

Leibniz, Gottfried Wilhelm (1646 – 1716)

grafik dan optimisasi

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