increasing

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6

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Increasing

IncreasingDecreasing

Features of +x3 Graphs

1

Stationary

Stationary

The original function is… f(x) is…y is…

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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Investigate the tangents of +x3 Graphs

2

The slope function is… f’(x) is…dy/dx is…

Slope values are decreasing

Slope valuesare increasing

Point of Inflection = slopes

stop decreasin

g and start

increasin

g

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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→slope function decreasing

Features of the Slope Function Graph

3

Reading the features of the graph of the slope function from the original function


→slope functionincreasing

Turning Point

of the slope

function: where slopes

turn from decreasing

to increasing

= min

slope function = 0 (cuts x-axis) dy/dx= 0

dy/dx= 0 slope function = 0 (cuts x-axis)

Slope Function: U shaped (positive cubic graph will have positive derivative graph) Minimum point at same x value as the point of inflection Cuts x-axis at the x values of the turning points

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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4

The slope function is… f’(x) is… dy/dx is…

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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Turning Point:

Decreasing to

increasing = min pt

dy/dx= 0; slope function = 0


Turning Point:

Decreasing to

increasing = min pt



SLOPE FUNCTIONy = f’(x)

ORIGINAL FUNCTIONy = f(x)

x

5

Also, we can read where the slope function is above and below the x-axis from the original function

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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Slopes are positive

Slope Function above x-axis

Slopes are negative

Slope Function below x-axis

Slopes are positive

Slope Function above x-axis

+ + + + + + + 0 - - - - - - - - - - 0 + + + + + + +

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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6

At what rate is the slope function changing? f’’(x) is… d2y/dx2 is...

How fast is the rate of decrease of the slopes?

How fast is the rate of increase of the slopes?

Finding the rate of change of the rate of change…. Finding the second derivative

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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Turning Point:

Decreasing to

increasing = min pt



7

A step further to investigate the tangents of the slope function.

Second Derivative Function is… f’’(x) is… d2y/dx2 is...

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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Turning Point:

Decreasing to

increasing = min pt



ORIGNAL FUNCTIONy = f(x)


-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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Turning Point:

Decreasing to

increasing = min pt




-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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Slope values are increasing→Second Derivative

Function is increasing

Slope valuesare increasing→Second Derivative

Function is increasing

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

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SECOND DERIVATIVE FUNCTIONy = f’’(x)

Slope=0 (d2y/dx2 = 0)Second Derivative

Function =0(cuts x-axis)


-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

x

y

10

Original Function, First Derivative Function, Second Derivative Function

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

x

y

-6 -5 -4 -3 -2 -1 1 2 3 4 5 6

-6-5-4-3-2-1

123456

x

y

y = f’’(x)

𝒍𝒐𝒄𝒂𝒍𝒎𝒂𝒙=𝒅𝟐𝒚𝒅 𝒙𝟐 <𝟎

𝒍𝒐𝒄𝒂𝒍𝒎𝒊𝒏=𝒅𝟐𝒚𝒅 𝒙𝟐 >𝟎

y = f’(x)

y = f(x)

increasing

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