Remember: Derivative=Slope of the Tangent Line

What is another way to find the slope of this line?The

DERIVATIVE!!!!)(' af

What is another way to find the slope of this line? xxf 2)('

2)1(2)1(')(' faf

Both ways give you the slope of the tangent to the curve at point A.

That means you can _____________________________.set them equal to each other

That means you can set them equal to each other:

axafy

xy

af

)(

12

2)('

That means you can set them equal to each other:

12

2

xy

)1(22 xy

Therefore,

)1(22 xyIs the slope of the tangent

line for f(x)=x2+1

y-f(a)=f’(a)(x-a)

Step 1: Find the point of contact by plugging in the x-

value in f(x). This is f(a).

39)3(4)3(3)3()( 2 faf

Step 2: Find f’(x). Plug in x-value for f’(a)

46)(' xxf 224)3(6)3(')(' faf

Step 3: Plug all known values into formula

y-f(a)=f’(a)(x-a)

))3((2239 xy

)3(2239 xy

• Find the equation of the tangent to y=x3+2x at:– x=2

– x=-1

– x=-2

f’(x)=0

Step 1: Find the derivative, f’(x)

Step 2: Set derivative equal to zero and solve, f’(x)=0

Step 3: Plug solutions into original formula to find y-value, (solution, y-

value) is the coordinates.

Note: If it asks for the equation then you will write y=y value found when

you plugged in the solutions for f’(x)=0

What do you notice about the labeled

minimum and maximum?

They are the coordinates where the tangent is horizontal

Where is the graph increasing?

{x| x<-3, x>1}

What is the ‘sign’ of the derivative for these

intervals?

-3 1

+ +

This is called a sign diagram

Where is the graph decreasing?

{x| -3<x<1}

What is the ‘sign’ of the derivative for this interval?

-3 1

+ + –

What can we hypothesize about how the sign of the derivative relates to the

graph?f’(x)=+, then graph

increasesf’(x)= – , then graph

decreases

We can see this:

When the graph is increasing then the gradient

of the tangent line is positive (derivative is +)

When the graph is decreasing then the

gradient of the tangent line is negative (derivative is - )

So back to the question…Why does the fact that the

relative max/min of a graph have horizontal tangents make sense?

A relative max or min is where the graph goes

from increasing to decreasing (max) or from decreasing to increasing (min). This means that

your derivative needs to change signs.

Okay…So what?

To go from being positive to negative, the derivative like any function must go through zero. Where the

derivative is zero is where the graph changes direction, aka the relative

max/min

Take a look at f(x)=x3. What is the coordinates of the point on the function where the derivative is equal to 0? Find

the graph in your calculator, is this coordinate a relative maximum or a

relative minimum?NO – the graph only flattened out then

continued in the same direction

This is called a HORIZONTAL INFLECTION

It is necessary to make a sign diagram to determine whether the coordinate where f’(x)=0 is a relative maximum, minimum, or a horizontal inflection.

Anywhere that f’(x)=0 is called a stationary point; a stationary point could be a relative

minimum, a relative maximum, or a horizontal inflection

• What do you know about the graph of f(x) when f’(x) is a) Positive b) Negative c) Zero

• What do you know about the slope of the tangent line at a relative extrema? Why is this so?

• Sketch a graph of f(x) when the sign diagram of f’(x) looks like

• What are the types of stationary points? What do they all have in common? What do the sign diagrams for each type look like?

-5 1

– – +

Stationary Point

? ?

3.7 – Critical Points & Extrema

Vocabulary• Critical Points – points on a graph in which

a line drawn tangent to the curve is horizontal or vertical– Maximum– Minimum– Point of Inflection

Maximum

• When the graph of a function is increasing to the left of x = c and decreasing to the right of x = c.

Minimum

• When the graph of a function is decreasing to the left of x = c and increasing to the right of x = c

Point of Inflection

• Not a maximum or minimum

• “Leveling-off Point”

• When a tangent line is drawn here, it is vertical