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Remember: Derivative=Slope of the Tangent Line

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What is the equation for the slope of the line tangent to the curve at point A using points A and B?

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What is the equation for the slope of the line tangent to the curve f(x)=x 2 +1 at point A using points A and B?

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

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

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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

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That means you can set them equal to each other:

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Therefore, Is the slope of the tangent line for f(x)=x 2 +1

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y-f(a)=f’(a)(x-a)

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Step 1: Find the point of contact by plugging in the x-value in f(x). This is f(a).

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

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Step 3: Plug all known values into formula y-f(a)=f’(a)(x-a)

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Find the equation of the tangent to y=x 3 +2x at: x=2 x=-1 x=-2

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f’(x)=0

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Step 1: Find the derivative, f’(x)

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

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Step 3: Plug solutions into original formula to find y-value, (solution, y- value) is the coordinates.

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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

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What do you notice about the labeled minimum and maximum? They are the coordinates where the tangent is horizontal

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Where is the graph increasing? {x| x 1} What is the ‘sign’ of the derivative for these intervals? -31 + + This is called a sign diagram

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Where is the graph decreasing? {x| -3<x<1} What is the ‘sign’ of the derivative for this interval? -31 + + – What can we hypothesize about how the sign of the derivative relates to the graph? f’(x)=+, then graph increases f’(x)= –, then graph decreases

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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 - )

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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.

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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

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Take a look at f(x)=x 3. 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

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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.

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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

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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 ??

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