Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points.

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Presentation transcript:

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points.

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases - As you can see, as I move from LEFT to RIGHT, the tangent line goes from a ( - ) value to a ( + ) value…

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases - As you can see, as I move from LEFT to RIGHT, the tangent line goes from a ( - ) value to a ( + ) value… Concave down – this occurs as the slope of the tangent line decreases - As you can see, as I move from LEFT to RIGHT, the tangent line goes from a ( + ) value to a ( - ) value…

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases Concave down – this occurs as the slope of the tangent line decreases

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Concave up – this occurs as the slope of the tangent line increases Concave down – this occurs as the slope of the tangent line decreases

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Inflection Point – occur where a graph changes from concave up to concave down, or vice versa

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Inflection Point – occur where a graph changes from concave up to concave down, or vice versa

Concavity and Inflection Points The second derivative will show where a function is concave up or concave down. It is also used to locate inflection points. Inflection Point – will occur where a graph changes from concave up to concave down, or vice versa Inflection point

Concavity and Inflection Points

2. Set up your intervals 0 1

Concavity and Inflection Points 2. Set up your intervals 0 1

Concavity and Inflection Points 2. Set up your intervals

Concavity and Inflection Points 2. Set up your intervals - place on number line

Concavity and Inflection Points 2. Set up your intervals

Concavity and Inflection Points 2. Set up your intervals Concave up CU

Concavity and Inflection Points 2. Set up your intervals Concave up CU Concave up CU

Concavity and Inflection Points 2. Set up your intervals Concave up CU Concave up CU Concave down CD

Concavity and Inflection Points 2. Set up your intervals Concave up CU Concave up CU Concave down CD Concave down CD

Concavity and Inflection Points

2. Set up your intervals

Concavity and Inflection Points 2. Set up your intervals

Concavity and Inflection Points 2. Set up your intervals - place on number line ** yes, it is possible that a critical point will also be an inflection point

Concavity and Inflection Points 2. Set up your intervals

Concavity and Inflection Points 2. Set up your intervals CU Concave up

Concavity and Inflection Points 2. Set up your intervals CU Concave up Concave down CD

Concavity and Inflection Points 2. Set up your intervals CU Concave up Concave down CD Concave up CU

Concavity and Inflection Points 2. Set up your intervals CU Concave up Concave down CD Concave up CU Concave up CU