Relating Graphs of f and f’

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Presentation transcript:

Relating Graphs of f and f’

Do Now: Answer each of the following questions. When the function, f, is increasing, what does that mean about the derivative, f’? When the function, f, is decreasing, what does that mean about the derivative, f’? When the function, f, reaches a maximum at a smooth point, what does that mean about the derivative, f’? When the function, f, reaches a minimum at a cusp, what does that mean about the derivative, f’? When the function, f, changes from concave up to concave down, what does that mean about the derivative, f’, and the second derivative, f’’? When the derivative, f’, has a positive slope, what does that mean about the second derivative, f’’?

When the function is…(and the graph of the function…) Then the derivative is… (and the graph of the derivative …) And the second derivative is… Increasing (has a positive slope) and concave up Increasing (has a positive slope) and is concave down Decreasing (has a negative slope) and concave up Decreasing (has a negative slope) and concave down At a relative maximum (smooth curve) and concave down At a relative minimum (smooth curve) and concave up At a relative maximum (cusp) and concave up At a relative minimum (cusp) and concave down Changing from concavity from down to up Changing from concavity from up to down

When the function is…(and the graph of the function…) Then the derivative is… (and the graph of the derivative …) And the second derivative is… Increasing (has a positive slope) and concave up Positive (the graph of f’ is above the x-axis and has a positive slope) Positive Increasing (has a positive slope) and is concave down Positive (the graph of f’ is above the x-axis and has a negative slope) Negative Decreasing (has a negative slope) and concave up Negative (the graph of f’ is below the x-axis and has a positive slope) Decreasing (has a negative slope) and concave down Negative (the graph of f’ is below the x-axis and has a negative slope) At a relative maximum (smooth curve) and concave down Zero (crossing the x-axis from above to below) At a relative minimum (smooth curve) and concave up Zero (crossing the x-axis from negative to positive) At a relative maximum (cusp) and concave up Undefined Positive/undefined At a relative minimum (cusp) and concave down Negative/undefined Changing from concavity from down to up At a minimum, could be positive or negative Zero -Crossing x-axis from below to above Changing from concavity from up to down At a maximum, could be positive or negative Zero – crossing x-axis from above to below axis

Using the previous information about the f, f’ and f’’, answer the following questions based on the graph of f on the right. On what interval(s) is f’ positive? On what interval(s) is f’ negative? On what interval(s) is f’’ positive? On what interval(s) is f’’ negative? On what interval(s) are both f’ and f’’ positive? On what interval(s) are both f’ and f’’ negative? On what interval(s) is f’ positive and f’’ negative? On what interval(s) is f’ negative and f’’ positive?

Using the previous information about the f, f’ and f’’, answer the following questions based on the graph of f’ on the right. For what value(s) of x does f have a horizontal tangent? For what value(s) fo x does f have a relative maximum? For what value(s) of x does f have a relative minimum? On what interval(s) is f increasing? On what interval(s) is f decreasing? For what value(s) of x does f have a point of inflection? On what intervals(s) is f concave up? On what interval(s) is f concave down?

Practice Problems Base your answers to each of the following questions based on the graph of f at right (justufy with a sentence): On what interval(s) is f’ positive? On what interval(s) is f’ negative? On what interval(s) is f’’ positive? On what interval(s) is f’’ negative? On what interval(s) are both f’ and f’’ positive? On what interval(s) are both f’ and f’’ negative? On what interval(s) is f’ positive and f’’ negative? On what interval(s) is f’ negative and f’’ positive?

Practice Problems Base your answers to the following questions based on the graph of f’ at right (justify with a sentence): For what value(s) of x does f have a horizontal tangent? For what value(s) fo x does f have a relative maximum? For what value(s) of x does f have a relative minimum? On what interval(s) is f increasing? On what interval(s) is f decreasing? For what value(s) of x does f have a point of inflection? On what intervals(s) is f concave up? On what interval(s) is f concave down?

Practice Base your answers to the following questions based on the graph of f’ at right (justify with a sentence): For what value(s) of x does f have a horizontal tangent? For what value(s) fo x does f have a relative maximum? For what value(s) of x does f have a relative minimum? On what interval(s) is f increasing? On what interval(s) is f decreasing? For what value(s) of x does f have a point of inflection? On what intervals(s) is f concave up? On what interval(s) is f concave down?

Practice Base your answers to the following questions based on the graph of f’ at right (justify with a sentence): For what value(s) of x does f have a horizontal tangent? For what value(s) fo x does f have a relative maximum? For what value(s) of x does f have a relative minimum? On what interval(s) is f increasing? On what interval(s) is f decreasing? For what value(s) of x does f have a point of inflection? On what intervals(s) is f concave up? On what interval(s) is f concave down?