Relating Graphs Of F And F’
- 1. Relating Graphs of f and f’
- 2. 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’’ ?
- 3. Changing from concavity from up to down Changing from concavity from down to up At a relative minimum (cusp) and concave down At a relative maximum (cusp) and concave up At a relative minimum (smooth curve) and concave up At a relative maximum (smooth curve) and concave down Decreasing (has a negative slope) and concave down Decreasing (has a negative slope) and concave up Increasing (has a positive slope) and is concave down Increasing (has a positive slope) and concave up And the second derivative is… Then the derivative is… (and the graph of the derivative …) When the function is…(and the graph of the function…)
- 4. Zero – crossing x-axis from above to below axis At a maximum, could be positive or negative Changing from concavity from up to down Zero -Crossing x-axis from below to above At a minimum, could be positive or negative Changing from concavity from down to up Negative/undefined Undefined At a relative minimum (cusp) and concave down Positive/undefined Undefined At a relative maximum (cusp) and concave up Positive Zero (crossing the x-axis from negative to positive) At a relative minimum (smooth curve) and concave up Negative Zero (crossing the x-axis from above to below) At a relative maximum (smooth curve) and concave down Negative Negative (the graph of f’ is below the x-axis and has a negative slope) Decreasing (has a negative slope) and concave down Positive Negative (the graph of f’ is below the x-axis and has a positive slope) Decreasing (has a negative slope) and concave up Negative Positive (the graph of f’ is above the x-axis and has a negative slope) Increasing (has a positive slope) and is concave down Positive Positive (the graph of f’ is above the x-axis and has a positive slope) Increasing (has a positive slope) and concave up And the second derivative is… Then the derivative is… (and the graph of the derivative …) When the function is…(and the graph of the function…)
- 5. 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?
- 6. 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?
- 7. 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?
- 8. 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?
- 9. 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?
- 10. 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?