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Calculus AB: Applications of the Derivative

Problems from "Using the First Derivative to Analyze Functions"

Using the First Derivative to Analyze Functions

Using the Second Derivative to Analyze Functions

Problem : f (x) = x 3 -4x 2 - 4 . Show where f (x) is increasing and where it is decreasing.

f'(x) = 3x 2 - 8x ;
f'(x) = 0 at x = 0 and x = . The sign of the derivative is reported on the number line below:
So, f is increasing on (- ∞, 0) and (,∞) , and it is decreasing on (0,)

Problem : f (x) = sin(x) . Where is f increasing and decreasing on the interval [0, 2Π]?

f'(x) = - cos(x) ;
f'(x) = 0 at x = and x = .
The sign of the derivative and the behavior of the function is indicated below:

Problem : Classify the critical points of f (x) = x 3 -3x 2 - 9x .

f'(x) = 3x 2 - 6x - 9 The sign of the derivative is shown in the figure below.
Because the derivative is positive to the left of x = - 1 and negative to the right, x = - 1 is a local maximum; x = 3 is a local minimum.

Problem : Below is the graph of the derivative of a function f . Find regions where f is increasing and decreasing, and classify the critical points.

The sign of f'(x) and the behavior of f is depicted below:
While b , d , f , and h are critical points, only d and h are local extrema. d is a local maximum and h is a local minimum.

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