Problem :
Sketch the graph of f (x) = .
f'(x) | = | ||
= | |||
= |
f''(x) | = | ||
= | |||
= |
= - ∞ | |||
= ∞ |
= + ∞ | |||
= - ∞ |
Problem : Give a possible equation of the function graphed below:
To find an appropriate equation, begin by pulling out relevant features of the graph. f has vertical asymptotes at x = 3 and x = - 3. Vertical asymptotes can occur where the denominator of a rational function is equal to zero. Thus, the function in question should look something like:f (x) = | |||
This is the same as | |||
f (x) = |
f (x) =
Finally, f (0) = - 3, so the equation must be modified to:f (x) =
Problem : Give a possible equation of the function graphed below:
As in the last problem, start with the vertical asymptotes:f (x) = | |||
f (x) = |
f (x) =
f (0) = 0, but this does not modify the equation of the function.Problem : Below is a sketch of the graph of f'(x). Use this to sketch a possible graph of f (x).
The information contained in the graph of f'(x) is enough to determine the sign and concavity of f (x). It is possible to clearly pick out intervals where f'(x) is positive or negative. Also, f''(x) is positive wherever f'(x) in increasing, and f''(x) is negative wherever f'(x) is decreasing. Putting this information together generates the following chart: This can be graphed in the following way:Take a Study Break!