Derivatives in Curve Sketching Derivatives can help graph many functions. The first derivative of a function is the slope of the tangent line for any point on the function! Therefore, it tells when the function is increasing, decreasing or where it has a horizontal tangent! Consider the following graph: Notice on the left side, the function is increasing and the slope of the tangent line is positive. At the vertex point of the parabola, the tangent is a horizontal line, meaning f '(x) = 0 and on the right side the graph is decreasing and the slope of the tangent line is negative! These observations lead to a generalization for any function f(x) that has a derivative on an interval I :
Here are some graphs of each of the observations made above! Some observations on the above graphs! 1) To be a minimum point, the graph must change direction from decreasing to increasing. 2) To be a maximum point, the graph must change direction from increasing to decreasing. 3) To be an inflection point, the graph doesn't change direction. In the above example ( one in middle) it is increasing before the f '(c) = 0 and it is still increasing after. You can also have one with the graph decreasing on both sides. Sample problems
Substitute these values into the original function to find the y values of the critical points. The points are (0, 4) and (2, 0)
The chart shows that (2, 0) is a local Max. and (0, 4) is a local min. You can tell because of the sign changes! c) Find the zeros of the original function. These are the xintercepts. You can use synthetic division and factoring to find the zeros! They are (2, 0) (double root) and (1, 0) d) Find the yintercept. This is the constant of the original function. (0, 4) e) Now take the limit as x goes to both infinities of the original function. f) Now put all these together and graph the function!
4x(x^{2}  4) = 0 4x(x  2)(x + 2) = 0
Substitute these into the original function and the points are (0, 9), (2, 7), (2, 7)
d) The y intercept is (0, 9) e) Find the limits on infinity. f) Put these together and graph the function! Quiz f(x) = x^{3} + 3x^{2}  4


Acknowledgement: The above is based on class materials from . 