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.
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 x-intercepts. You can use synthetic division and factoring to find the zeros! They are (-2, 0) (double root) and (1, 0)
d) Find the y-intercept. 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(x2 - 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!
f(x) = x3 + 3x2 - 4
|Acknowledgement: The above is based on class materials from .|