Download MATH 1830 Section 5

MATH 1830
Section 5.1
First Derivatives and Graphs
A. Increasing and Decreasing Functions
Recall: 1. The derivative of a function at a point gives the slope of the tangent line to
the function at that point.
2. A line that has positive slope ___________________________________.
3. A line that has negative slope __________________________________.
4. A horizontal line has a slope ___________________________________.
Graph: y  x5  23x 4  109 x3  690 x 2  1089 x  527 window:[2,11,1] by [300,300,50]
Think of this graph as a roller coaster track, the floor of the car as it moves from left to
right along the track represents the tangent line at each point. Using this analogy, we can
see that the slope of the tangent will be positive when the car travels uphill and y is
increasing, and the slope of the tangent will be negative when the car travels downhill
and y is decreasing. In this case it is also true that the slope will be zero at "peaks and
valleys."
The derivative f '(x) can change signs from positive to negative (or negative to positive)
at points where f '(x) =0, and also at points where f '(x) does not exist. The values of x
where this occurs are called critical numbers. By definition, the critical numbers for a
function f are those numbers c in the domain of f for which f '(c) = 0 or f '(c) does not
exist, but f(c) does exist.
First Derivative Test & A Sign Diagram
1. Locate on a number line those values of x for which f '(c) = 0 or f '(c) does not exist.
These points determine several open intervals.
2. Choose a value of x in each of the intervals determined in step 1. Use these values to
decide whether f '(x) > 0 or f '(x) < 0.
a. If f '(x) > 0 (slope is +), then f is increasing on the interval and
b. If f '(x) < 0 (slope is ), then f is decreasing on the interval.
3. Use + or  to indicate whether f is increasing or decreasing on the interval.
Find the derivative of the prior function. y'  5 x 4  92 x3  570 x 2  1380 x  1089