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 4.3 Derivatives and the Shapes of Graphs
Many of the applications of calculus depend on our ability to deduce facts about a function f from information concerning its derivatives. Because f ' (x) represents the slope of the curve y = f(x) at the point (x, f(x)), it gives us information about f(x). In particular, if the slope of the tangent line is positive, it tells us that the function is increasing, and if the slope of the tangent line is negative, it tells us that the function is decreasing.
Mar 26­10:53 AM
1
The results of the first derivative test are illustrated in the following diagrams.
Determine the intervals on which the function is increasing and decreasing, and then find the local maximum and minimum values for f(x) = 2x3 + 3x2 ­ 72x + 17
Mar 28­2:49 PM
2
Determine the intervals on which the function is increasing and decreasing, and then find the local maximum and minimum values for f(x) = cos2 x ­ 2sin x, 0 ≤ x ≤ 2π
Mar 28­4:17 PM
3
We have determined the information that f '(x) gives us about f(x), but what about f ''(x)? Consider the graphs below. You can see in case (a) that the slopes of the tangent lines are getting increasingly steep, which means that f '(x) is increasing, so its derivative (f ''(x)) must be positive. In case (b), the slopes of the tangent lines are getting less steep, which means that f '(x) is decreasing, so its derivative (f ''(x)) must be negative. The point where the graph changes concavity is called an inflection point. Mar 28­4:30 PM
4
Determine the intervals on which the function is concave up and concave down, and then find any inflection points for f(x) = 2x3 + 3x2 ­ 72x + 17
Determine the intervals on which the function is concave up and concave down, and then find any inflection points for f(x) = x4 ­ 24x2 Mar 28­4:40 PM
5
Besides telling us the concavity of a graph of a function, the second derivative can also be used in some cases to find values of c at which a local maximum or minimum occurs. This seems plausible if we consider the graph below.
Use the second derivative test to find the local extrema of f(x) = x4 ­ 24x2 Mar 28­5:22 PM
6
Note that the second derivative test is inconclusive if f ''(c) = 0 or when f ''(c) does not exist, as in the following example.
f(x) = x5 ­ 5x3
Mar 28­5:32 PM
7
Determine the intervals on which the function is increasing and decreasing, concave up and concave down, and find any local extrema and inflection points for
Then use the information you've found to sketch the graph of f(x).
Mar 29­1:36 PM
8
Let's relate the information that f ' and f '' give us to a different kind of problem: Use the graph below to determine the x­coordinates of the inflection points, assuming the graph shown is the graph of a) f
b) f '
c) f '' Mar 29­1:36 PM
9
Find the vertical and horizontal asymptotes of the function, determine the intervals on which the function is increasing or decreasing, find the local maximum and minimum points, find the intervals on which the function is concave up or concave down, and find any inflection points. Then use the information you've found to sketch the graph of f(x).
Mar 29­1:37 PM
10