Antiderivatives Graphically & Numerically

[UMich1344]

Homework Statement

Using the graph of g’ in the figure below and the fact that g(0) = 50, sketch the graph of g(x). Give the coordinates of all critical points and inflection points of g.

http://s52.photobucket.com/albums/g14/CCieslak1689/?action=view&current=graph.jpg

2. Attempt and Question(s)

My first sketch of the graph had local maximums at x=50 and x=40 and a local minimum at x=15.

I had inflection points at x=10 and x=20.
The graph was concave down on the intervals (0, 10) and (20, 40).
It was concave up on the interval 10<x<20.

Hopefully I have my sketch of the graph right up to this point. My only doubt about it is the fact that the graph of the derivative has sharp corners and isn't smooth. I do not know if this needs to be accounted for and, if it does, I am completely lost.

Also, I'm wondering what steps I can take in order to find the coordinates of the critical points and inflection points. I'm confident I have the x-values correct but can't figure out how to solve for the y-values.

 

[mighty2000]

3. Response

Hello,

You are on the right track with your sketch of the graph. The sharp corners in the graph of the derivative do not necessarily need to be accounted for in your sketch of the graph of g(x). However, if you wanted to make the graph smoother, you could use a smoothing function or a curve fitting tool to approximate the graph of g(x).

To find the coordinates of the critical points, you can set the derivative equal to zero and solve for x. The resulting x-values will be the x-coordinates of the critical points. You can then substitute these x-values into the original function g(x) to find the corresponding y-values.

To find the coordinates of the inflection points, you can set the second derivative equal to zero and solve for x. The resulting x-values will be the x-coordinates of the inflection points. Again, you can substitute these x-values into the original function g(x) to find the corresponding y-values.

I hope this helps. Good luck with your graph!