Derivative/Antiderivative Graphs

[t_n_p]

Homework Statement

http://img149.imageshack.us/img149/1751/untitled3zt8.jpg

If g(x) is an antiderivative of f(x) and g(0)=2, detemine the x-coordinates of the points on the graph of g(x) corresponding to its local max and min values.

The Attempt at a Solution

I understand that stationary points occur when y=0 on the f(x) graph (i.e. at x=1 and x=3.5). I also know that at x=1 it's a maximum and at x=3.5 it's a minimum, but how do I find the corresponding y-values?

 

[HallsofIvy]

By "integrating". You could, of course, find the equation for f itself since it is "piecewise linear" but that isn't necessary. The integral of f is the area under the graph and, from x= 0 to x= 1, that is just a the area of a triangle. Don't forget to add the initial value, g(0)= 2. One you know g(1), you can find g(3.5) by "adding" the area under the curve from 1 to 3.5. I put "adding" in quotes because that value is actually negative, since the curve is below the x-axis.

To find g(1), Find the area of the triangle with base 1 and height 1 and add that to g(0)= 2. To find g(3.5), find the area of the trapezoid with height 1, bases 2.5 and 1, and subtract that from g(1).

 

[t_n_p]

ah gotcha thanks!

A turning point on the graph of f refers to a point of inflexion on the graph g, so does that mean that g(x) is increasing for (0,1), (3.5,4) and (4,6)?

Also, how would I find where g(x) is concave up/down?

 

[Hootenanny]

Well, what does the second derivative of g(x) tell you?

 

[t_n_p]

Well, what does the second derivative of g(x) tell you?

Well I do know f"(x)>0 represents concave up and f"(x)<0 represents concave down, but how can I get f"(x) (i.e. the derivative of g)

P.S. are my intervals for increasing g(x) correct?

 

[Office_Shredder]

But...

detemine the x-coordinates of the points on the graph of g(x) corresponding to its local max and min values.

Couldn't you just save yourself a lot of time and just answer the question instead?

 

[t_n_p]

But...



Couldn't you just save yourself a lot of time and just answer the question instead?

Yeah I know, just wanted to further my knowledge

 

[Hootenanny]

Well I do know f"(x)>0 represents concave up and f"(x)<0 represents concave down, but how can I get f"(x) (i.e. the derivative of g)

P.S. are my intervals for increasing g(x) correct?

Yes, your intervals look good to me. You don't need to get f''(x), you only need f'(x). Since g''(x) = f'(x), he gradient of f(x) on your plot represents the second derivative of g(x). Does that make sense?

 

[t_n_p]

Yes, your intervals look good to me. You don't need to get f''(x), you only need f'(x); since g''(x) = f'(x), so the gradient of f(x) on your plot represents the second derivative of g(x). Does that make sense?

So g(x) is concave up for positive gradients of f(x), that is (3,4) and concave down for negative gradients of f(x), that is (1,2) and (4,6).

 

[Hootenanny]

So g(x) is concave up for positive gradients of f(x), that is (3,4) and concave down for negative gradients of f(x), that is (1,2) and (4,6).

Looks good to me

 

[t_n_p]

Finally, points of inflexion on the graph of g occur when there are turning points on the graph of f, that is when x=4.

I now need to use all this info and the fundamental theorem of calculus to find values of g(x) at appropriate points to draw g(x). I suppose this is where the y-coordinates come in handy! I'll be back later with my attempted graph!

 

[t_n_p]

I've got a rough idea of what it looks like already, but with the graph being concave down for (1,2) and concave up for (3,4), what's happening between the interval of (2,3)?

Also how would I go about finding the y value for x=4?
will it be the y value at x=3.5 plus the area of the small triangle, i.e.
0.75+(0.5X0.5X1)?

 

[Hootenanny]

I've got a rough idea of what it looks like already, but with the graph being concave down for (1,2) and concave up for (3,4), what's happening between the interval of (2,3)?

Well, the second derivative test is inconclusive. However, you can say that g'(x)=-1 for {x : 2 < x < 3}, therefore the slope at this point is constant and the graph is a straight line y = -x.

Also how would I go about finding the y value for x=4?
will it be the y value at x=3.5 plus the area of the small triangle, i.e.
0.75+(0.5X0.5X1)?

I haven't checked your numbers, but you method is correct

 

[t_n_p]

http://img379.imageshack.us/img379/6467/asdfra4.jpg

think i got it right