Need Help with Slopes

MooseofDoom · 2008-08-11 07:03:50

I was wondering if anyone knows what to do for the following math problems for my AP Calculus class:

For the following functions, you will find the average rate of change over the specified intervals and use the results to estimate the instantaneous rate of change at a point.

Find the rates of change:
A) Between x=1 and x=2, x=1 and x=1.5, x=1 and x=1.23, x=1 and x=1.1, and x=1 and x=1.05. Use this to estimate the instantaneous rate of change at x=1.
B) Between x=-2 and x=-1, x=-2 and x=-1.5, x=-2 and x=-1.75, x=-2 and x=-1.9, and x=-2 and x=-1.95. Use this to estimate the instantaneous rate of change at x=-2

Then the first problem on the W/S is as follows:

Functions
1. y=-x²+4x

Can anyone explain to me how to solve this?

Thanks
~MooseofDoom

mathsyperson · 2008-08-11 10:25:48

At x=1, y = -(1)² + 4(1) = 3.
At x=2, y = -(2)² + 4(2) = 4.

x has moved 1 and y has moved 1, so the rate of change is 1/1 = 1.

Moving the endpoint to x=1.5 gives that y = 3.75.
This time x has moved 0.5 and y has moved 0.75, so the rate of change is 0.75/0.5 = 1.5.

Next we try at x=1.23. Here y = 3.4071, so the rate of change is 0.4071/0.23 = 1.77.

This table shows the full results:

End-point | y-value | change in x | change in y | rate of change
--------------------------------------------------------------------------------
2 | 4 | 1 | 1 | 1
1.5 | 3.75 | 0.5 | 0.75 | 1.5
1.23 | 3.4071 | 0.23 | 0.4071 | 1.77
1.1 | 3.19 | 0.1 | 0.19 | 1.9
1.05 | 3.0975 | 0.05 | 0.0975 | 1.95

As this table shows, as the end-point approaches 1, the rate of change appears to approach 2.
So we can estimate that the instantaneous rate of change for y = -x² + 4x, at x=1, is 2.

You can estimate its instantaneous rate of change at x=-2 in much the same way.
Draw up the table and see where the rates head to.

MooseofDoom · 2008-08-11 16:20:20

Hmm... Wow thank you so much I think I get it now! Thanks again! For x=-2 is the change from x=-2 to x=-1 equal to 1 or -1? A change in rate can only be positive right?

Last edited by MooseofDoom (2008-08-11 16:43:28)

mathsyperson · 2008-08-12 00:06:50

To calculate the change from a to b, you do b-a.
So the change from x=-2 to x=-1 is 1.

Changes in x are almost always positive, since convention says you go from left to right, but it's possible to have negative changes in y (it won't happen in your question though).

If you want to see an example of where rates of change are negative, you could try finding the instantaneous rate of change of your function when x=5.

MooseofDoom · 2008-08-12 08:06:29

hmm that's what I figured. That's what physics taught me with delta LOL. Thanks again!

Math Is Fun Forum

#1 2008-08-11 07:03:50

Need Help with Slopes

#2 2008-08-11 10:25:48

Re: Need Help with Slopes

#3 2008-08-11 16:20:20

Re: Need Help with Slopes

#4 2008-08-12 00:06:50

Re: Need Help with Slopes

#5 2008-08-12 08:06:29

Re: Need Help with Slopes

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