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

MooseofDoom
Member
Registered: 2008-08-10
Posts: 13

Need Help with Slopes

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


π≈ 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 982 148 086 513 282 306 647 093 844 609 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930...

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

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Need Help with Slopes

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.


Why did the vector cross the road?
It wanted to be normal.

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

MooseofDoom
Member
Registered: 2008-08-10
Posts: 13

Re: Need Help with Slopes

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?what

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


π≈ 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 982 148 086 513 282 306 647 093 844 609 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930...

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

mathsyperson
Moderator
Registered: 2005-06-22
Posts: 4,900

Re: Need Help with Slopes

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.


Why did the vector cross the road?
It wanted to be normal.

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

MooseofDoom
Member
Registered: 2008-08-10
Posts: 13

Re: Need Help with Slopes

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


π≈ 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169 399 375 105 820 974 944 592 307 816 406 286 208 998 628 034 825 342 117 067 982 148 086 513 282 306 647 093 844 609 550 582 231 725 359 408 128 481 117 450 284 102 701 938 521 105 559 644 622 948 954 930...

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