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Tuesday, December 20, 2005

Scribe post - fun with functions

In todays class we basically focused on two problems. The first one was as follows: The are of a trapezoid is given by: A=1/2(B+b)h. A trapezoid is bounded by the lines y=1/2x+4, x=4, x=10, and the x-axis. Find it's area.

The first thing we should do is draw a graph with these lines on it. We know what the x axis is, so that is already on the graph. We know that x=4 is a vertical line where the y coordinate, or "output" is 4. The same thing for x=10: it's a vertical line where the output is 10. To figure out the last line(y=1/2x+4) we know that the y-intercept is 4, so we plot a point on the graph where y=4. Then we also know that that slope is 1/2, so we RISE one, and RUN two. Now we have our trapeziod.

To figure out the area of the trapezoid, we need to know the lengths of h, B, and b. The trapeziod that is drawn is a right triangle, but is tipped on it's side. The height(h) is the x axis. To figure this out, we take the x coordinate of ten from one point, and subtract 4 , which is the x coordinate of the other point, which gives us 6. So we now know that the height is six.

To figure out b, which we know is at coordinates (4, x). We don't know the y coordinate, or the "output". But because we have the input of 4, we substitute that into the equation y=1/2x+4, which will give us the y coordinate of 6. SO, between coordinates (4, 0) and (4,6) there is six points. So b=6.

We now need to figure out B. We do the same procedure for this that we did for b. So we substitute as the input in the equation, whic gives us 9 as the output or y coordinate. Between (10, 0) and (10,9) there is nine points so B=9.

Now we know all the information we need to do the formula. So, substituting all our values in the right places would result in: A=1/2(9+6)(6), which gives us 45 units^2!

The second question that was given was as follows: A cellophone contract costs(C) $20/month plus 10 cents/minute(t). Monthly changes are calculated using the formula: C(t)=0.10x+20.

We we first told to draw a graph. A helpful way to do this is to do a table of graphs, using incriments of 10, instead of one. So, use 0, 10, 20, 30....etc as inputs and see what you get as outputs. Then you plot the points and connect the dots.

Then we were told to find how much it would cost if we were to talk on the phone for 320 minutes. To do this, we simply substitute 320 for x, (C(t)=0.10(320)+20 which would result in the answer of $52.

Then, the next part of the question was if there was no teim used on the phone, would there still be a cost? In other words, if the input is zero, is the output also zero? If you plug in zero for x, for find that the output is 20. In other words, even if you were not to talk on the phone for that month, you would have to still pay $20.

We were then asked what the slope is, and what it means. The slope is 0.10, but even though decimals are delightful, fractions are our friends, so the slope is 1/10. The slope represents cost per minute, or rate of change of the cost.

The domain of this function is [0, 44640] because there are 24 hours in a day, 31 days in December, and 60 minutes in an hour. (24)(31)(60)=44640. This is the MOST amount of time that could be used in one month.

The range of this function is [20, 4484} because when the input is 0, the output is still twenty, and we CANNOT HAVE NEGATIVE TIME. There are 60 minutes in an hour, and 24 hours in a day, and (60)(24)=4484.

Thats about it folks. Homework for tonight is Exercise 55. FUN WITH FUNCTIONS! yay.
The scribe for tomorrow is..kimc!!!! good luck!

Oh wait...we have a pre test tomorrow eh? well then I guess she's scribe um....when we get back! happy holidays everyone!

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