009.  `query 9

Question:  `qSymbolic calculation of slope, preliminary exercise

What was the function, between which two points were you to calculate the average slope and how did you get this slope?

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Given Solution: 

** For the function y = .1 x^2 - 3 between x = -2 and x = 7 we get:

slope = (y2 - y1) / (x2 - x1).

For x1 = 2 and x2 = 7 we have y2 = .1 * 7^2 - 3 = 1.9 and y1 = .1 * 2^2 - 3 = -2.6, so

slope = (1.9 - (-2.6) ) / ( 7 - (-2) ) = 4.5 / 9 = 1/2. **

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Question:  `qproblem 2 symbolic expression for slope, fn depth(t).

What is the expression for the slope between the two specified t values?

Your solution: 

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Given Solution: 

** The function is given a name:  depth(t).

t values are 10 and 30.

So rise = depth(30) - depth(10) and run = 30 - 10 = 20.

Thus slope = [ depth(30) - depth(10) ] / 20 . **

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Question:  `qWhat is the rise between the two specified t values?

Your solution: 

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Given Solution: 

** The rise is the change in depth.  The two depths are depth(10) and depth(30).

The change in depth is final depth - initial depth, which gives us the expression

depth(30)-depth(10) **

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Question:  `qWhat is the run between the two specified t values?

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Given Solution: 

** run = 30 - 10 = 20 **

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Question:  `qWhat therefore is the slope and what does it mean?

Your solution: 

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Given Solution: 

** rise = depth(30)-(depth(10) indicates change in depth.

run = 30 - 10 = 20 = change in clock time.

Slope = rise / run = (depth(30) - depth(10) ) / 20, which is the average rate at which depth changes with respect to clock time between t = 10 and t = 30. **

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Question:  `q     problem 5 graph points corresponding to load1 and load2

What are the coordinates of the requested graph points?

Your solution: 

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Given Solution: 

** The horizontal axis is the 'load' asix, the vertical axis is the springLength axis. 

The load axis coordinates are load1 and load2.

The corresponding spring lengths are springLength(load1) and springLength(load2).

The springLength axis coordinates are springLength(load1) and springLength(load2).

The graph points are thereofore  (load1, springLength(load1) ) and  (load2, springLength(load2) ).  **

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Question:  `qWhat is your expression for the average slope of the graph between load1 and load2?

Your solution: 

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Given Solution: 

** rise = springLength(load2) - springLength(load1)

run = load2 - load1 so

slope = [ springLength(load2) - springLength(load1)] / (load2 - load1). **

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Question:  `q     problem 6  symbolic expression for slope of depth function

Your solution: 

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Given Solution: 

** the name of the function is depth(t).

We need the slope between t = t1 and t = t2.

The depths are depth(t1) and depth(t2).

Thus rise is depth(t2) - depth(t1) and run is t2 - t1.

Slope is [ depth(t2) - depth(t1) ] / (t2 - t1). **

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Question:  `q    problem 8  average rate from formula  f(t) = 40 (2^(-.3 t) ) + 25 intervals of partition (10,20,30,40)

What average rate do you get from the formula?  Show your steps.

Your solution: 

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Given Solution: 

**  ave rate = change in depth / change in t.  For the three intervals we get

 (f(20)-f(10))/(20-10) = (25.625 - 30 )/(20 - 10) = -4.375 / 10 = -.4375

(f(30)-f(20))/(30-20) = (25.07813 - 25.625)/(30 - 20) = -.5469 / 10 = -.05469.

(f(40)-f(30))/(40-30) = (25.00977 - 25.07813)/(40 - 30) = -.0684 / 10 = -.00684. **

    Add comments on any surprises or insights you experienced as a result of this assignment.

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STUDENT RESPONSE: 

Ummm I know the slope formula is (y2-y1)/(x2-x1), but I always just put the number into the expression in the order I see them, but that is ok because I keep the order and get the correct answere because the y2,y1,x2,x1 or all relative.  I am correct in doing this? 

INSTRUCTOR COMMENT:

 In other words you use (y1 - y2) / (x1 - x2) instead of (y2 - y1) / (x2 - x1).

It's more conventional to regard, say, 10 as x1 and 20 as x2, so f(20) is y2 and f(10) is y1.  If you start from the lower x number and change to the higher the difference is higher - lower, and this is the way we usually think about changes.  According to this convention we calculate change in y as y2 - y1 and change in x as x2 - x1. 

You are doing (y1 - y2) / (x1 - x2) and you get a negative change in x, a negative denominator, and if you are thinking about change from the first quantity to the second this is backwards.

However as you say both numerator and denominator follow the same order so you still get the right answer, since (y1-y2)/(x1-x2)= (y2-y1) / (x2-x1).  **

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