Query 9

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course MTH 163

Things are looking up...Oct. 27, @ 5:19 PM

009. `query 9

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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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Your solution:

- I used 0.1 x ^ 2 + 3 from the previous exercise, and got ½ for the slope.

confidence rating #$&*: 3

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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?

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Your solution:

- For x= 10, y = 7. For x = 30, y = 87, giving coordinates of (10, 7) and (30, 87). I got slope of 80/20, coming to 40.

confidence rating #$&*: 3

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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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Self-critique (if necessary): I had actually used the function y = f(x) = 0.1x^2 + 3

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

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Your solution:

- 87-7 (or otherwise the change in depth between depth(30) and depth(10) = 80

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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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Your solution:

- 20.

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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?

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Your solution:

- My answer was 40, but to reiterate this is from using the function f(x) = 0.1x^2 + 3. Either way, it is the change of depth between depth(30) and depth(10), over the change in time 30 - 10, or 20.

confidence rating #$&*: 3

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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?

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Your solution:

- My slope for this problem concluded to be 1.3, working from:

- (3, -2.1) and (10, 7) for my coordinates. The rise was the change between the two y’s, which is 9.1, and the run was 7; therefore slope = 9.1 / 7, or 1.3.

confidence rating #$&*: 3

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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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Self-critique (if necessary): Could you elaborate on this solution, without the values that I went through? I was working under the assumption to continue using the aforementioned function, f(x) = 0.1x^2 + 3. The parentheses are giving me a headache.

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Self-critique Rating:

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In this problem we're basically naming things.

The graph is of springLength vs. load. So the first coordinate of a point on the graph represents the load, and the load is measured along what we normally think of as the x axis.

Spring length is a function of the load. If we change the load the length of the spring changes in what we assume to be a consistent manner.

The function can therefore be called springLength (load).

If the load is load1, then we will get the spring length by substituting this load into our function, which will output the value of the spring length. So

springLength(load1).

represents the length of the spring when the load is load1.

The spring length is the coordinate we would associate with the y coordinate. So our graph point corresponding to load 1 is

(load1, springLength(load1)).

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

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Your solution:

- 9.1/7, or 1.3

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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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We're still naming things, not using a specific function.

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

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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.

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Your solution: I could loosely follow your solution, but I need a bit of extra clarity in order to solidify my understanding.

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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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Your solution:

- The manner in which the graphs are shown, without designated points KIND OF make sense, but they aren’t crystal clear yet. I can connect the idea of what I think it means, but so far it’s proven to be a bit confusing.

@&

It takes awhile to learn the name things. Moving from the concrete to something a little more abstract (but more powerful).

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Given Solution:

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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&#Good work. See my notes and let me know if you have questions. &#