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2.6, #6. Given Problem. |
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Part(a)(i).
This is the definition that the book calls definition 1. It is just another form of the one below, called definition 2. |
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Use the definition of the function to write the problem in terms of the function. Note that in this problem, a = -1. |
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Factor the numerator as the difference of two cubes, and cancel the two like factors. Then substitute to find the limit. |
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Part (a)(ii)
This is definition 2 and the definition used by this website in the notes section. |
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Again, we follow the definition of the function. |
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We multiply out the cubes,
and simplify. |
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We cannot take this limit the way it is, so we try to simplify further. An h is factored out of the numerator. |
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The h's canel and we can now take the limit. |
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And finally, we can take the limit at the desired point when x = -1. |
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Part (b).
We use the slope-intercept form of a line. We already know the slope
(3), We also know the point (-1,-1).
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We now have the equation of the line tangent to the function at the point (-1,-1). |
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Part (c).
This is the first plot for the interval (-2,0) |
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This is the second plot for the interval (-1.2,-.8) |
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This is the thrid plot for the interval (-1.05,-.95) |
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2.6, #24, Given Problem. |
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Part (a)(i).
To find average rate of change calculate the slope of the line containing the two points in question. |
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Part (a)(ii).
Same process as above with two new points. |
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Part (a)(iii).
Same process as above with two new points. |
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Part (b).
We average the two previous answers to get an estimated instantaneous rate of change in 1996. |
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Scatterplot of given data points along with estimated tangent line to the point (1996,1015). |
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My own estimate of two points that this estimated tangent line intersects: (1995,650), and (1997,1370). Find the slope. |
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