Use the definitional formula given to find the derivative of the function.
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Problem 1
Given y = f(x) = x2 + 2x +3
a) Use the definitional formula given below to find the derivative of the function.
b) Find the value of the derivative at x = 3.
Problem 2
Given, y = f(x) = 2 x3 - 3x2 + 4x +5
a) Use the Power function to find derivative of the function.
b) Find the value of the derivative at x = 4.
Problem 3
The revenue and cost functions for producing and selling quantity x for a certain production facility are given below.
R(x) = 16x - x2
C(x) = 20 + 4x
a) Determine the profit function P(x).
b) Use Excel to graph the functions R(x), C(x) and P(x) for the interval 0? x ? 12. Copy and paste the graph below. Note: Use Scatter plot with smooth lines and markers.
c) Compute the break-even quantities.
d) Determine the average cost at the break-even quantities.
e) Determine the marginal revenue R'(x).
f) Determine the marginal cost C'(x)
g) At what quantity is the profit maximized?
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Solution Summary
Revenue, cost and profit as a function of quantity are graphed in the attached Excel file. The calculations for the solution are formatted in ten pages. These are attached in two files. The files are identical in content, only differ in format. The first is in MS Word format, while the other is in Adobe pdf format. Therefore you can choose the format that is most suitable to you.
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The Excel file is attached as well.
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We will use the binomial expansion:
(1.1)
Where
(1.2)
Therefore:
(1.3)
When we take the derivative of we have to find the limit:
(1.4)
Since and x is finite, all the terms in the sum that are multiplied by h disappear.
The only term that survives corresponds to and it is:
(1.5)
We can see this explicitly for n=2:
(1.6)
And for n=3:
(1.7)
Derivative of a sum of functions is the sum of the individual derivatives:
(1.8)
A derivative of a constant ...
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