Homework 04

Assessment: Each item is worth 2 points. The total for this homework is 20 points. There are no partial points. The best, rather than just correct, solutions are presented well and have a logical flow in the reasoning. Write the solutions for yourself, your future self, and an audience.

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Exercise A

1. Use a linear approximation about \(x=1\) to estimate \((1.03)^{1/3}\).
2. Use a quadratic approximation about \(x=1\) to estimate \((1.03)^{1/3}\).
3. Find the actual value of \((1.03)^{1/3}\) and determine which of the approximations is more accurate.

Exercise B

The curve \[3xe^{xy^2}-2y=3x^2+y^2\] implicitly defines \(y=f(x)\). What is the quadratic approximation to \(y\) about \(x=1\)?

Exercise C

Recall that we extended the linear approximation of \(f(x)\) about \(x_0\) to obtain the quadratic approximation of \(f(x)\) about \(x_0\).

1. Develop the cubic approximation using a similar argument. Take \(P_2(x)\) as the starting point and introduce \(P_3(x)=P_2(x)+c_3 (x-x_0)^3\). Determine \(c_3\) by ensuring that \(f(x_0)=P_3(x_0)\) and that the first three derivatives of \(f\) agree with \(P_3\) at \(x=x_0\).
2. Return to Exercise A and obtain a cubic approximation of \((1.03)^{1/3}\).
3. Obtain a cubic approximation of \(e^x\) about \(x=0\).

Exercise D

Suppose a firm is selling frisbees and it faces linear demand \(q=2000-20p\). The total costs of the firm is given by \(C(q)=0.05q^2+10000\).

1. Using what you have learned in Application 04, when will the firm stop adjusting production in order to maximize profits?
2. Using what you learned in Application 05, how are revenues affected by given demand conditions?
3. Using what you learned in Application 06, what would be the pricing rule of the firm if it maximizes profits?