Differentials

J. Zachary Klingensmith

8 Differentials

8.1 Differentials

Practice Problems

Problem 8.1.1: Find the differential $dy$ of the function below and use it to estimate the change in $y$ when $x=2$ and $dx=0.1$ . Check the approximation algebraically.

$y=x^2+2x-1$

Answer:

$dy = (2x+2)dx$

Approximated change is 0.6, actual change is 0.61.

Problem 8.1.2: Find the differential $dy$ of the function below and use it to estimate the change in $y$ when $x=5$ and $dx=0.1$ . Check the approximation algebraically.

$y=(x^3-1)^5$

Answer:

$dy =\left[15x^2(x^3-1)^4\right]dx$

Approximated change is 8,865,801,600.

Problem 8.1.3: Find the differential $dy$ of the function below and use it to estimate the change in $y$ when $x=2$ and $dx=0.1$ . Check the approximation algebraically.

$y=\frac{x+1}{x-1}$

Answer:

$dy = \frac{-2}{(x-1)^2}dx$

Approximated change is -0.2, actual change is -0.19.

Problem 8.1.4: The revenue function for a firm is given below. Find the marginal revenue function and use it to estimate the change in revenue when $Q=500$ and $dQ=10$ . Check the approximation algebraically.

$R(Q)=1000Q+20Q^2-.005Q^3$

Answer:

$dR = (1000+40Q-0.015Q^2)dQ$

Approximated change is 172,500, actual change is 173,745.

Problem 8.1.5: A firm has estimated the following revenue and cost functions:

$R(Q)=400Q-Q^2$

$C(Q)=175+7Q$

Create a differential that measures the estimated change in profit and use it to estimate the change when $Q=50$ and $dQ=10$ . Check the approximation algebraically.

Answer:

$d\pi = (-2Q+383)dQ$

Approximated change is 2,930, actual change is 2,830.

Problem 8.1.6: A manufacturer has determined that the total cost $C$ (in dollars) of operating a factory is

$C(Q)=0.5Q^2+10Q+7200.$

Find the marginal cost function and use it to estimate the change in revenue when $Q=100$ and $dQ=10$ . Check the approximation algebraically.

Answer:

$dTC=(Q+10)dQ$

Approximated change is 1,100, actual change is 1,150.

Problem 8.1.7: Suppose that a firm has determined that its (inverted) demand function and cost function can be given as, respectively:

$P(Q)=70-0.01Q$

$C(Q)=8000+50Q+0.03Q^2$

Create a differential that measures the estimated change in profit and use it to estimate the change when $Q=150$ and $dQ=5$ . Check the approximation algebraically.

Answer:

$d\pi=(-0.08Q+20)dQ$

Approximated change is 40, actual change is 39.

Problem 8.1.8: The demand equation for a product is given below. Calculate the price-elasticity of demand when $P=70$ .

$Q_D=15000-50P$

Answer: -0.30

Problem 8.1.9: The demand equation for a product is given below. Calculate the price-elasticity of demand when $P=5$ . What if $P=10$ ? At what price is revenue maximized?

$Q_D=400-P^2$

Answer: -0.13; -2.57; 11.55

External Resources

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8.2 Total differentials

Practice Problems

Problem 8.2.1: Find the differential $dz$ for the function below:

$z=5x^2-2xy+3y^2-5.$

Use it to approximate the change in $z$ at the point $(5,1)$ when $\Delta x=0.1$ and $\Delta y=0.2.$ Compare the approximation to the actual change in $z$ .

Answer:

$dz=(10x-2y)dx+(-2x+6y)dy$

Approximated change is 4, actual change is 4.13.

Problem 8.2.2: Find the differential $dz$ for the function below:

$z=(x^2y-xy^3)^5.$

Use it to approximate the change in $z$ at the point $(-1,3)$ when $\Delta x=0.25$ and $\Delta y=-0.15.$ Compare the approximation to the actual change in $z$ .

Answer:

$dz=5(x^2y-xy^3)^4(2xy-y^3)dx+(5(x^2y-xy^3)^4(x^2-3xy^2)dy$

Approximated change is -50,422,500.

Problem 8.2.3: Find the differential $dz$ for the function below:

$z=\frac{x^2-1}{xy-4}.$

Use it to approximate the change in $z$ at the point $(3,3)$ when $\Delta x=0.1$ and $\Delta y=0.1.$ Compare the approximation to the actual change in $z$ .

Answer:

*** QuickLaTeX cannot compile formula:
dz=\left(\frac{x^2y-8x+y}{(xy-4)^2}\right)dx+\left(\frac{-x^3+x}{(xy-4)^2\right)dy

*** Error message:
File ended while scanning use of \frac .
Emergency stop.

Approximated change is -0.072, actual change is -0.065.

External Resources

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Student Companion for Mathematical Economics Copyright © by J. Zachary Klingensmith is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License, except where otherwise noted.

8.1 Differentials

Practice Problems

External Resources

8.2 Total differentials

Practice Problems

External Resources

License

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