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

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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.

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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.

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