# 8 Differentials

## 8.1 Differentials

### Practice Problems

**Problem 8.1.1: **Find the differential of the function below and use it to estimate the change in when and . Check the approximation algebraically.

Answer:

Approximated change is 0.6, actual change is 0.61.

**Problem 8.1.2: **Find the differential of the function below and use it to estimate the change in when and . Check the approximation algebraically.

Answer:

Approximated change is 8,865,801,600.

**Problem 8.1.3: **Find the differential of the function below and use it to estimate the change in when and . Check the approximation algebraically.

Answer:

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 and . Check the approximation algebraically.

Answer:

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

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

Create a differential that measures the estimated change in profit and use it to estimate the change when and . Check the approximation algebraically.

Answer:

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

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

Find the marginal cost function and use it to estimate the change in revenue when and . Check the approximation algebraically.

Answer:

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:

Create a differential that measures the estimated change in profit and use it to estimate the change when and . Check the approximation algebraically.

Answer:

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 .

Answer: -0.30

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

Answer: -0.13; -2.57; 11.55

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## 8.2 Total differentials

### Practice Problems

**Problem 8.2.1: **Find the differential for the function below:

Use it to approximate the change in at the point when and Compare the approximation to the actual change in .

Answer:

Approximated change is 4, actual change is 4.13.

**Problem 8.2.2: **Find the differential for the function below:

Use it to approximate the change in at the point when and Compare the approximation to the actual change in .

Answer:

Approximated change is -50,422,500.

**Problem 8.2.3: **Find the differential for the function below:

Use it to approximate the change in at the point when and Compare the approximation to the actual change in .

Answer:

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

### External Resources

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