# Lesson Worksheet: Real-World Applications of Derivatives Mathematics

In this worksheet, we will practice applying the power, product, quotient, and chain rules for differentiation to business, economics, management sciences, social sciences, and life sciences.

**Q1: **

A population study on a conservation forest has determined that the population of a certain species of rabbits depends on the population of gray foxes in the same forest. If there are foxes in the forest, then the expected number of rabbits is given by , where . Find the rate of change in the expected population of this species of rabbits with respect to the population of gray foxes when there are 200 gray foxes in the forest. Round your answer to the nearest whole number.

**Q2: **

A company manufacturing shoes makes $100,000 in total cumulative profit by the beginning of the year 2015. The companyβs total cumulative profit years from the beginning of the year 2015 is thousand dollars.

Find the rate of change of the companyβs profit at the beginning of the year 2020. Round your answer to the nearest dollar.

- A/year
- B/year
- C/year
- D/year
- E/year

**Q3: **

Consider a savings account with an annual interest rate of , compounded continuously. The initial account balance is $12,000, and the balance in dollars after years is given by .

Find the rate of change of the account balance with respect to time in 5 years and in 10 years. Round your answers to the nearest cent.

- A$ 3,852.08 /year, $9,892.33/year
- B$ 770.42 /year, $989.23/year
- C$ 1,217.10 /year, $3,639.18/year
- D$ 15,408.31 /year, $19,784.66/year

**Q4: **

The demand for salmon in a fish market depends on the market price. When the market price is on a given day, the number of salmon in demand in the fish market on that day is given by , where .

Find the rate of change of the demand for salmon in the fish market with respect to the price of salmon when it sells for $20 on a given day. Round your answer to the nearest hundredth.

- A salmon/dollar
- B469.67 salmon/dollar
- C salmon/dollar
- D salmon/dollar
- E999.50 salmon/dollar

**Q5: **

A population study on a conservation forest develops a model that can predict the population of a certain species of rabbit in the forest. Due to the presence of predators in the forest, the rabbit population is estimated by a periodic function. Initially, there are 2,000 rabbits in the forest, and the rabbit population months after the start of the study is estimated by , where .

Find the rate of change in the rabbit population 2 months after the start of the study. Round your answer to the nearest integer.

- A131 rabbits/month
- B2,131 rabbits/month
- C rabbits/month
- D785 rabbits/month
- E2,433 rabbits/month

**Q6: **

Consider a savings account with an annual interest rate , compounded monthly. If the initial account balance is 1,000, the balance in dollars after 3 years is given by .

Find the rate of change of the account balance with respect to interest rate when and when . Round your answers to the nearest cent.

- A$39,287.60/interest rate, $48,133.40/interest rate
- B$91.17/interest rate, $112.35/interest rate
- C$1,094.05/interest rate, $1,348.18/interest rate
- D$3,273.97/interest rate, $4,011.12/interest rate

**Q7: **

The population of a certain species of rabbit in a conservation area is predicted to be thousands in five years, where is the maximum level of population, in thousands, of the rabbit species sustainable in the conservation area. Find the rate of change of the predicted population with respect to when the maximum level of population sustainable in this area is 20 thousands rabbits. Round your answer to the nearest hundredth.

**Q8: **

The balance of a loan, in dollars, years after its inception, is given by . Find the rate of change of the loan balance, with respect to , 3 years after the loanβs inception. Round your answer to the nearest cent.

- A/year
- B/year
- C/year
- D/year

**Q9: **

A factory determines that the cost, in dollars, of the factoryβs monthly operation when making smartphones is given by , where . Use the marginal cost function to approximate the cost of making the 101st smartphone. Round your answer to the nearest dollar.

**Q10: **

A factory determines that the cost, in dollars, of the factoryβs monthly operation of making smartphones is given by , where . The rate of change in the average cost per smartphone when the factory manufactures 100 smartphones = . Round your answer to the nearest dollar.

A team determines that if the factory manufactures smartphones per month, the recommended retail price per smartphone, in dollars, is , where .

Assuming that all manufactured smartphones are sold at the recommended price, the rate of change in the monthly profit when 100 smartphones are manufactured = . Round your answer to the nearest dollar.