Chapter 1 Business Calculus: Functions, Linear Models, and Financial Applications – Guided Study

Q1. Find the domain of each function.

Background

Topic: Functions and Domains

This question tests your understanding of how to determine the set of all possible input values (the domain) for different types of functions, which is foundational in calculus and business applications.

Key Terms and Formulas

• Domain: The set of all real numbers for which a function is defined.

• For polynomials, the domain is usually all real numbers.

• For rational functions, exclude values that make the denominator zero.

• For square roots, the expression inside must be non-negative.

Step-by-Step Guidance

1. For each function, identify its type (polynomial, rational, etc.).

2. Check for any restrictions: denominators that could be zero, square roots of negative numbers, or logarithms of non-positive numbers.

3. For (a) $F(x) = 2x^3 - x^2 + 3$, note that this is a polynomial.

4. For (b) $f(x) = x - 2x + 4$, simplify the expression first to see its form.

5. For (c) $g(x) = 7 - x$, recognize the function type and consider any restrictions.

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Q2. Price-Demand Function for Memory Chips

Background

Topic: Linear Functions and Revenue Modeling

This question involves interpreting and graphing a linear price-demand function, estimating prices for given demands, and constructing a revenue function, which are key skills in business calculus for modeling and analysis.

Key Terms and Formulas

• Price-Demand Function: $p(x) = 75 - 3x$

• Revenue Function: $R(x) = x \cdot p(x)$

• Domain: $1 \leq x \leq 20$ (where $x$ is in millions of chips)

Step-by-Step Guidance

1. For (a), plot the points for $x$ from 1 to 20 using $p(x) = 75 - 3x$.

2. Sketch the graph by connecting the plotted points, noting the linear decrease.

3. For (b), substitute $x = 7$ and $x = 11$ into the price-demand function to estimate the price per chip.

4. For (c), write the revenue function as $R(x) = x \cdot (75 - 3x)$ and simplify.

5. For (d), plot the revenue function for $x$ in the given domain and sketch its graph, noting its shape (parabola opening downward).

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Q3. Price-Supply and Price-Demand Equations for Barley

Background

Topic: Linear Equations and Market Equilibrium

This question tests your ability to model supply and demand with linear equations, find their intersection (equilibrium), and graph these relationships, which are essential in business and economics.

Key Terms and Formulas

• Linear Equation: $p = mx + b$

• Equilibrium Point: The point where supply equals demand.

Step-by-Step Guidance

1. For (a), use the two supply data points to find the slope $m$ and intercept $b$ for the supply equation.

2. For (b), repeat the process for the demand data points to find the demand equation.

3. For (c), set the supply and demand equations equal to each other to find the equilibrium quantity and price.

4. Graph both equations and mark the equilibrium point on the graph.

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Q4. Linear Depreciation of a Bulldozer

Background

Topic: Linear Depreciation

This question involves finding a linear equation to model depreciation and using it to estimate future values, a common application in business calculus.

Key Terms and Formulas

• Linear Depreciation: $V = mt + b$

• Given two points: $(t_1, V_1)$ and $(t_2, V_2)$

Step-by-Step Guidance

1. Identify the two points: at $t = 0$, $V = 224,000$; at $t = 8$, $V = 100,000$.

2. Calculate the slope $m$ using $m = \frac{V_2 - V_1}{t_2 - t_1}$.

3. Write the linear equation $V = mt + b$ using the slope and one point.

4. For (b), substitute $t = 12$ into your equation to estimate the value after 12 years.

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Q5. Cost and Average Cost Functions for Surfboards

Background

Topic: Linear Cost Functions and Average Cost

This question asks you to model total and average costs using linear functions, which is fundamental in business calculus for cost analysis.

Key Terms and Formulas

• Cost Function: $C(x) = mx + b$

• Average Cost Function: $C_{avg}(x) = \frac{C(x)}{x}$

• Fixed Cost: Cost that does not change with output.

• Variable Cost: Cost that changes with output.

Step-by-Step Guidance

1. Use the given fixed cost and total cost at $x = 20$ to set up two equations.

2. Solve for the variable cost per board (the slope $m$).

3. Write the total cost function $C(x)$.

4. For the average cost, divide $C(x)$ by $x$ to get $C_{avg}(x)$.

5. Consider the behavior of $C_{avg}(x)$ as $x$ increases, and identify any asymptotes.

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Q6. Continuous Compounding Investment

Background

Topic: Exponential Growth and Continuous Compounding

This question tests your ability to use the formula for continuous compounding to find the future value of an investment, a key financial application in business calculus.

Key Terms and Formulas

• Continuous Compounding Formula: $A = Pe^{rt}$

• $P$ = principal (initial investment)

• $r$ = annual interest rate (as a decimal)

• $t$ = time in years

• $A$ = amount after $t$ years

Step-by-Step Guidance

1. Identify the values: $P = 10,000$, $r = 0.0395$, $t = 12$.

2. Plug these values into the formula $A = Pe^{rt}$.

3. Calculate the exponent $rt$.

4. Compute $e^{rt}$ and multiply by $P$ to get $A$.

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Q7. Quarterly Compounding Investment

Background

Topic: Compound Interest (Non-Continuous)

This question involves using the compound interest formula for investments compounded a specific number of times per year, which is a common business calculus application.

Key Terms and Formulas

• Compound Interest Formula: $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• $P$ = principal

• $r$ = annual interest rate (decimal)

• $n$ = number of compounding periods per year

• $t$ = number of years

Step-by-Step Guidance

1. Identify the values: $P = 2,500$, $r = 0.07$, $n = 4$, $t = 15$.

2. Plug these values into the formula $A = P \left(1 + \frac{r}{n}\right)^{nt}$.

3. Calculate $\frac{r}{n}$ and $nt$.

4. Raise $1 + \frac{r}{n}$ to the $nt$ power and multiply by $P$.

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Q8. Present Value for Future Education Fund (Daily Compounding)

Background

Topic: Present Value and Compound Interest

This question asks you to find the present value needed to reach a future goal with daily compounding, a practical financial application in business calculus.

Key Terms and Formulas

• Compound Interest Formula (solving for P): $A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Rearranged: $P = \frac{A}{\left(1 + \frac{r}{n}\right)^{nt}}$

• $A$ = future value ($40,000$)

• $r$ = annual interest rate (decimal)

• $n$ = number of compounding periods per year (daily = 365)

• $t$ = number of years (17)

Step-by-Step Guidance

1. Identify the values: $A = 40,000$, $r = 0.055$, $n = 365$, $t = 17$.

2. Plug these values into the rearranged formula for $P$.

3. Calculate $\frac{r}{n}$ and $nt$.

4. Compute $\left(1 + \frac{r}{n}\right)^{nt}$ and divide $A$ by this value to find $P$.

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Q9. Doubling Time with Annual Compounding

Background

Topic: Exponential Growth and Doubling Time

This question tests your ability to determine how long it takes for an investment to double with a given annual compound interest rate, a classic business calculus problem.

Key Terms and Formulas

• Compound Interest Formula: $A = P(1 + r)^t$

• For doubling: $A = 2P$

• To solve for $t$: $2 = (1 + r)^t$

• Take the natural logarithm of both sides to solve for $t$.

Step-by-Step Guidance

1. Set up the equation for doubling: $2 = (1 + r)^t$ with $r = 0.2136$.

2. Take the natural logarithm of both sides: $\ln(2) = t \cdot \ln(1 + r)$.

3. Solve for $t$ by dividing both sides by $\ln(1 + r)$.

4. Calculate $\ln(2)$ and $\ln(1 + r)$, then divide to find $t$.

Try solving on your own before revealing the answer!