Q: A company’s cost function is $$C(q) = 5000 + 2q. If $$they produce 100 units, what is the total cost?

$$C(100) = 5000 + 2 \times 100 = 5200$$

Q: If the revenue function is $$R(q) = 4q$$ and the cost function is $$C(q) = 5000 + 2q$$, what is the profit when 100 units are sold?

$$\pi(100) = 4 \times 100 - (5000 + 2 \times 100) = 400 - 5200 = -4800$$

Question 1

A company finds that for every $$1 increase in price, the quantity demanded drops by 5 units. If 50 units are sold at 200$$, what is the correct linear demand function $$q = D(p)$$?

Accepted Answer

$$q = -5p + 1050$$

Question 2

Given the supply function $$q = S(p) = 50p + 500$$ and the demand function $$q = -5p + 1050$$, what is the equilibrium price $$p$$?

Accepted Answer

$$p = 10$$

Question 3

If $$1,200 is $$invested at an annual interest rate of $$7\%$$ compounded monthly for $$10$$ years, which formula correctly represents the account balance?

Accepted Answer

$$A = 1200\left(1 + \frac{0.07}{12}\$$right)^{120}$$

Question 4

A business has a cost function $$C(q) = 5000 + 2q$$ and a revenue function $$R(q) = 4q. $$What is the break-even quantity?

Accepted Answer

$$q = 2500$$

Question 5

Suppose $$1,000 is $$invested at a $$5\%$$ annual interest rate compounded continuously. How many years will it take for the account to reach $$5,000$$?

Accepted Answer

$$t = \frac{\ln(5)}{0.05}$$

Question 6

Which of the following best describes the fixed cost in the function $$C(q) = 5000 + 2q$$?

Accepted Answer

$$5000$$

Question 7

If the price per item is $$4 in $$the revenue function $$R(q) = 4q$$, what does this represent?

Accepted Answer

The price at which the commodity is sold

Question 8

Given the profit function $$\pi(q) = 2q - 5000$$, what does $$\pi(10)$$ represent?

Accepted Answer

The profit when 10 units are sold

Question 9

A company wants to double its investment using continuous compounding at a rate $$r. $$What is the formula for the doubling time $$t$$?

Accepted Answer

$$t = \frac{\ln(2)}{r}$$

Question 10

Which equation correctly models the account balance $$A(t)$$ for an initial investment $$P at $$annual rate $$r$$ compounded $$n$$ times per year for $$t$$ years?

Accepted Answer

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

Question 11

If a linear function passes through points $$(x_1$, y_1$)$$ and $$(x_2$, y_2$)$$, what is the formula for its slope $$a$$?

Accepted Answer

$$a = \frac{y_2$ - y_1$}{x_2$ - x_1$}$$

Question 12

A budget constraint is given by $$4x + 2y \leq 100. If$$ $$x is $$the number of books and $$y is $$the number of pens, what does the slope of the constraint represent?

Accepted Answer

The rate at which pens can be traded for books

Question 13

Which of the following is NOT a property of exponential functions?

Accepted Answer

They always have a constant slope

Question 14

If the demand function is $$q = -5p + 1050$$ and the supply function is $$q = 50p + 500$$, what is the equilibrium quantity $$q$$?

Accepted Answer

$$q = 1000$$

Question 15

Which formula would you use to solve for $$t in $$the equation $$A(t) = P e$$^{rt}$$ when A(t$$)$$, P$, and r$ are known?

Accepted Answer

$$t = \frac{\ln\left(\frac{A(t)}{P}\right)}{r}$$

Question 16

A company’s cost function is $$C(q) = 5000 + 2q. If $$they produce 100 units, what is the total cost?

Accepted Answer

$$C(100) = 5000 + 2 	imes 100 = 5200$$

Question 17

If a function is exponential and passes through $$(0, 100)$$ and $$(2, 400)$$, what is its equation?

Accepted Answer

$$y = 100 \cdot 2^x$$

Question 18

Which equation would you use to solve for the time $$t it $$takes for an investment to triple with continuous compounding at rate $$r$$?

Accepted Answer

$$t = \frac{\ln(3)}{r}$$

Question 19

If the revenue function is $$R(q) = 4q$$ and the cost function is $$C(q) = 5000 + 2q$$, what is the profit when 100 units are sold?

Accepted Answer

$$\pi(100) = 4 	imes 100 - (5000 + 2 	imes 100) = 400 - 5200 = -4800$$

Business Calculus Exam 1 Review – Step-by-Step Guidance

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