Q: A population of bacteria grows according to the model $$P(t) = 200e$$^{0.03t}$$, where t$ is in hours. What is the population’s instantaneous rate of change at t = 10$ hours?

$$P'(10) = 6e$$^{0.3}$$ \times 200$$

Q: Which of the following is the correct definition of the derivative of a function $$f(x) at$$ $$x = a$$?

$$\lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$$

Q: A company’s cost function is $$C(x) = 1000 + 20x + 0.5x^2$. What is the marginal cost when x = 40$ units?

$$C'(40) = 20 + 2 \times 40 = 100$$

Question 1

A company’s revenue, in dollars, from selling x units of a product is modeled by the function $$R(x) = 5000(1 - e$$^{-0.02x}$$). $$What is the instantaneous rate of change of revenue with respect to units sold when $$x = 50$$?

Accepted Answer

$$R'(50) = 100e$$^{-1}$$

Question 2

Which of the following best describes the average rate of change of a function $$f(x)$$ over the interval $$[a, b]$$?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Question 3

A population of bacteria grows according to the model $$P(t) = 200e$$^{0.03t}$$, where t$ is in hours. What is the population’s instantaneous rate of change at t = 10$ hours?

Accepted Answer

$$P'(10) = 6e$$^{0.3}$$ 	imes 200$$

Question 4

A pharmaceutical company wants to maximize profit, given by $$P(x) = -2x^2$ + 120x - 1000$$, where $$x is $$the number of units produced. What is the optimal number of units to produce for maximum profit?

Accepted Answer

$$x = 60$$

Question 5

Which of the following is the correct definition of the derivative of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 6

A company’s cost function is $$C(x) = 1000 + 20x + 0.5x^2$. What is the marginal cost when x = 40$ units?

Accepted Answer

$$C'(40) = 20 + 2 	imes 40 = 100$$

Question 7

If the demand for a product is given by $$p = 100 - 2x$$, where $$p is $$the price and $$x is $$the quantity sold, what is the revenue function $$R(x)$$?

Accepted Answer

$$R(x) = 100x - 2x^2$

Question 8

Which of the following statements about exponential growth is correct?

Accepted Answer

The rate of change increases as the quantity increases.

Question 9

Suppose the rate at which water flows into a tank is given by $$r(t) = 5 + 2t$$ liters per minute. How much water flows into the tank from $$t = 0 to$$ $$t = 10$$ minutes?

Accepted Answer

$$\$$int_0$$^{10} (5 + 2t) dt = 150$$ liters

Question 10

A company’s profit is maximized when the marginal revenue equals marginal cost. If $$R(x) = 50x - x^2$ and C(x) = 10x + 20$, what is the production level x$ that maximizes profit?

Accepted Answer

$$x = 20$$

Question 11

Which of the following is the correct interpretation of the definite integral $$\int_a^b f(x) dx in a $$business context?

Accepted Answer

It represents the accumulated change in $$f(x)$$ from $$x = a to$$ $$x = b.$$

Question 12

If $$f(x) = \ln(x)$$, what is $$f'(x)$$?

Accepted Answer

$$f'(x) = \frac{1}{x}$$

Question 13

A company’s revenue is modeled by $$R(x) = 100x e$$^{-0.01x}$$. What is the value of x$ that maximizes revenue?

Accepted Answer

$$x = 100$$

Question 14

Which of the following is a property of logarithmic functions?

Accepted Answer

They are only defined for positive real numbers.

Question 15

A company’s cost function is $$C(x) = 500 + 15x. $$What is the average cost per unit when 50 units are produced?

Accepted Answer

$$\frac{C(50)}{50} = 25$$

Question 16

If $$f(x) = 3x^2$ - 4x + 7$$, what is the equation of the tangent line to $$f(x) at$$ $$x = 2$$?

Accepted Answer

$$y = 8x - 5$$

Question 17

A company’s profit function is $$P(x) = -x^2$ + 40x - 300. $$What is the maximum profit?

Accepted Answer

$$P(20) = 100$$

Question 18

Which of the following best describes a linear function?

Accepted Answer

A function of the form $$f(x) = mx + b$$

Question 19

If the marginal cost of producing $$x$$ units is $$MC(x) = 4x + 10$$, what is the total cost function $$C(x)$$, assuming fixed costs are $$500$$?

Accepted Answer

$$C(x) = 2x^2$ + 10x + 500$$

Brief Survey of Calculus I (MATH-M 119) - Course Overview and Learning Objectives

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