Question 1

Which of the following best describes a function?

Accepted Answer

A relation where each input is paired with exactly one output.

Question 2

Given the function $$f(x) = 2x^2$ - 3x + 5$$, what is the value of $$f(4)$$?

Accepted Answer

$$29$$

Question 3

Which transformation occurs when the graph of $$y = f(x) is $$changed to $$y = f(x-3) + 2$$?

Accepted Answer

Shift right by 3 units and up by 2 units.

Question 4

If $$f(x) = x^2$ and g(x) = 2x + 1$, what is (f$$ \circ g)(x)$$?

Accepted Answer

$$(2x + 1)^2$

Question 5

Which of the following is the correct definition of a limit?

Accepted Answer

The value that $$f(x)$$ approaches as $$x$$ approaches a specific value.

Question 6

Evaluate $$\lim_{x 	o 2} (3x + 4).$$

Accepted Answer

$$10$$

Question 7

Which of the following functions is continuous everywhere?

Accepted Answer

$$f(x) = e^x$$

Question 8

Find the derivative of $$f(x) = 5x^3$.

Accepted Answer

$$15x^2$$

Question 9

A company’s cost function is $$C(x) = 100 + 5x. $$What is the marginal cost?

Accepted Answer

$$5$$

Question 10

If $$f(x) = e$$^{2x}$$, what is f$$'(x)$$?

Accepted Answer

$$2e^{2x}$$

Question 11

Find the derivative of $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 12

If $$y = x^2$ e^x$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$2x e^x + x^2$ e^x$$

Question 13

A tank is being filled with water at a rate of $$5t^2$$ liters per minute. How many liters are added in the first 3 minutes?

Accepted Answer

$$\$$int_0$$^3 5t^2$ \, dt = 45$$

Question 14

Which of the following is the correct application of the chain rule?

Accepted Answer

If $$y = f(g(x))$$, then $$\frac{dy}{dx} = f'(g(x)) \cdot g'(x).$$

Question 15

Find the critical points of $$f(x) = x^3$ - 3x^2$ + 4.$$

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 16

A business wants to maximize profit given $$P(x) = -2x^2$ + 40x - 300. $$What is the value of $$x$$ that maximizes profit?

Accepted Answer

$$x = 10$$

Question 17

Evaluate $$\int 6x \, dx.$$

Accepted Answer

$$3x^2 + C$

Question 18

What does the Fundamental Theorem of Calculus state?

Accepted Answer

If $$F is an $$antiderivative of $$f on$$ $$[a, b]$$, then $$\int_a^b f(x) \, dx = F(b) - F(a).$$

Question 19

A company’s revenue function is $$R(x) = 50x - x^2$. What is the marginal revenue when x = 10$?

Accepted Answer

$$30$$

Business Calculus: Core Topics and Chapter Structure

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