Question 1

A company produces widgets. The cost function is $$C(x) = 5x + 200$$, and the revenue function is $$R(x) = 10x. $$What is the break-even point?

Accepted Answer

$$x = 40$$

Question 2

Evaluate the limit: $$\lim_{x 	o 3} (x^2$ - 2x + 1)$$ using direct substitution.

Accepted Answer

$$4$$

Question 3

Find the derivative of $$f(x) = x^3$ - 2x^2$ + 5x$$ using the power rule.

Accepted Answer

$$f'(x) = 3x^2$ - 4x + 5$$

Question 4

Find the derivative of $$f(x) = \sqrt{x}.$$

Accepted Answer

$$f'(x) = \frac{1}{2\sqrt{x}}$$

Question 5

Find the derivative of $$f(x) = \frac{1}{x^2$}.$$

Accepted Answer

$$f'(x) = -2x$$^{-3}$$

Question 6

Find the derivative of $$f(x) = x^2$ e^x$$ using the product rule.

Accepted Answer

$$f'(x) = 2x e^x + x^2$ e^x$$

Question 7

Find the derivative of $$f(x) = \frac{x^2$ + 1}{x}$$ using the quotient rule.

Accepted Answer

$$f'(x) = \frac{x^2$ - 1}{x^2$}$$

Question 8

Find the derivative of $$f(x) = (3x + 2)^4$ using the chain rule.

Accepted Answer

$$f'(x) = 4(3x + 2)^3$ \cdot 3$$

Question 9

Find the second derivative of $$f(x) = x^3$ - 2x^2$ + 5x.$$

Accepted Answer

$$f''(x) = 6x - 4$$

Question 10

Find the derivative of $$f(x) = e$$^{2x}$$.

Accepted Answer

$$f'(x) = 2e$$^{2x}$$

Question 11

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

Accepted Answer

$$f'(x) = \frac{2x}{x^2$ + 1}$$

Question 12

A function $$f(x)$$ has a relative minimum at $$x = 2$$ and a relative maximum at $$x = 5. $$Which of the following statements is true?

Accepted Answer

The function decreases before $$x = 2$$, increases between $$x = 2$$ and $$x = 5$$, then decreases after $$x = 5.$$

Question 13

Find the coordinates of the inflection point for $$f(x) = x^3$ - 3x^2$ + 4.$$

Accepted Answer

$$(2, 0)$$

Question 14

A company’s profit function is $$P(x) = -2x^2$ + 40x - 100. $$How many units maximize profit?

Accepted Answer

$$x = 10$$

Question 15

Given the revenue function $$R(x) = 50x - x^2$, find the marginal revenue at x = 10$.

Accepted Answer

$$MR = 30$$

Question 16

If the demand function is $$p(x) = 100 - 2x$$, what is the elasticity at $$x = 20$$?

Accepted Answer

$$E = \frac{2 \cdot 20}{100 - 2 \cdot 20} = \frac{40}{60} = \frac{2}{3}$$

Question 17

Differentiate implicitly: $$x^2$$ + $$y^2 = 25$. Find $$\frac{dy}{dx}$$.

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 18

Find the integral $$\int (3x^2$ + 4x) dx$$ using the power rule.

Accepted Answer

$$x^3$$ + $$2x^2 + C$

Question 19

Find the definite integral $$\$$int_0$$^2 (x^2$ + 1) dx.$$

Accepted Answer

$$\frac{14}{3}$$

Question 20

Evaluate the partial derivative $$\frac{\partial f}{\partial x}$$ for $$f(x, y) = x^2$ y + 3xy^2$.

Accepted Answer

$$2xy + 3y^2$

Business Calculus Final Exam Study Guide – Step-by-Step Guidance

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