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

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

Q: Which of the following statements about limits is TRUE?

If $$f(x) is $$continuous at $$x = a$$, then $$\lim_{x \to a} f(x) = f(a).$$

Question 1

Which of the following best describes the limit of a function $$f(x) as$$ $$x$$ approaches $$a$$?

Accepted Answer

The value that $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$a$$

Question 2

Evaluate the following limit: $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$

Accepted Answer

$$4$$

Question 3

A company’s profit $$P(x) in $$thousands of dollars is given by $$P(x) = -2x^2$ + 12x - 16$$, where $$x is $$the number of units sold in thousands. What is the marginal profit when $$x = 2$$?

Accepted Answer

$$P'(2) = 4$$

Question 4

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

Accepted Answer

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

Question 5

If $$f(x) = 3x^2$ - 5x + 2$$, what is $$f'(x)$$?

Accepted Answer

$$6x - 5$$

Question 6

A medication’s concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

$$C'(2) = -5e$$^{-1}$$

Question 7

Which of the following limits does NOT exist?

Accepted Answer

$$\lim_{x 	o 0} \frac{1}{x}$$

Question 8

A business’s cost function is $$C(x) = 500 + 20x$$, where $$x is $$the number of units produced. What is the marginal cost?

Accepted Answer

$$20$$

Question 9

Find the derivative of $$f(x) = \frac{1}{x}$$ using the power rule.

Accepted Answer

$$-x^{-2}$$

Question 10

If $$f(x) = x^3$ - 6x^2$ + 9x + 1$$, at what value of $$x$$ does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 3$$

Question 11

A company’s revenue function is $$R(x) = 100x - x^2$. What is the maximum revenue, and at what value of x$ does it occur?

Accepted Answer

Maximum revenue is $$2500 at$$ $$x = 50$$

Question 12

Evaluate $$\lim_{x 	o \infty} \frac{5x^2$ + 3x}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 13

Which of the following is the correct derivative of $$f(x) = e$$^{2x}$$?

Accepted Answer

$$2e^{2x}$$

Question 14

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

Accepted Answer

$$\frac{1}{x}$$

Question 15

A population of bacteria grows according to $$P(t) = 100e$$^{0.03t}$$, where t$ is in hours. What is the instantaneous rate of growth at t = 10$ hours?

Accepted Answer

$$3e^{0.3}$$ 	imes 100$$

Question 16

Which of the following statements about limits is TRUE?

Accepted Answer

If $$f(x) is $$continuous at $$x = a$$, then $$\lim_{x 	o a} f(x) = f(a).$$

Question 17

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

Accepted Answer

$$\frac{1}{2\sqrt{x}}$$

Question 18

A company’s demand function is $$p = 50 - 2x$$, where $$p is $$the price per unit and $$x is $$the number of units sold. What is the price elasticity of demand when $$x = 10$$?

Accepted Answer

$$-2$$

Question 19

If $$f(x) = x^2$ and g(x) = 2x + 1$, what is the derivative of h(x) = f(g(x$$))$$?

Accepted Answer

$$4x + 2$$

Question 20

Evaluate $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Business Calculus: Exponents, Radicals, Rational Expressions, and Limits

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