Question 1

Evaluate the following limit using L'Hôpital's Rule: $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 2

Which of the following best describes a vertical asymptote for the function $$f(x)$$?

Accepted Answer

A value $$x = a$$ where $$f(x)$$ approaches $$\infty or$$ $$-\infty as$$ $$x$$ approaches $$a.$$

Question 3

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

Accepted Answer

$$5x^4$$

Question 4

A company’s profit $$P(x) is $$given by $$P(x) = -2x^2$ + 40x - 100. At $$what value of $$x is $$the profit maximized?

Accepted Answer

$$x = 10$$

Question 5

Which of the following is the correct application of the product rule for differentiation for $$f(x) = u(x)v(x)$$?

Accepted Answer

$$f'(x) = u(x)v'(x) + u'(x)v(x)$$

Question 6

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

Accepted Answer

$$2e^{2x}$$

Question 7

If the demand function is $$p = 100 - 2q$$, what is the elasticity of demand when $$q = 20$$?

Accepted Answer

$$\frac{4}{3}$$

Question 8

Which of the following is the correct antiderivative of $$f(x) = x^n$$ for $$n 
eq -1$$?

Accepted Answer

$$\frac{x$$^{n+1}$$}{n+1} + C$$

Question 9

A medication decays in the bloodstream according to $$C(t) = C_0$ e$$^{-kt}$$. If C_0$ = 100 mg $$and $$k = 0.2$$, what is the concentration after 5 hours?

Accepted Answer

$$100 e$$^{-1}$$ mg

Question 10

Evaluate the definite integral $$\$$int_0$$^2 (3x^2$) dx.$$

Accepted Answer

$$12$$

Question 11

Which of the following statements about critical numbers is correct?

Accepted Answer

A critical number is where the derivative is zero or undefined.

Question 12

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

Accepted Answer

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

Question 13

Which of the following best describes the Fundamental Theorem of Calculus?

Accepted Answer

It relates differentiation and integration as inverse processes.

Question 14

Find the slope of the tangent line to $$y = x^3$ - 2x at$$ $$x = 1.$$

Accepted Answer

$$1$$

Question 15

Which of the following is the correct solution to the differential equation $$\frac{dy}{dt} = ky$$ with initial condition $$y(0) = y_0$?

Accepted Answer

$$y = y_0$ e$$^{kt}$$

Business Calculus: Limits, Derivatives, and Integrals – Course Outline and Key Concepts

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