Q: Which of the following is the correct application of L’Hôpital’s Rule to $$\lim_{x \to 0} \frac{\sin(x)}{x}$$?

$$\lim_{x \to 0} \frac{\cos(x)}{1} = 1$$

Question 1

Which of the following best describes the derivative of a function $$f(x) at a $$point $$x = a$$?

Accepted Answer

The slope of the tangent line to the graph of $$f(x) at$$ $$x = a.$$

Question 2

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

Accepted Answer

$$4$$

Question 3

A car's position at time $$t is $$given by $$s(t) = 3t^2$ + 2t + 1. $$What is the instantaneous velocity of the car at $$t = 2$$ seconds?

Accepted Answer

$$s'(2) = 14$$

Question 4

Which of the following functions is NOT continuous at $$x = 0$$?

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

A pharmaceutical company models the concentration $$C of a $$drug in the bloodstream by $$C(t) = 5e$$^{-0.2t}$$, where t$ is in hours. What is the rate of change of the concentration at t = 3$ hours?

Accepted Answer

$$C'(3) = -1e$$^{-0.6}$$

Question 7

Which of the following statements about the Mean Value Theorem is TRUE?

Accepted Answer

If $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}.$$

Question 8

Find the derivative of $$y = \sin(x^2$)$$ with respect to $$x.$$

Accepted Answer

$$y' = 2x \cos(x^2$)$$

Question 9

A company’s profit $$P(x)$$ from selling $$x$$ units is given by $$P(x) = -2x^2$ + 40x - 100. $$How many units should the company sell to maximize profit?

Accepted Answer

$$x = 10$$

Question 10

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

Accepted Answer

$$16$$

Question 11

Which of the following is the correct application of L’Hôpital’s Rule to $$\lim_{x 	o 0} \frac{\sin(x)}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{\cos(x)}{1} = 1$$

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

It relates the process of differentiation and integration.

Question 14

A tank is being filled with water at a rate given by $$R(t) = 4t$$ liters per minute. How much water is added to the tank between $$t = 0$$ and $$t = 5$$ minutes?

Accepted Answer

$$\$$int_0$$^5 4t \, dt = 50$$ liters

Question 15

Which of the following is the correct antiderivative of $$f(x) = 6x^2$?

Accepted Answer

$$2x^3 + C$

Question 16

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). If$$ $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$, what can you conclude about $$f(x) on$$ $$[a, b]$$?

Accepted Answer

$$f(x) is $$increasing on $$[a, b].$$

Question 17

Find $$\frac{dy}{dx} if$$ $$x^2$$ + $$y^2 = 25$ (use implicit differentiation).

Accepted Answer

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

Question 18

A ball is thrown upward with a velocity $$v(t) = 20 - 9.8t (in m/s). At $$what time does the ball reach its maximum height?

Accepted Answer

$$t = 2.04$$ seconds

Question 19

Which of the following is a correct statement about the continuity of a function $$f(x) at$$ $$x = a$$?

Accepted Answer

$$\lim_{x 	o a} f(x) = f(a)$$

Calculus I Syllabus and Topic Overview – MAT 122 (Nassau Community College)

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