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

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

Accepted Answer

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

Question 4

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

Accepted Answer

$$v(2) = 14$$

Question 5

Which rule is used to differentiate the function $$f(x) = (2x^3$ + 5x)(x^2$ - 1)$$?

Accepted Answer

Product Rule

Question 6

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

Accepted Answer

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

Question 7

A medication is administered and its concentration in the bloodstream is modeled by $$C(t) = 10e$$^{-0.5t}$$. What is the rate of change of the concentration at t = 2$ hours?

Accepted Answer

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

Question 8

Which of the following statements is true about the Mean Value Theorem for derivatives?

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 9

Evaluate $$\int 3x^2$ dx.$$

Accepted Answer

$$x^3 + C$

Question 10

A company wants to minimize the cost $$C(x) = x^2$ - 8x + 20$$ for producing $$x$$ units. What is the optimal number of units to produce?

Accepted Answer

$$x = 4$$

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

Find the equation of the tangent line to $$y = x^2$ at x = 1$.

Accepted Answer

$$y = 2x - 1$$

Question 13

Which of the following is the correct antiderivative of $$f(x) = \cos(x)$$?

Accepted Answer

$$F(x) = \sin(x) + C$$

Question 14

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

Accepted Answer

$$40$$ liters

Question 15

Which of the following is the correct use of the Chain Rule for $$f(x) = \sin(3x)$$?

Accepted Answer

$$f'(x) = 3\cos(3x)$$

Question 16

If $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$increasing on that interval.

Question 17

Evaluate $$\$$int_0$$^1 2x dx.$$

Accepted Answer

$$1$$

Question 18

Which of the following is the correct substitution for evaluating $$\int x e$$^{x^2}$$ dx$$?

Accepted Answer

Let $$u = x^2$, then du = 2x dx$

Calculus I Syllabus and Course Structure – MATH 2600 (CT State Community College)

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