Q: Suppose $$f(x) is $$continuous at $$x = 2. $$Which of the following statements must be true?

$$\lim_{x \to 2} f(x)$$ exists and equals $$f(2).$$

Q: Which of the following is the correct application of L'Hôpital's Rule to $$\lim_{x \to 0} \frac{e^x - 1}{x}$$?

$$\lim_{x \to 0} \frac{e^x}{1} = 1$$

Question 1

Which of the following best describes the concept of a limit as introduced in Calculus I?

Accepted Answer

The value a function approaches as the input approaches a certain point.

Question 2

Suppose $$f(x) is $$continuous at $$x = 2. $$Which of the following statements must be true?

Accepted Answer

$$\lim_{x 	o 2} f(x)$$ exists and equals $$f(2).$$

Question 3

Evaluate the following limit: $$\lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 4

A car's position at time $$t is $$given by $$s(t) = 4t^2$ + 2t. $$What is the instantaneous velocity at $$t = 3$$?

Accepted Answer

$$26$$

Question 5

Which rule is used to differentiate the function $$f(x) = x^2$ \sin x$$?

Accepted Answer

Product Rule

Question 6

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

Accepted Answer

$$\frac{2x}{x^2$ + 1}$$

Question 7

Which of the following is an example of an implicit differentiation problem?

Accepted Answer

Finding $$\frac{dy}{dx} if$$ $$x^2$$ + $$y^2 = 25$

Question 8

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

Accepted Answer

$$-2.5e^{-1}$$

Question 9

Which of the following is the correct application of the Chain Rule for $$y = \sin(3x^2$)$$?

Accepted Answer

$$y' = \cos(3x^2$) \cdot 6x$$

Question 10

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

Accepted Answer

$$4$$

Question 11

Which theorem guarantees that a continuous function on $$[a, b]$$ attains both a maximum and a minimum?

Accepted Answer

Extreme Value Theorem

Question 12

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

Accepted Answer

$$8$$

Question 13

Which of the following is a correct statement of the Fundamental Theorem of Calculus?

Accepted Answer

If $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Question 14

A tank is being filled with water at a rate given by $$R(t) = 5 + 2t$$ liters per minute. How much water is added between $$t = 0$$ and $$t = 3$$ minutes?

Accepted Answer

$$24$$ liters

Question 15

Which of the following is the correct $$u-$$substitution for $$\int 2x \cos(x^2$) dx$$?

Accepted Answer

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

Question 16

A ball is thrown upward with velocity $$v(t) = 20 - 9.8t. $$What is the maximum height reached by the ball?

Accepted Answer

$$\frac{400}{19.6}$$ meters

Question 17

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

Accepted Answer

$$2e^{2x}$$

Question 18

If $$f(x) is $$differentiable on $$[a, b]$$, which theorem guarantees there exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

Question 19

Which of the following is the correct application of L'Hôpital's Rule to $$\lim_{x 	o 0} \frac{e^x - 1}{x}$$?

Accepted Answer

$$\lim_{x 	o 0} \frac{e^x}{1} = 1$$

MAT 271-100 Calculus I Syllabus and Course Structure Study Guide

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