Q: Which of the following is the correct application of L'Hospital's Rule?

$$\lim_{x \to 0} \frac{\sin x}{x}$$

Question 1

Let $$f(x) = 3x^2$ - 2x + 5. $$What is the derivative $$f'(x)$$?

Accepted Answer

$$f'(x) = 6x - 2$$

Question 2

A ball is thrown vertically upward with an initial velocity of $$20 m/s. $$Using $$s(t) = -4.9t^2$ + 20t + 2$$, what is the velocity after $$2$$ seconds?

Accepted Answer

$$v(2) = 0.4 m/s$$

Question 3

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

Accepted Answer

$$4$$

Question 4

Which of the following functions is continuous everywhere?

Accepted Answer

$$f(x) = \sin(x)$$

Question 5

If $$y = e$$^{3x}$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

$$3e^{3x}$$

Question 6

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

Accepted Answer

$$y = 2x - 1$$

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 concentration at t = 2$ hours?

Accepted Answer

$$-5e^{-1}$$

Question 8

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

Accepted Answer

If $$f is $$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 $$\$$int_0$$^1 (3x^2$ + 2x) dx.$$

Accepted Answer

$$2.5$$

Question 10

Suppose $$f(x) is $$differentiable and $$f'(x) \u003e 0$$ for all $$x in$$ $$(a, b). $$Which of the following is true?

Accepted Answer

$$f(x) is $$increasing on $$(a, b)$$

Question 11

Find the critical points of $$f(x) = x^3$ - 3x^2$ + 2.$$

Accepted Answer

$$x = 1, 2$$

Question 12

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

$$21$$ liters

Question 13

Which of the following is the correct application of L'Hospital's Rule?

Accepted Answer

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

Question 14

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 15

A population of bacteria grows according to $$P(t) = 100e$$^{0.2t}$$. What is the population's instantaneous rate of growth at t = 5$ hours?

Accepted Answer

$$20e^{1}$$

Question 16

Find $$\int x e$$^{x^2}$$ dx.$$

Accepted Answer

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

Unit 2 Calculus Practice Exercises Overview

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