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

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

Question 1

Suppose the position of a car moving along a straight road is given by $$s(t) = 3t^2$ + 2t + 1$$, where $$s is in $$meters and $$t is in $$seconds. What is the instantaneous velocity of the car at $$t = 4$$ seconds?

Accepted Answer

$$26 m/s$$

Question 2

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

Accepted Answer

$$4$$

Question 3

Which of the following statements best describes the Intermediate Value Theorem?

Accepted Answer

If $$f is $$continuous on $$[a, b]$$ and $$N is $$any number between $$f(a)$$ and $$f(b)$$, then there exists $$c in$$ $$(a, b)$$ such that $$f(c) = N.$$

Question 4

A chemical reaction's temperature $$T (in $$degrees Celsius) at time $$t (in $$minutes) is modeled by $$T(t) = 100e$$^{-0.05t}$$ + 20. $$What is the rate of change of temperature at $$t = 10$$ minutes?

Accepted Answer

$$-5e^{-0.5}$$ degrees/min$

Question 5

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

Accepted Answer

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

Question 6

Find the derivative of $$f(x) = x^2$ \sin(x).$$

Accepted Answer

$$2x\sin(x) + x^2$\cos(x)$$

Question 7

Evaluate $$\displaystyle \$$int_0$$^1 3x^2$ dx$$ using the Fundamental Theorem of Calculus.

Accepted Answer

$$1$$

Question 8

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

Accepted Answer

$$10$$

Question 9

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

Accepted Answer

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

Question 10

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

Accepted Answer

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

Question 11

A ladder 10 ft long is leaning against a wall. If the bottom of the ladder slides away from the wall at a rate of $$1 ft/sec$$, how fast is the top of the ladder sliding down the wall when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-\frac{3}{4} ft/sec$$

Question 12

Which of the following is the correct statement of the Mean Value Theorem?

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 13

Evaluate $$\displaystyle \int \cos(3x) dx.$$

Accepted Answer

$$\displaystyle \frac{1}{3}\sin(3x) + C$$

Question 14

Which of the following best describes the relationship between the derivative and the integral as stated by the Fundamental Theorem of Calculus?

Accepted Answer

Differentiation and integration are inverse processes.

Question 15

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

Accepted Answer

$$y = x - 2$$

Question 16

A population of bacteria grows according to $$P(t) = 500e$$^{0.2t}$$, where t$ is in hours. What is the rate of growth of the population at t = 5$ hours?

Accepted Answer

$$100e^{1}$$

Question 17

Evaluate $$\displaystyle \lim_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 18

Which of the following is the correct antiderivative of $$f(x) = 4x^3$ - 2x$$?

Accepted Answer

$$x^4$$ - $$x^2 + C$

Question 19

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and $$f(1) = 2$$, $$f(4) = 8. $$According to the Intermediate Value Theorem, which of the following must be true?

Accepted Answer

There exists $$c in$$ $$(1, 4)$$ such that $$f(c) = 5$$

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