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

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

Question 1

Suppose the position of a car moving along a straight road is given by $$s(t) = 3t^2$ + 2t + 5$$, 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 functions is NOT continuous at $$x = 0$$?

Accepted Answer

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

Question 4

A tank is being filled with water such that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V(t) = 5t^3$. At what rate is the volume increasing when t = 2$ minutes?

Accepted Answer

$$60 L/$$min$

Question 5

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

Accepted Answer

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

Question 6

Evaluate $$\displaystyle \lim_{x 	o 0} \frac{\sin(5x)}{x}.$$

Accepted Answer

$$5$$

Question 7

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

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 8

A spherical balloon is being inflated so that its radius increases at a rate of $$2 cm/$$min. How fast is the volume of the balloon increasing when the radius is $$3 cm$$? (Recall: $$V = \frac{4}{3}\pi r^3$)

Accepted Answer

$$72\pi cm$^3/$$min

Question 9

Find $$\displaystyle \$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 10

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

Accepted Answer

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

Question 11

Which of the following is the antiderivative of $$f(x) = 2x + \cos(x)$$?

Accepted Answer

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

Question 12

A company wants to design a rectangular enclosure using 100 meters of fencing to maximize the area. What should the dimensions of the enclosure be?

Accepted Answer

$$25 m by$$ $$25 m$$

Question 13

Which of the following is the correct derivative of $$y = \ln(x^2$ + 1)$$?

Accepted Answer

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

Question 14

Evaluate $$\displaystyle \int x e$$^{x^2}$$ dx$$ using substitution.

Accepted Answer

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

Question 15

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

Accepted Answer

It relates differentiation and integration, stating that if $$F is an $$antiderivative of $$f on$$ $$[a, b]$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Question 16

If $$f(x) = x^3$ - 3x^2$ + 4$$, at what $$x-$$value does $$f(x)$$ have a local minimum?

Accepted Answer

$$x = 2$$

Question 17

Which of the following is the correct derivative of $$y = \arctan(x)$$?

Accepted Answer

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

Question 18

A medication is administered so that the amount in the bloodstream at time $$t (in $$hours) is $$A(t) = 100e$$^{-0.5t}$$. What is the rate at which the medication is leaving the bloodstream at t = 2$ hours?

Accepted Answer

$$-30.33 mg/hr$$

Question 19

Which of the following is the correct linear approximation of $$f(x) = \sqrt{x} at$$ $$x = 4$$?

Accepted Answer

$$f(x) \approx 2 + \frac{1}{4}(x - 4)$$

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