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

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

Question 1

Which of the following is the precise definition of the limit $$\lim_{x 	o a} f(x) = L$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists a $$\delta \u003e 0$$ such that if $$0 \u003c |x - a| \u003c \delta$$, then $$|f(x) - L| \u003c \epsilon.$$

Question 2

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

Accepted Answer

$$1$$

Question 3

Which of the following functions is continuous everywhere?

Accepted Answer

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

Question 4

If $$f(x) is $$differentiable at $$x = a$$, which of the following statements is always true?

Accepted Answer

$$f(x) is $$continuous at $$x = a$$

Question 5

Find the derivative of $$f(x) = x^3$ + 2x.$$

Accepted Answer

$$f'(x) = 3x^2$ + 2$$

Question 6

Which rule is used to find the derivative of $$f(x) = \sin(x^2$)$$?

Accepted Answer

Chain Rule

Question 7

Given $$f(x) = x^2$ and g(x$$) = \cos x$$, what is $$\frac{d}{dx}[f(x)g(x)]$$?

Accepted Answer

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

Question 8

If $$f(x) = \ln(x)$$, what is $$f''(x)$$?

Accepted Answer

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

Question 9

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

Accepted Answer

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

Question 10

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a)  0. $$What theorem guarantees there is a $$c in$$ $$(a, b)$$ such that $$f(c) = 0$$?

Accepted Answer

Intermediate Value Theorem

Question 11

Find $$\$$int_0$$^\pi \sin x \, dx.$$

Accepted Answer

$$0$$

Question 12

Which of the following is the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

$$\frac{d}{dx} \int_a^x f(t) \, dt = f(x)$$

Question 13

Evaluate $$\int x e^x \, dx$$ using integration by parts.

Accepted Answer

$$x e^x - e^x + C$$

Question 14

Which formula gives the area between two curves $$y = f(x)$$ and $$y = g(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b [f(x) - g(x)] \, dx$$

Question 15

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

Accepted Answer

$$\$$int_0$$^2 3t^2$ \, dt = 12$$ liters

Question 16

Which of the following is the correct formula for arc length of $$y = f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\int_a^b \sqrt{1 + [f'(x)]^2} \, dx$$

Question 17

Which integration technique is most appropriate for $$\int x \cos x \, dx$$?

Accepted Answer

Integration by Parts

Question 18

If $$f(x)$$ has a critical point at $$x = c$$, which of the following must be true?

Accepted Answer

$$f'(c) = 0 or$$ $$f'(c)$$ does not exist

Question 19

Suppose $$f(x) is $$continuous on $$[a, b]$$ and differentiable on $$(a, b). $$Which theorem guarantees there is a $$c in$$ $$(a, b)$$ such that $$f'(c) = \frac{f(b) - f(a)}{b - a}$$?

Accepted Answer

Mean Value Theorem

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