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

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

Question 1

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

Accepted Answer

$$4$$

Question 2

Which of the following statements best describes the precise (epsilon-delta) 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 3

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

Accepted Answer

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

Question 4

Find the derivative of $$f(x) = x^3$ - 5x + 2.$$

Accepted Answer

$$f'(x) = 3x^2$ - 5$$

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

$$26$$

Question 7

Use the chain rule to find $$\frac{d}{dx} \left( \sin(x^2$) \right).$$

Accepted Answer

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

Question 8

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

Accepted Answer

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

Question 9

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

Accepted Answer

$$30$$ liters/min$

Question 10

Find the maximum value of $$f(x) = -x^2$ + 4x + 1.$$

Accepted Answer

$$9$$

Question 11

Which of the following is a 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 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

$$12$$

Question 14

Which of the following is an antiderivative of $$f(x) = 2x$$?

Accepted Answer

$$x^2 + C$

Question 15

Use the substitution $$u = x^2$ to evaluate $$\int 2x \$$cos(x^2) dx$.

Accepted Answer

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

Question 16

A medication is administered such that the rate of absorption is modeled by $$A(t) = 5e$$^{-0.2t}$$. What is the total amount absorbed from t = 0$ to t = 10$?

Accepted Answer

$$\$$int_0$$^{10} 5e$$^{-0.2t}$$ dt$$

Question 17

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

Accepted Answer

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

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