Q: Which of the following limits can be evaluated using L'Hôpital's Rule?

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

Question 1

Which of the following is the correct definition of a critical value for a function $$f(x)$$?

Accepted Answer

A value $$x = c$$ where $$f'(c) = 0 or$$ $$f'(c)$$ does not exist.

Question 2

Given $$f(x) = x^3$ - 3x^2$ + 2$$, on which interval is $$f(x)$$ increasing?

Accepted Answer

$$(0, 2)$$

Question 3

If $$f''(x) \u003e 0$$ for all $$x in an $$interval, what can you conclude about $$f(x) on $$that interval?

Accepted Answer

$$f(x) is $$concave up.

Question 4

A company’s profit function is $$P(x) = -2x^2$ + 8x + 10$$, where $$x is $$the number of units sold. What is the maximum profit?

Accepted Answer

$$P(4) = 26$$

Question 5

Which of the following limits can be evaluated using L'Hôpital's Rule?

Accepted Answer

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

Question 6

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

Accepted Answer

$$x^3 + C$

Question 7

Given $$\$$int_0$$^2 (4x) \, dx$$, what is the value of the definite integral?

Accepted Answer

$$16$$

Question 8

Which of the following is the correct application of the Fundamental Theorem of Calculus to find the area under $$f(x) = x^2$ from x = 1$ to x = 3$?

Accepted Answer

$$\$$int_1$$^3 x^2$ \, dx = \left[ \frac{x^3$}{3} \right]_1^3$

Question 9

Evaluate $$\int 2x \cos(x^2$) \, dx$$ using substitution.

Accepted Answer

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

Question 10

A tank is being filled with water at a rate given by $$r(t) = 5t$$ liters per minute. How much water is added between $$t = 0$$ and $$t = 4$$ minutes?

Accepted Answer

$$40$$ liters

Question 11

Which of the following expressions represents the area between the curves $$y = x^2$ and y = x$ from x = 0$ to x = 1$?

Accepted Answer

$$\$$int_0$$^1 (x - x^2$) \, dx$$

Question 12

If $$f'(x) = 0$$ and $$f''(x) \u003c 0 at$$ $$x = c$$, what does $$x = c$$ represent?

Accepted Answer

A local maximum

Question 13

Which of the following is an example of an indeterminate form?

Accepted Answer

$$\frac{0}{0}$$

Question 14

Given $$f(x) = 2x + 1$$ and $$f(0) = 1$$, what is the constant of integration when finding the antiderivative?

Accepted Answer

$$C = 0$$

Question 15

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

Accepted Answer

$$1$$

Question 16

Which substitution is appropriate for evaluating $$\int x \sqrt{1 + x^2$} \, dx$$?

Accepted Answer

Let $$u = 1 + x^2$

Question 17

A medication is administered at a rate $$r(t) = 10e$$^{-0.5t}$$ mg/hour. What is the total amount administered from t = 0$ to t = 2$ hours?

Accepted Answer

$$\$$int_0$$^2 10e$$^{-0.5t}$$ \, dt$$

Question 18

Which of the following best describes the use of Riemann sums?

Accepted Answer

Approximating the area under a curve by summing areas of rectangles

Question 19

If $$F(x) is an $$antiderivative of $$f(x)$$, which statement is true according to the Fundamental Theorem of Calculus?

Accepted Answer

$$F'(x) = f(x)$$

Calculus Study Guide: Derivatives, Integrals, and Applications

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