Q: Which of the following best describes the limit $$\lim_{x \to 2} \frac{x^2$ - 4}{x - 2}$$?

The limit is $$4$$ because $$\lim_{x \to 2} \frac{x^2$ - 4}{x - 2} = \lim_{x \to 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2} x+2 = 4.$$

Q: A bacteria population $$P(t)$$ grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t=10$?

$$5e^{0.5} = 0.05$$ \times $$100e^{0.5}$$

Question 1

Which of the following best describes the limit $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2}$$?

Accepted Answer

The limit is $$4$$ because $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2} = \lim_{x 	o 2} \frac{(x-2)(x+2)}{x-2} = \lim_{x 	o 2} x+2 = 4.$$

Question 2

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

Accepted Answer

$$\frac{3}{5}$$

Question 3

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

Accepted Answer

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

Question 4

The derivative of $$f(x) = x^3$ - 2x + 5 is$$:

Accepted Answer

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

Question 5

If $$y = \sin(x^2$)$$, what is $$\frac{dy}{dx}$$?

Accepted Answer

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

Question 6

A bacteria population $$P(t)$$ grows according to $$P(t) = 100e$$^{0.05t}$$. What is the instantaneous rate of growth at t=10$?

Accepted Answer

$$5e^{0.5} = 0.05$$ 	imes $$100e^{0.5}$$

Question 7

Given $$y^2$$ + $$x^2 = 25$, find $$\frac{dy}{dx}$$ using implicit differentiation.

Accepted Answer

$$\frac{dy}{dx} = -\frac{x}{y}$$

Question 8

A ladder 10 ft long is leaning against a wall. If the bottom slides away from the wall at 2 ft/sec, how fast is the top sliding down the wall when the bottom is 6 ft from the wall?

Accepted Answer

$$-1.5 ft/sec$$

Question 9

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 10

Find the $$x-$$coordinate of the critical point(s) of $$f(x) = x^3$ - 3x^2$ + 2.$$

Accepted Answer

$$x = 0$$ and $$x = 2$$

Question 11

Suppose $$f(x) = \ln(x^2$ + 1). $$What is $$f'(x)$$?

Accepted Answer

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

Question 12

Evaluate $$\lim_{x 	o 0} \frac{\sin(3x)}{x}.$$

Accepted Answer

$$3$$

Question 13

A company wants to design a cylindrical can that holds 500 mL of liquid. What is the minimum surface area possible for such a can?

Accepted Answer

When the height equals twice the radius, i.e., $$h = 2r$$; minimum surface area is $$2\pi r^2$ + 2\pi r h$$ with $$r = \sqrt[3]{\frac{500}{2\pi}}.$$

Question 14

Use L'Hôpital's Rule to evaluate $$\lim_{x 	o 0} \frac{e^x - 1}{x}.$$

Accepted Answer

$$1$$

Question 15

Find the linear approximation of $$f(x) = \sqrt{x} at$$ $$x = 4$$ and use it to estimate $$\sqrt{4.1}.$$

Accepted Answer

$$\sqrt{4.1} \approx 2.025$$

Question 16

Evaluate $$\$$int_0$$^1 (3x^2$ + 2x) dx.$$

Accepted Answer

$$2$$

Question 17

Which of the following is the correct application of the Fundamental Theorem of Calculus?

Accepted Answer

If $$F(x) is an $$antiderivative of $$f(x)$$, then $$\int_a^b f(x) dx = F(b) - F(a).$$

Question 18

Find the area between the curves $$y = x^2$ and y = x + 2$ from x = 0$ to x = 2$.

Accepted Answer

$$\frac{10}{3}$$

Question 19

Find the volume of the solid generated by revolving the region bounded by $$y = x^2$ and y = 0$ from x = 0$ to x = 1$ about the x$-axis.

Accepted Answer

$$\frac{\pi}{5}$$

MAC2311 Calculus I: Structured Study Guide Based on Course Schedule

Guida di studio - Domande di pratica

Study guide - Flashcard