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

A car’s position at time $$t (in $$seconds) is given by $$s(t) = 3t^2$ + 2t + 1 (in $$meters). What is the instantaneous velocity of the car at $$t = 4$$ seconds?

Accepted Answer

$$26 m/s$

Question 2

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

Accepted Answer

$$4$$

Question 3

Which of the following statements best describes the Mean Value Theorem for derivatives?

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 4

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

Accepted Answer

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

Question 5

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

Accepted Answer

$$60 L/$$min$

Question 6

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

Accepted Answer

$$18$$

Question 7

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 8

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

Accepted Answer

$$\frac{-x}{y}$$

Question 9

A company wants to design a cylindrical can that holds $$500 cm$^3 of $$liquid. What is the minimum surface area required? (Let $$r be $$the radius and $$h$$ the height.)

Accepted Answer

When $$r = \sqrt[3]{\frac{250}{\pi}}$$ and $$h = 2r$$

Question 10

Which of the following is the correct statement 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 11

Find the limit: $$\lim_{x 	o \infty} \frac{5x^2$ + 3}{2x^2$ - x}.$$

Accepted Answer

$$\frac{5}{2}$$

Question 12

Which of the following best describes a critical point of a function $$f(x)$$?

Accepted Answer

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

Question 13

Suppose $$f(x) = e$$^{2x}$$. What is the linear approximation of f(x$$)$$ at x = 0$?

Accepted Answer

$$1 + 2x$$

Question 14

A ladder $$10 ft $$long is leaning against a wall. If the bottom slides away from the wall at $$1 ft/sec$$, how fast is the top sliding down when the bottom is $$6 ft $$from the wall?

Accepted Answer

$$-0.75 ft/sec$$

Question 15

Evaluate $$\int x \cos(x) \, dx$$ using integration by parts.

Accepted Answer

$$x \sin(x) + \cos(x) + C$$

Question 16

Which of the following is the correct substitution for evaluating $$\int 2x \sqrt{x^2$ + 1} \, dx$$?

Accepted Answer

Let $$u = x^2$ + 1$$

Calculus I: Core Concepts and Learning Objectives

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