Q: A function $$f(x) is $$continuous at $$x = 2 if$$:

$$\displaystyle \lim_{x \to 2} f(x)$$ exists, $$f(2)$$ exists,$$\displaystyle \lim_{x \to 2} f(x) = f(2)$$

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

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

Question 1

Which of the following statements best describes the meaning of the limit $$\displaystyle \lim_{x 	o a} f(x) = L$$?

Accepted Answer

As $$x$$ approaches $$a$$, $$f(x)$$ approaches $$L$$, regardless of the value of $$f(a).$$

Question 2

Evaluate the following limit: $$\displaystyle \lim_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 3

A function $$f(x) is $$continuous at $$x = 2 if$$:

Accepted Answer

$$\displaystyle \lim_{x 	o 2} f(x)$$ exists, $$f(2)$$ exists,$$\displaystyle \lim_{x 	o 2} f(x) = f(2)$$

Question 4

Which of the following is an example of a removable discontinuity?

Accepted Answer

A function with a hole at $$x = 3$$

Question 5

Given $$f(x) = x^2$ + 3x$$, use the limit definition of the derivative to find $$f'(2).$$

Accepted Answer

$$11$$

Question 6

Which rule would you use to differentiate $$f(x) = x^2$ \sin x$$?

Accepted Answer

Product Rule

Question 7

Find the derivative of $$f(x) = \ln(x^2$ + 1).$$

Accepted Answer

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

Question 8

A car’s position is given by $$s(t) = 3t^2$ + 2t + 1. $$What is the car’s acceleration at time $$t = 2$$?

Accepted Answer

$$6$$

Question 9

Suppose $$y^2$$ + $$x^2 = 25$. Using implicit differentiation, find $$\displaystyle \frac{dy}{dx}$$.

Accepted Answer

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

Question 10

Which of the following is a correct application of the Mean Value Theorem for a function $$f(x)$$ continuous on $$[a, b]$$ and differentiable on $$(a, b)$$?

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f'(c) = \displaystyle \frac{f(b) - f(a)}{b - a}$$

Question 11

Evaluate $$\displaystyle \$$int_0$$^1 3x^2$ dx.$$

Accepted Answer

$$1$$

Question 12

Which of the following best describes the Fundamental Theorem of Calculus, Part 1?

Accepted Answer

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

Question 13

A tank is being filled with water so that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V(t) = 5t^2$. At what rate is the water entering the tank at t = 4$ minutes?

Accepted Answer

$$40$$ liters/min

Question 14

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

Accepted Answer

$$2$$

Question 15

Which of the following is the correct antiderivative of $$f(x) = 4x^3$?

Accepted Answer

$$x^4 + C$

Question 16

Suppose $$f(x) is $$continuous on $$[1, 4]$$ and $$f(1) = 2$$, $$f(4) = 8. $$According to the Intermediate Value Theorem, which of the following must be true?

Accepted Answer

For some $$c in$$ $$(1, 4)$$, $$f(c) = 5$$

Question 17

A medication is administered so that the amount in the bloodstream at time $$t is$$ $$A(t) = 100e$$^{-0.2t}$$. What is the rate at which the medication is leaving the bloodstream at t = 5$ hours?

Accepted Answer

$$-20e^{-1}$$

Question 18

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

Accepted Answer

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

Question 19

Suppose $$f(x) = x^3$ - 3x^2$ + 2. $$Find all critical points of $$f(x).$$

Accepted Answer

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

Question 20

A company wants to design a rectangular garden with a fixed perimeter of $$40$$ meters. What dimensions maximize the area?

Accepted Answer

A square with sides $$10$$ meters

Single Variable Calculus I: Core Topics and Learning Outcomes

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