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

$$\lim_{x \to c} f(x)$$ exists, $$f(c) is $$defined, and $$\lim_{x \to c} f(x) = f(c)$$

Q: A ball is thrown upward. Its position is given by $$s(t) = 10t - 4.9t^2$. What is its velocity at t = 2$ seconds?

$$10 - 9.8 \times 2 = -9.6 m/s$$

Question 1

Which of the following statements about the limit $$\lim_{x 	o a} f(x) is $$TRUE?

Accepted Answer

The limit exists if and only if the left-hand and right-hand limits are equal.

Question 2

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

Accepted Answer

$$3$$

Question 3

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

Accepted Answer

$$\lim_{x 	o c} f(x)$$ exists, $$f(c) is $$defined, and $$\lim_{x 	o c} f(x) = f(c)$$

Question 4

Which rule should be used to differentiate $$y = x^2$ e$$^{x}$$?

Accepted Answer

Product Rule

Question 5

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

Accepted Answer

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

Question 6

A ball is thrown upward. Its position is given by $$s(t) = 10t - 4.9t^2$. What is its velocity at t = 2$ seconds?

Accepted Answer

$$10 - 9.8 	imes 2 = -9.6 m/s$$

Question 7

If $$f(x) = \ln(x^2$ + 1)$$, what is $$f'(x)$$?

Accepted Answer

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

Question 8

Which of the following is NOT a valid application of the Mean Value Theorem?

Accepted Answer

Finding the maximum value of a function on a closed interval.

Question 9

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

Accepted Answer

$$-\dfrac{x}{y}$$

Question 10

A tank is being filled so that the volume $$V (in $$liters) at time $$t (in $$minutes) is $$V = 100t^2$. How fast is the volume increasing when t = 3$?

Accepted Answer

$$600$$ liters/min

Question 11

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 12

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

Accepted Answer

$$x = 1$$ and $$x = 2$$

Question 13

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

Accepted Answer

$$f(x) = |x|$$

Question 14

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

Accepted Answer

$$1$$

Question 15

Which of the following best describes the relationship between differentiability and continuity?

Accepted Answer

If a function is differentiable at a point, it must be continuous at that point.

Question 16

A population grows according to $$P(t) = 500e$$^{0.02t}$$. What is the instantaneous rate of change of the population at t = 10$?

Accepted Answer

$$0.02 	imes 500e$$^{0.2}$$

Question 17

Which of the following is the correct derivative of $$f(x) = 	an$$^{-1}$$(x)$$?

Accepted Answer

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

Question 18

Solve the differential equation $$\dfrac{dy}{dx} = 3y$$ with initial condition $$y(0) = 2.$$

Accepted Answer

$$y(x) = 2e$$^{3x}$$

Question 19

Which of the following is the correct formula for the derivative of $$f(x) = e$$^{x^2}$$?

Accepted Answer

$$2x e$$^{x^2}$$

Calculus I: Limits, Derivatives, Integrals, and Applications – 4-Week Study Guide

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