Q: For the function $$f(x)$$ shown in the graph, at which $$x-$$value does $$f(x)$$ fail to be continuous, and which condition of continuity is violated?

At $$x = 3$$, $$\lim\limits_{x \to 3} f(x) \neq f(3)$$

Q: Find $$\frac{d}{dx}(\tan x \cdot \frac{1}{x^2$}).$$

$$\sec^2$ x \cdot \frac{1}{x^2$} - 2 \tan x \cdot \frac{1}{x^3$}$$

Question 1

Which of the following is NOT a condition for a function $$f(x) to be $$continuous at $$x = c$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = c$$

Question 2

Given $$f(x) = x^2$, evaluate $$\lim\limits_{x 	o 3} f(x)$$.

Accepted Answer

$$9$$

Question 3

Which of the following best describes the formal definition of the limit $$\lim\limits_{x 	o c} f(x) = L$$?

Accepted Answer

For every $$\epsilon \u003e 0$$, there exists $$\delta \u003e 0$$ such that $$|f(x) - L| \u003c \epsilon$$ whenever $$0 \u003c |x - c| \u003c \delta$$

Question 4

Evaluate $$\lim\limits_{x 	o 2} \frac{x^2$ - 4}{x - 2}.$$

Accepted Answer

$$4$$

Question 5

A function $$f(x) is $$NOT differentiable at $$x = c if$$:

Accepted Answer

Any of the above

Question 6

Given $$f(x) = \sqrt{x+1}$$, find $$\lim\limits_{x 	o 4} f(x).$$

Accepted Answer

$$\sqrt{5}$$

Question 7

Using the definition of the derivative, find $$\frac{d}{dx}(4x^2$ - 4x + 1).$$

Accepted Answer

$$8x - 4$$

Question 8

A spherical balloon is inflated with gas at the rate of $$100$$ cubic feet per minute. How fast is the radius increasing when the radius is $$4$$ feet?

Accepted Answer

$$\frac{25}{4\pi} ft/$$min

Question 9

For the function $$f(x)$$ shown in the graph, at which $$x-$$value does $$f(x)$$ fail to be continuous, and which condition of continuity is violated?

Accepted Answer

At $$x = 3$$, $$\lim\limits_{x 	o 3} f(x) 
eq f(3)$$

Question 10

Find $$\frac{d}{dx}(x^2$ \cot x).$$

Accepted Answer

$$2x \cot x - x^2$ \csc^2$ x$$

Question 11

Evaluate $$\lim\limits_{x 	o 0} \frac{\sin x}{x}.$$

Accepted Answer

$$1$$

Question 12

If $$f(x) is $$defined as $$f(x) = \frac{1}{x}$$, which of the following statements is true about $$f(x) at$$ $$x = 0$$?

Accepted Answer

$$f(x) is $$not defined at $$x = 0$$

Question 13

Find $$\frac{d}{dx}(	an x \cdot \frac{1}{x^2$}).$$

Accepted Answer

$$\sec^2$ x \cdot \frac{1}{x^2$} - 2 	an x \cdot \frac{1}{x^3$}$$

Question 14

The position of a particle moving along a straight line at any time $$t is $$given by $$x(t) = 4 \sin(2t) + 2 \cos(3t). $$What is the acceleration of the particle when $$t = \frac{\pi}{5}$$?

Accepted Answer

$$-8 \sqrt{3} + 18$$

Question 15

For the function $$y = x^2$ + \sqrt{x}$$, find $$\frac{dy}{dx}.$$

Accepted Answer

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

Question 16

If $$f(x)$$ has infinite oscillations as $$x$$ approaches $$c$$, which property is violated?

Accepted Answer

Differentiability

Question 17

Evaluate $$\lim\limits_{x 	o 0} \frac{1 - \cos x}{x}.$$

Accepted Answer

$$0$$

Question 18

A particle moves along the $$x-$$axis so that at any time $$t$$, its position is given by $$x(t) = t^3$ - 3t^2$ + 5t - 1. $$For what values of $$t is $$the particle at rest?

Accepted Answer

$$t = 1$$ and $$t = 3$$

Question 19

Find $$\frac{d}{dx}(\ln(x^2$ + x)).$$

Accepted Answer

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

Limits, Continuity, and Differentiation: Calculus I Study Guide

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