$$\lim\limits_{x \to 0} f(x) = f(0)$$

$$\lim\limits_{x \to 0} f(x) = f(0)$$

Question 1

Which of the following is a necessary condition for a function $$f(x) to be $$continuous at $$x=0$$?

Accepted Answer

$$\lim\limits_{x 	o 0} f(x) = f(0)$$

Question 2

Given $$f(x) = \dfrac{(x-1)(x-3)}{(x+1)(x-1)}$$, what is the simplified form of $$f(x)$$ for $$x 
eq 1$$?

Accepted Answer

$$\dfrac{x-3}{x+1}$$

Question 3

Which of the following is the correct derivative of $$\sin(x)$$?

Accepted Answer

$$\cos(x)$$

Question 4

The Intermediate Value Theorem (IVT) guarantees that if $$f is $$continuous on $$[a, b]$$ and $$N is $$between $$f(a)$$ and $$f(b)$$, then:

Accepted Answer

There exists $$c in$$ $$(a, b)$$ such that $$f(c) = N$$

Question 5

Which of the following is the correct derivative of $$	an(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 6

If $$f'(x)  0 on an $$interval, what can you say about the graph of $$f(x) on $$that interval?

Accepted Answer

The graph is decreasing and concave up

Question 7

Which of the following is the correct derivative of $$\csc(x)$$?

Accepted Answer

$$-\csc(x)\cot(x)$$

Question 8

Suppose $$f(x) is $$continuous on $$[1, 3]$$ and $$f(1) = -2$$, $$f(3) = 4. $$According to the IVT, which of the following must be true?

Accepted Answer

There exists $$c in$$ $$(1, 3)$$ such that $$f(c) = 0$$

Question 9

Evaluate $$\dfrac{d}{dx} \left( \sec(x) \right).$$

Accepted Answer

$$\sec(x)	an(x)$$

Question 10

Which of the following is the correct derivative of $$\cot(x)$$?

Accepted Answer

$$-\csc^2$(x)$$

Question 11

Given $$f(x) = \sqrt{x}$$, what is $$f'(x)$$?

Accepted Answer

$$\dfrac{1}{2\sqrt{x}}$$

Question 12

If $$f(x) is $$continuous at $$x=0$$, which of the following must be true?

Accepted Answer

$$\lim\limits_{x 	o 0} f(x) = f(0)$$

Question 13

Which of the following is the correct derivative of $$\cos(x)$$?

Accepted Answer

$$-\sin(x)$$

Question 14

If $$f'(x) \u003e 0$$ and $$f''(x) \u003c 0 on an $$interval, what does this indicate about the graph of $$f(x)$$?

Accepted Answer

The graph is increasing and concave down

Question 15

Which of the following is the correct derivative of $$	an(x)$$?

Accepted Answer

$$\sec^2$(x)$$

Question 16

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b)$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is some $$c in$$ $$(a, b)$$ such that $$f(c) = 0$$

Calculus Study Notes: Limits, Derivatives, and Applications

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