Q: Which of the following describes a vertical asymptote for the function $$f(x)$$?

$$\\lim_{x \to a} f(x) = \infty or$$ $$-\infty$$

Q: Which of the following is the correct definition of continuity at a point $$x = a$$?

$$f(a) is $$defined and $$\\lim_{x \to a} f(x) = f(a)$$

Q: State the definition of the derivative $$f'(a).$$

$$f'(a) = \\lim_{h \to 0} \\frac{f(a+h) - f(a)}{h}$$

Question 1

A car travels along a straight road. Its position at time $$t (in $$hours) is given by $$s(t) = 4t^2$ + 2t (in $$miles). What is the average velocity of the car between $$t = 1$$ and $$t = 3$$ hours?

Accepted Answer

$$16$$ miles/hour

Question 2

Given the function $$f(x) = x^2$ - 3x + 2$$, what is the slope of the secant line between $$x = 1$$ and $$x = 4$$?

Accepted Answer

$$4$$

Question 3

Using a table of values, you find that as $$x$$ approaches $$2$$ from both sides, $$f(x)$$ approaches $$5. $$What is $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

$$5$$

Question 4

The graph of $$g(x)$$ shows that as $$x$$ approaches $$-1$$ from the left, $$g(x)$$ approaches $$3$$, but from the right, $$g(x)$$ approaches $$-2. $$What can you conclude about $$\lim_{x 	o -1} g(x)$$?

Accepted Answer

The limit does not exist

Question 5

Which of the following describes a vertical asymptote for the function $$f(x)$$?

Accepted Answer

$$\lim_{x 	o a} f(x) = \infty or$$ $$-\infty$$

Question 6

Evaluate $$\lim_{x 	o 3} (x^2$ - 2x)$$ using direct substitution.

Accepted Answer

Question 7

Find $$\lim_{x 	o 2} \frac{x^2$ - 4}{x - 2} by $$factoring.

Accepted Answer

$$4$$

Question 8

Evaluate $$\lim_{x 	o 1} \frac{\sqrt{x+3} - 2}{x - 1}$$ using the conjugate method.

Accepted Answer

$$\frac{1}{4}$$

Question 9

If $$\lim_{x 	o a^-} f(x) = 2$$ and $$\lim_{x 	o a^+} f(x) = 5$$, what is $$\lim_{x 	o a} f(x)$$?

Accepted Answer

Does not exist

Question 10

Suppose $$f(x)$$ satisfies $$g(x) \leq f(x) \leq h(x)$$ for all $$x$$ near $$a$$, and $$\lim_{x 	o a} g(x) = L = \lim_{x 	o a} h(x). $$What does the Squeeze Theorem tell us about $$\lim_{x 	o a} f(x)$$?

Accepted Answer

$$\lim_{x 	o a} f(x) = L$$

Question 11

Which of the following is the correct definition of continuity at a point $$x = a$$?

Accepted Answer

$$f(a) is $$defined and $$\lim_{x 	o a} f(x) = f(a)$$

Question 12

Let $$f(x) = \frac{1}{x}. Is$$ $$f(x)$$ continuous at $$x = 0$$?

Accepted Answer

No, because $$f(0) is $$not defined

Question 13

A patient’s body temperature $$T(t) (in$$ $$^\circC) $$changes continuously from $$36.5 to$$ $$39.0$$ over $$t in$$ $$[0, 5]$$ hours. By the Intermediate Value Theorem, which statement is true?

Accepted Answer

There is a time $$c in$$ $$[0, 5]$$ when $$T(c) = 37.5$$

Question 14

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

Accepted Answer

$$2.5$$

Question 15

What is $$\lim_{x 	o \infty} \frac{1}{x^r}$$ for $$r \u003e 0$$?

Accepted Answer

$$0$$

Question 16

State the definition of the derivative $$f'(a).$$

Accepted Answer

$$f'(a) = \lim_{h 	o 0} \frac{f(a+h) - f(a)}{h}$$

Question 17

Given $$f(x) = x^2$, calculate f$$'(2)$$ using the definition of the derivative.

Accepted Answer

$$4$$

Question 18

In the context of a falling object, what does $$f'(a)$$ represent if $$f(t) is $$the height at time $$t$$?

Accepted Answer

The instantaneous velocity at time $$t = a$$, in meters/second

Question 19

Which of the following is NOT a valid notation for the derivative of $$f(x)$$?

Accepted Answer

$$f(x)'$$

Question 20

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

Accepted Answer

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

Question 21

Given the graph of $$f(x)$$, which of the following statements is true if $$f'(x) is $$positive and $$f''(x) is $$negative at $$x = a$$?

Accepted Answer

$$f(x) is $$increasing and concave down at $$x = a$$

Limits, Continuity, and Derivatives: Essential Concepts and Techniques

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