Question 1

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

Accepted Answer

As $$x$$ approaches $$a$$ from both sides, $$f(x)$$ approaches $$L.$$

Question 2

Which technique should you try first when evaluating $$ \lim_{x 	o 2} (x^2$ - 4) $$?

Accepted Answer

Direct Substitution

Question 3

Evaluate $$ \lim_{x 	o 3} \frac{x^2$ - 9}{x - 3} $$ using the appropriate algebraic technique.

Accepted Answer

$$6$$

Question 4

Which of the following is an example of the rationalization technique?

Accepted Answer

Multiplying numerator and denominator by the conjugate of a radical expression.

Question 5

Suppose $$f(x) = \sqrt{x}$$, find the difference quotient $$\frac{f(x+h) - f(x)}{h}.$$

Accepted Answer

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

Question 6

A car travels 50 miles in 1 hour. What is the average velocity?

Accepted Answer

$$50$$ miles/hour

Question 7

Given $$f(x) = x^2$, what is the instantaneous rate of change at x = 3$?

Accepted Answer

$$6$$

Question 8

Which of the following is NOT a reason a function is not differentiable at a point?

Accepted Answer

The function is linear at that point.

Question 9

Find $$\lim_{x 	o 1} \frac{\sqrt{x} - 1}{x - 1}$$ using rationalization.

Accepted Answer

$$\frac{1}{2}$$

Question 10

If $$R(x) = 0.001x^2$ + 150x$$, what is the average revenue from selling between 300 and 305 units?

Accepted Answer

$$151.495$$

Question 11

Given $$f(x) = x^2$ + 2x$$, find the slope of the tangent line at $$x = 1.$$

Accepted Answer

$$4$$

Question 12

Which of the following best describes the difference quotient for a function $$f(x)$$?

Accepted Answer

$$\frac{f(x+h) - f(x)}{h}$$

Question 13

If $$f(x) = |x|$$, at which point is $$f(x)$$ not differentiable?

Accepted Answer

$$x = 0$$

Question 14

Which of the following is the correct formula for the average rate of change of $$f(x)$$ from $$x = a to$$ $$x = b$$?

Accepted Answer

$$\frac{f(b) - f(a)}{b - a}$$

Question 15

Suppose $$f(x) = x^2$. What is the equation of the tangent line at x = 2$?

Accepted Answer

$$y = 4x - 4$$

Algebraic Limits, Average Rates of Change, and Differentiation in Business Calculus

Study Guide - Practice Questions