$f(4)$ is undefined, but $\lim_{x \to 4} f(x)$ may exist$

$\frac{\sqrt{x+h} - \sqrt{x}}{h}$ \text{ rationalized } = \frac{1}{\sqrt{x+h} + \sqrt{x}}$

Question 1

Which of the following best describes the limit of a function $f(x)$ as $x$ approaches $a$?

Accepted Answer

The value that $f(x)$ approaches as $x$ gets arbitrarily close to $a$

Question 2

Given $f(x) = 2x + 1$, what is $\lim_{x 	o 1} f(x)$?

Accepted Answer

$3$

Question 3

If $f(x)$ has a hole at $x = 4$, which of the following is true?

Accepted Answer

$f(4)$ is undefined, but $\lim_{x 	o 4} f(x)$ may exist$

Question 4

Which of the following functions is NOT continuous everywhere on $\mathbb{R}$?

Accepted Answer

$f(x) = \frac{1}{x}$

Question 5

Evaluate $\lim_{x 	o 0} \frac{x^2 - 4}{x - 2}$.

Accepted Answer

$2$

Question 6

The average rate of change of $f(x) = x^2$ from $x = 1$ to $x = 3$ is:

Accepted Answer

$5$

Question 7

Which of the following is the correct difference quotient for $f(x) = x^2$?

Accepted Answer

$\frac{f(x+h) - f(x)}{h} = \frac{(x+h)^2 - x^2}{h}$

Question 8

If $f(x) = x^3$, what is $f'(x)$ using the power rule?

Accepted Answer

$3x^2$

Question 9

For $f(x) = \sqrt{x}$, what is the simplified form of the difference quotient $\frac{f(x+h) - f(x)}{h}$?

Accepted Answer

$\frac{\sqrt{x+h} - \sqrt{x}}{h}$ 	ext{ rationalized } = \frac{1}{\sqrt{x+h} + \sqrt{x}}$

Question 10

Which of the following is the correct application of the product rule for differentiation?

Accepted Answer

$(fg)' = f'g + fg'$

Question 11

Given $f(x) = x^2$ and $g(x) = 3x$, what is the derivative of $f(x) + g(x)$?

Accepted Answer

$2x + 3$

Question 12

If $f(x) = \frac{1}{x}$, what is $f'(x)$ using the power rule?

Accepted Answer

$-x^{-2}$

Question 13

Which of the following is true about the derivative at a point where the function has a sharp corner (cusp)?

Accepted Answer

The derivative does not exist

Question 14

For $f(x) = x^2 + 2x$, what is the equation of the tangent line at $x = 2$?

Accepted Answer

$y = 6x - 4$

Question 15

If $f(x) = x^2$ and $g(x) = x + 1$, what is $\frac{d}{dx}[f(g(x))]$ using the chain rule?

Accepted Answer

$2(x+1)$

Question 16

A ball is thrown vertically upward with initial velocity $v_0$. Its height is given by $s(t) = -4.9t^2 + v_0 t$. What is the velocity at time $t$?

Accepted Answer

$-4.9t + v_0$

Question 17

Which of the following is the correct application of the quotient rule for differentiation?

Accepted Answer

$\left(\frac{f}{g}\right)' = \frac{f'g - fg'}{g^2}$

Question 18

If $f(x) = x^4$, what is the second derivative $f''(x)$?

Accepted Answer

$12x^2$

Question 19

For the function $f(x) = |x|$, where does the derivative fail to exist?

Accepted Answer

$x = 0$

Business Calculus: Limits, Continuity, and Differentiation Study Notes

Study Guide - Practice Questions