Q: Which of the following is the correct definition of the derivative of a function $f$ at a point $x$?

$f'(x) = \lim_{h \to 0} \dfrac{f(x+h) - f(x)}{h}$

Q: Given $h(x) = \begin{cases} 3x-1 & x 2 \\ 6 & x = 2 \end{cases}$, is $h(x)$ continuous at $x=2$?

No, because $\lim_{x \to 2} h(x) \neq h(2)$

Q: A function $f(x)$ is differentiable at $x = a$ if:

$\lim_{h \to 0} \dfrac{f(a+h) - f(a)}{h}$ exists

Q: Which of the following statements about limits is FALSE?

If $\lim_{x \to a} f(x)$ exists and $f(a)$ is defined, then $f(x)$ is continuous at $x=a$.

Question 1

Which of the following is the correct definition of the derivative of a function $f$ at a point $x$?

Accepted Answer

$f'(x) = \lim_{h 	o 0} \dfrac{f(x+h) - f(x)}{h}$

Question 2

Suppose $f(x)$ is defined as $f(x) = \dfrac{1}{x+2}$. What is $f'(x)$ using the definition of the derivative?

Accepted Answer

$f'(x) = -\dfrac{1}{(x+2)^2}$

Question 3

A function $f(x)$ is continuous at $x = a$ if and only if:

Accepted Answer

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

Question 4

Given $h(x) = \begin{cases} 3x-1 & x < 2 \ 7-x & x > 2 \ 6 & x = 2 \end{cases}$, is $h(x)$ continuous at $x=2$?

Accepted Answer

No, because $\lim_{x 	o 2} h(x) 
eq h(2)$

Question 5

Evaluate $\lim_{x 	o 4} \dfrac{x^2 - 4x}{x^2 - 6x + 8}$.

Accepted Answer

$2$

Question 6

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

Accepted Answer

$(uv)' = u'v + uv'$

Question 7

If $f(x) = x^3 + 2x^2 - x^{1/2} + 2$, what is $f'(x)$?

Accepted Answer

$3x^2 + 4x - \dfrac{1}{2}x^{-1/2}$

Question 8

A company’s cost function is $C(x) = 5x^2 + 3x + 100$. What is the marginal cost when $x = 10$?

Accepted Answer

$C'(10) = 103$

Question 9

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

Accepted Answer

$\left(\dfrac{u}{v}\right)' = \dfrac{u'v - uv'}{v^2}$

Question 10

Find the derivative of $f(x) = (x^2 + 1)(\sqrt{x} + 1)$ using the product rule.

Accepted Answer

$f'(x) = 2x(\sqrt{x} + 1) + (x^2 + 1)\left(\dfrac{1}{2}x^{-1/2}\right)$

Question 11

If $f(x) = \dfrac{x+1}{x+10}$, what is $f'(x)$ using the quotient rule?

Accepted Answer

$f'(x) = \dfrac{9}{(x+10)^2}$

Question 12

Which of the following is the correct application of the chain rule for $y = (x^2 + x + 4)^{1/2}$?

Accepted Answer

$\dfrac{dy}{dx} = \dfrac{2x+1}{2(x^2 + x + 4)^{1/2}}$

Question 13

A function $f(x)$ is differentiable at $x = a$ if:

Accepted Answer

$\lim_{h 	o 0} \dfrac{f(a+h) - f(a)}{h}$ exists

Question 14

Suppose $f(x)$ models the temperature (in $^0$C) of a chemical reaction over time $x$ (in minutes) and $f'(5) = -2$. What does this mean in context?

Accepted Answer

At 5 minutes, the temperature is decreasing at a rate of 2 $^0$C per minute.

Question 15

If $f(x) = 6y^{1/2} - 8y^{3/2}$, what is $f'(y)$?

Accepted Answer

$3y^{-1/2} - 12y^{1/2}$

Question 16

Which of the following statements about limits is FALSE?

Accepted Answer

If $\lim_{x 	o a} f(x)$ exists and $f(a)$ is defined, then $f(x)$ is continuous at $x=a$.

Question 17

A hospital models the number of bacteria in a culture as $N(t) = 1000e^{0.05t}$, where $t$ is in hours. What is the instantaneous rate of change of bacteria at $t=0$?

Accepted Answer

$N'(0) = 50$

Question 18

If $f(x) = 8w^3 - 18w^2 + 7w - 5$, what is $f'(w)$?

Accepted Answer

$24w^2 - 36w + 7$

Business Calculus: Limits, Continuity, and Differentiation (MAT 114 Test 1 Review)

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