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

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 $$h(x) = \begin{cases} 3x-1 & x  2 \ 6 & x = 2 \end{cases}$$, 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 $$^0C) of a $$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 $$^0C $$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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