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

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

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

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

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