Q: Given the piecewise function $$f(x) = \begin{cases} x^2$ + 1, & x \u003c 0 \\ x - 2, & x \geq 0 \end{cases}$$, at which value of $$x is$$ $$f(x)$$ discontinuous, and what are the left and right limits at that point?

$$x = 0$$; \lim_{x \to 0^-} f(x) = 1$$, $$\lim_{x \to 0^+} f(x) = -2$

Question 1

Given the piecewise function $$f(x) = \begin{cases} x^2$ + 1, & x \u003c 0 \ x - 2, & x \geq 0 \end{cases}$$, at which value of $$x is$$ $$f(x)$$ discontinuous, and what are the left and right limits at that point?

Accepted Answer

$$x = 0$$; \lim_{x 	o 0^-} f(x) = 1$$, $$\lim_{x 	o 0^+} f(x) = -2$

Question 2

Evaluate $$\lim_{x 	o -2} \frac{x^2$ - x - 6}{x + 2}.$$

Accepted Answer

$$-5$$

Question 3

For the function $$f(x) = x^2$ + 3x - 1$$, what is the average rate of change from $$x = 1 to$$ $$x = 3$$?

Accepted Answer

$$7$$

Question 4

Given $$f(x) = x^2$ - 5x$$, use the definition of the derivative to find $$f'(x). $$Which of the following is correct?

Accepted Answer

$$f'(x) = 2x - 5$$

Question 5

Find the slope of the tangent line to $$f(x) = x^3$ - 2 at$$ $$x = 1.$$

Accepted Answer

$$3$$

Question 6

A country's GDP is modeled by $$f(t) = 0.1t^3$ + 0.5t^2$ + 2t + 20$$, where $$t = 0$$ corresponds to 2007. What was the GDP in 2010?

Accepted Answer

$$33.2$$ billion

Question 7

Given $$C(x) = -0.04x^2$ + 80x + 75$$, what is the marginal cost when $$x = 100$$ devices?

Accepted Answer

$$72$$

Question 8

For $$f(x) = 2x^3$ - 3x^2$ - 12x$$, at which points is the tangent line horizontal?

Accepted Answer

$$(-1, 7)$$ and $$(2, -20)$$

Question 9

Evaluate $$\lim_{x 	o \infty} (2 - \frac{1}{x}).$$

Accepted Answer

$$2$$

Question 10

Given $$f(x) = e$$^{x}$$ - 2$$, what is $$\lim_{x 	o -\infty} f(x)$$?

Accepted Answer

$$-2$$

Question 11

For $$f(x) = x^5$ - 8x^3$ + 3x^2$ - x + 8$$, what is $$f'(x)$$?

Accepted Answer

$$5x^4$$ - $$24x^2 + 6x - 1$

Question 12

Suppose $$y = x + \sqrt{x} + x$$^{-2}$$ - 2x$$^{-3}$$. What is $$\frac{dy}{dx}$$?

Accepted Answer

$$1 + \frac{1}{2\sqrt{x}} - 2x$$^{-3}$$ + 6x$$^{-4}$$

Question 13

If $$f(x) is $$not continuous at $$x = -1$$ because $$\lim_{x 	o -1} f(x)$$ does not exist, which of the following is true?

Accepted Answer

The left and right limits at $$x = -1$$ are not equal.

Question 14

Find $$\lim_{x 	o 1} \frac{5x^2$ - 7x + 2}{x - 1}.$$

Accepted Answer

$$3$$

Question 15

Given $$f(x) = x^2$ + 3x - 1$$, use limits to find the instantaneous rate of change at $$x = 1.$$

Accepted Answer

$$5$$

Question 16

If the cost function is $$C(x) = -0.04x^2$ + 80x + 75$$, what does the marginal cost at $$x = 100$$ represent?

Accepted Answer

The approximate cost to produce the 101st device.

Question 17

For $$f(x) = 2x^2$ + 1$$, what is the equation of the tangent line at $$x = -1$$?

Accepted Answer

$$y = -4x - 1$$

Question 18

Evaluate $$\lim_{x 	o 1} \left( \frac{1}{x} + 2 ight).$$

Accepted Answer

$$3$$

Question 19

If $$f(x) is $$discontinuous at $$x = 1$$ because $$\lim_{x 	o 1} f(x) 
eq f(1)$$, what type of discontinuity is this?

Accepted Answer

Removable discontinuity

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