Q: A function $$f is $$said to be continuous from the right at $$x = c if$$:

$$\lim_{x \to c^+} f(x) = f(c)$$

Question 1

Which of the following is NOT a requirement for a function $$f to be $$continuous at $$x = c$$?

Accepted Answer

$$f(x) is $$differentiable at $$x = c.$$

Question 2

Given $$f(x) = x + 2$$, which of the following statements is true about its continuity?

Accepted Answer

$$f(x) is $$continuous for all $$x.$$

Question 3

Consider $$g(x) = \frac{x^2$ - 4}{x - 2}. $$Which statement best describes the continuity of $$g(x) at$$ $$x = 2$$?

Accepted Answer

$$g(x) is $$discontinuous at $$x = 2$$ because $$g(2) is $$not defined.

Question 4

Which of the following functions is continuous for all real numbers?

Accepted Answer

$$f(x) = x^3$ + 2x - 5$$

Question 5

A bakery’s profit function is $$P(x) = -\frac{1}{1000}x^2$ + 6x - 5000. $$For which values of $$x is $$the profit positive?

Accepted Answer

$$1000 \u003c x \u003c 5000$$

Question 6

Which of the following is a partition number for the function $$f(x) = \frac{x+1}{x-2}$$?

Accepted Answer

$$x = 2$$ only

Question 7

If $$f(x) is $$continuous on $$(a, b)$$ and $$f(x) \u003e 0$$ for all $$x in$$ $$(a, b)$$, what can you conclude about the sign of $$f(x) on$$ $$(a, b)$$?

Accepted Answer

$$f(x) is $$always positive on $$(a, b).$$

Question 8

Which of the following statements about rational functions is true?

Accepted Answer

A rational function is continuous for all $$x$$ except where the denominator is zero.

Question 9

Suppose $$f(x) is $$continuous everywhere and $$f(-1) = 0$$, $$f(2) = 0. $$What can you say about $$f(x) on $$the interval $$(-1, 2)$$?

Accepted Answer

$$f(x)$$ could change sign in $$(-1, 2).$$

Question 10

Which of the following is NOT a property of continuous functions?

Accepted Answer

The composition of two continuous functions is always discontinuous.

Question 11

For $$h(x) = \frac{|x|}{x}$$, which of the following is true?

Accepted Answer

$$h(x) is $$discontinuous at $$x = 0.$$

Question 12

Which of the following intervals describes where $$f(x) = \sqrt{x+1} is $$continuous?

Accepted Answer

$$[-1, \infty)$$

Question 13

A function $$f is $$said to be continuous from the right at $$x = c if$$:

Accepted Answer

$$\lim_{x 	o c^+} f(x) = f(c)$$

Question 14

Which of the following best describes a partition number for a function $$f$$?

Accepted Answer

A value where $$f is $$discontinuous or $$f(x) = 0.$$

Question 15

If $$f(x) is $$continuous on $$[a, b]$$, which of the following must be true?

Accepted Answer

$$f(x) is $$continuous on $$(a, b)$$ and continuous from the right at $$a$$ and from the left at $$b.$$

Continuity and Sign Charts in Business Calculus

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