Question 1

Given $$f(x) = e$$^{-	an^2(x)}$$, what is the domain of f(x$$)$$?

Accepted Answer

$$x \in \mathbb{R} \setminus \{\frac{\pi}{2} + k\pi : k \in \mathbb{Z}\}$$

Question 2

For $$f(x) = e$$^{-	an^2(x)}$$, at which points does f(x$$)$$ have discontinuities?

Accepted Answer

At $$x = \frac{\pi}{2} + k\pi$$, $$k \in \mathbb{Z}$$

Question 3

Classify the discontinuity of $$f(x) = e$$^{-	an^2(x)}$$ at x$$ = \frac{\pi}{2}$$.

Accepted Answer

Infinite

Question 4

Which of the following best describes a removable discontinuity?

Accepted Answer

A point where the left and right limits exist and are equal, but the function is not defined or not equal to the limit

Question 5

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What theorem guarantees there is $$c \in (a, b)$$ such that $$f(c) = 0$$?

Accepted Answer

Intermediate Value Theorem

Question 6

Given $$f(x) = e$$^{-	an^2(x)}$$, which of the following intervals contains a removable discontinuity?

Accepted Answer

None; all discontinuities are infinite

Question 7

If $$f(x) = e$$^{-	an^2(x)}$$, what is $$\lim_{x 	o \frac{\pi}{2}^-} f(x)$$?

Accepted Answer

$$0$$

Question 8

Which of the following is a necessary condition for the Intermediate Value Theorem to apply to a function $$f on$$ $$[a, b]$$?

Accepted Answer

$$f is $$continuous on $$[a, b]$$

Question 9

Consider $$f(x) = e$$^{-	an^2(x)}$$. Which of the following statements is true?

Accepted Answer

$$f(x) is $$discontinuous at $$x = \frac{\pi}{2} + k\pi$$, $$k \in \mathbb{Z}$$

Question 10

Suppose $$g(x) is $$defined as $$g(x) = 	an^2$(x). $$For which values of $$x is$$ $$g(x)$$ undefined?

Accepted Answer

$$x = \frac{\pi}{2} + k\pi$$, $$k \in \mathbb{Z}$$

Question 11

A function $$h(x) is $$continuous on $$[1, 4]$$ with $$h(1) = -2$$ and $$h(4) = 5. $$What can you conclude?

Accepted Answer

There exists $$c \in (1, 4)$$ such that $$h(c) = 0$$

Question 12

Which of the following best describes an infinite discontinuity?

Accepted Answer

The function approaches infinity as $$x$$ approaches a certain value

Question 13

If $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003e 0$$, $$f(b) \u003c 0$$, which theorem guarantees a root in $$(a, b)$$?

Accepted Answer

Intermediate Value Theorem

Question 14

Which of the following is NOT a requirement for the Intermediate Value Theorem?

Accepted Answer

The function must be differentiable on the interval

Question 15

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) = f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

For any $$N$$ between $$f(a)$$ and $$f(b)$$, there exists $$c \in (a, b)$$ such that $$f(c) = N$$

Continuity, Discontinuity, and the Intermediate Value Theorem (IVT)

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