Question 1

Consider the function $$f(x)$$ defined as follows:

$$\begin{cases}
|x + 4|, & x \leq -4 \
\sqrt{x + 4} - 2, & -4 \u003c x \u003c 0 \
x^2$ + 2, & x \geq 0
\end{cases}$$

Which of the following statements is TRUE about the continuity of $$f(x) at$$ $$x = -4$$?

Accepted Answer

The function is continuous at $$x = -4.$$

Question 2

Given $$f(x) = \sin(\pi x) - 4 \log_{10}(x^2$ + 1)$$, which theorem guarantees that $$f(x)$$ takes the value $$-2$$ for some $$x \in \mathbb{R}$$?

Accepted Answer

Intermediate Value Theorem

Question 3

For the function $$g(x) = \begin{cases} |4x|, & x \u003c 3 \ 2x^2$/3 + 6, & x \geq 3 \end{cases}$$, does the derivative exist at $$x = 3$$?

Accepted Answer

No, because the left and right derivatives at $$x = 3$$ are not equal.

Question 4

Suppose $$f(x) = \begin{cases} x^2$, & x \leq 2 \ mx + b, & x \u003e 2 \end{cases}. $$What conditions must $$m$$ and $$b$$ satisfy for $$f(x) to be $$differentiable at $$x = 2$$?

Accepted Answer

$$m = 4$$, $$b = 4$$

Question 5

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

Accepted Answer

$$f$$ must be continuous on $$[a, b].$$

Question 6

A chemical reaction's temperature $$T(t) (in$$ $$^ C) $$over time $$t (in $$minutes) is modeled by a continuous function. If $$T(0) = 20$$ and $$T(10) = 80$$, what does the Intermediate Value Theorem guarantee?

Accepted Answer

There is a time $$t$$ where $$T(t) = 50.$$

Question 7

For $$f(x) = \sqrt{4 - x^2$} - (2 - \cos(\pi x))$$, why does the equation $$f(x) = 0$$ have a solution?

Accepted Answer

Because $$f(x) is $$continuous and takes both positive and negative values on an interval.

Question 8

Let $$h(x) = |x|. $$Which of the following statements is TRUE about the derivative of $$h(x) at$$ $$x = 0$$?

Accepted Answer

The derivative does not exist because the left and right limits are not equal.

Question 9

Suppose $$f(x) is $$defined as $$f(x) = \sqrt{x + 4} - 2$$ for $$-4 \u003c x \u003c 0. $$What is $$\lim_{x 	o -4^+} f(x)$$?

Accepted Answer

$$0$$

Question 10

A patient’s blood concentration of a drug is modeled by a continuous function $$C(t)$$, with $$C(1) = 5 mg/L $$and $$C(3) = 15 mg/L. $$What can you conclude?

Accepted Answer

There is a time $$t in$$ $$(1, 3)$$ where $$C(t) = 10 mg/L.$$

Question 11

For $$f(x) = x^2$ for x$$ \leq 2$$ and f(x) = mx + b$ for x$$ \u003e 2$$, what is the value of $$\lim_{x 	o 2^-} f(x)$$?

Accepted Answer

$$4$$

Question 12

Which of the following best describes the left-hand derivative of $$g(x) = |4x| at$$ $$x = 3$$?

Accepted Answer

$$4$$

Question 13

If $$f(x) is $$continuous at $$x = a$$ but not differentiable at $$x = a$$, which of the following could be true?

Accepted Answer

$$f(x)$$ has a sharp corner at $$x = a.$$

Question 14

Suppose $$f(x) is $$continuous on $$[a, b]$$ and $$f(a) \u003c 0 \u003c f(b). $$What does the Intermediate Value Theorem guarantee?

Accepted Answer

There exists $$c \in (a, b)$$ such that $$f(c) = 0.$$

Calculus Assignment Guidance: Continuity, Intermediate Value Theorem, and Differentiability

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