Q: Suppose $$f(x) is $$such that $$8x - 20 \leq f(x) \leq x^2$ + 4x - 16$$ for all $$x$$ near $$2. If$$ $$f(x)$$ models the temperature (in $$^\circC) in a $$chemical reaction, what can you conclude about the temperature as $$x \to 2$$?

The temperature approaches $$-4^\circC as$$ $$x \to 2.$$

Question 1

Let $$f(x) = 6x^2$ + 6. $$Compute the following limit:

$$
\lim_{h 	o 0} \frac{f(-2 + h) - f(-2)}{h}
$$
What does this limit represent?

Accepted Answer

The derivative $$f'(x)$$ evaluated at $$x = -2.$$

Question 2

Given $$f(x) = 6x^2$ + 6$$, evaluate:

$$
\lim_{h 	o 0} \frac{f(-2 + h) - f(-2)}{h}
$$
Choose the correct value.

Accepted Answer

$$-24$$

Question 3

Suppose $$8x - 20 \leq f(x) \leq x^2$ + 4x - 16. $$Using the Squeeze Theorem, compute:

$$
\lim_{x 	o 2} f(x)
$$

Accepted Answer

$$-4$$

Question 4

Which of the following best describes the Squeeze Theorem as used in the context of the limit $$
\lim_{x 	o 2} f(x)
$$ where $$8x - 20 \leq f(x) \leq x^2$ + 4x - 16$$?

Accepted Answer

If two functions bound $$f(x)$$ from above and below and have the same limit at a point, then $$f(x)$$ has the same limit at that point.

Question 5

Given $$f(x) = 6x^2$ + 6$$, what is the equation of the tangent line to $$f(x) at$$ $$x = -2$$?

Accepted Answer

$$y = -24(x + 2) + 30$$

Question 6

A chemical reaction's rate is modeled by $$f(x) = 6x^2$ + 6$$, where $$x is $$time in seconds. What is the instantaneous rate of change at $$x = -2$$ seconds?

Accepted Answer

$$-24$$ units/second

Question 7

Suppose $$f(x) is $$bounded by $$8x - 20 \leq f(x) \leq x^2$ + 4x - 16$$ for all $$x$$ near $$2. $$Which of the following is a necessary condition for applying the Squeeze Theorem to find $$\lim_{x 	o 2} f(x)$$?

Accepted Answer

Both $$8x - 20$$ and $$x^2 + 4x - 16$ must have the same limit as x$$ 	o 2$$.

Question 8

If $$f(x) = 6x^2$ + 6$$, what is $$f'(x)$$, the derivative of $$f(x)$$ with respect to $$x$$?

Accepted Answer

$$12x$$

Question 9

A medical researcher models the spread of a virus with $$f(x) = 6x^2$ + 6. If$$ $$x is $$days since exposure, what does $$\lim_{h 	o 0} \frac{f(-2 + h) - f(-2)}{h}$$ represent in this context?

Accepted Answer

The instantaneous rate of change in the number of cases at day $$-2.$$

Question 10

Suppose $$f(x) is $$such that $$8x - 20 \leq f(x) \leq x^2$ + 4x - 16$$ for all $$x$$ near $$2. If$$ $$f(x)$$ models the temperature (in $$^\circC) in a $$chemical reaction, what can you conclude about the temperature as $$x 	o 2$$?

Accepted Answer

The temperature approaches $$-4^\circC as$$ $$x 	o 2.$$

Question 11

Which of the following steps is NOT necessary when using the Squeeze Theorem to evaluate $$\lim_{x 	o 2} f(x)$$ given $$8x - 20 \leq f(x) \leq x^2$ + 4x - 16$$?

Accepted Answer

Checking if $$f(x) is $$differentiable at $$x = 2.$$

Question 12

If $$f(x) = 6x^2$ + 6$$, what is the average rate of change of $$f(x)$$ from $$x = -2 to$$ $$x = 0$$?

Accepted Answer

$$6$$

Question 13

Given $$f(x) = 6x^2$ + 6$$, which of the following is the correct difference quotient for $$f at$$ $$x = a$$?

Accepted Answer

$$\frac{f(a + h) - f(a)}{h}$$

Question 14

Suppose $$f(x) is $$bounded as $$8x - 20 \leq f(x) \leq x^2$ + 4x - 16$$ for all $$x$$ near $$2. If$$ $$f(x)$$ represents the height (in meters) of a projectile at time $$x$$ seconds, what is the height at $$x = 2$$?

Accepted Answer

It could be any value between $$-4$$ and $$-4$$ meters, so $$-4$$ meters.

Question 15

If $$f(x) = 6x^2$ + 6$$, what is $$f(-2)$$?

Accepted Answer

$$30$$

Calculus Quiz Guidance: Limits and Squeeze Theorem

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