Q: Let $$f(x) = \begin{cases} 7x, & x \u003c 4 \\ 3x + 1, & x \geq 4 \end{cases}. $$Find $$\lim_{x \to 4^-} f(x)$$ and $$\lim_{x \to 4^+} f(x).$$

$$\lim_{x \to 4^-} f(x) = 28$$, $$\lim_{x \to 4^+} f(x) = 13$$

Question 1

A pharmaceutical company is monitoring the concentration of a drug in a patient's bloodstream, modeled by $$h(t) = \cot t. $$What is the average rate of change of $$h(t)$$ over the interval $$\left[ \frac{3\pi}{4}, \pi \right]$$?

Accepted Answer

$$-\frac{4}{\pi}$$

Question 2

Given $$R(t) = 40t$$^{-1}$$, what is the average rate of change of R(t$$)$$ over the interval $$[6,12]$$?

Accepted Answer

$$\frac{1}{3}$$

Question 3

A ball is thrown upward and its height is modeled by $$y = 5 - 9x^2$. At the point P(-4$$, -139)$$, what is the slope of the curve and the equation of the tangent line at P$?

Accepted Answer

Slope: $$72$$, Tangent: $$y = 72x + 149$$

Question 4

Evaluate the following limit, which models the instantaneous rate of change of a chemical reaction: $$\lim_{x 	o 8} (8x + 10).$$

Accepted Answer

$$62$$

Question 5

A patient’s temperature is modeled by $$y = (15 - y)^5$ as y$$ 	o 12$$. What is $$\lim_{y 	o 12} (15 - $$y)^5$$?

Accepted Answer

$$243$$

Question 6

Find $$\lim_{x 	o 16} \frac{x - 16}{x^2$ - 256}$$, which could represent the rate of change of a population at $$x = 16.$$

Accepted Answer

$$\frac{1}{32}$$

Question 7

A function $$f(x) is $$defined for all real values of $$x in$$ $$[-1,1]. $$Can anything be said about the existence of $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

Nothing can be said about the existence of the limit. Even though the function is defined at $$x=0$$, it may oscillate.

Question 8

Evaluate $$\lim_{x 	o 169} \frac{\sqrt{x} - 13}{x - 169}$$, which could represent the rate of change of the side length of a square as its area changes.

Accepted Answer

$$\frac{1}{26}$$

Question 9

Find $$\lim_{x 	o 5} \frac{\sqrt{x^2$ + 11} - 6}{x - 5}$$, which could represent the rate of change of the diagonal of a rectangle as its side changes.

Accepted Answer

$$\frac{5}{6}$$

Question 10

Evaluate $$\lim_{x 	o 0} \frac{5 + 2x + \sin x}{6 \cos x}$$, which could represent the average rate of change of a signal as $$x$$ approaches zero.

Accepted Answer

$$\frac{5}{6}$$

Question 11

A function $$f(x) is $$defined as $$f(x) = x^2$ and x = 7$. Evaluate $$\lim_{h 	o 0} \frac{f(x + h) - f(x)}{h}$$.

Accepted Answer

$$-14$$

Question 12

Given $$f(x) = \sqrt{x}$$ and $$x = 19$$, evaluate $$\lim_{h 	o 0} \frac{f(x + h) - f(x)}{h}$$, which models the instantaneous rate of change of $$f at$$ $$x=19.$$

Accepted Answer

$$\frac{1}{2\sqrt{19}}$$

Question 13

A function $$f(x) is $$defined as $$f(x) = \sqrt{5}$$ for all $$x. $$What is $$\lim_{x 	o 0} f(x)$$?

Accepted Answer

$$\sqrt{5}$$

Question 14

A researcher sketches the interval $$(a, b) on $$the $$x-$$axis with the point $$c$$ inside the interval. If $$a = 1$$, $$b = 9$$, $$c = 7$$, what is the largest value of $$\delta \u003e 0$$ such that for all $$x$$, $$a \u003c x \u003c b$$ whenever $$0 \u003c |x - c| \u003c \delta$$?

Accepted Answer

$$2$$

Question 15

Let $$f(x) = \begin{cases} 7x, & x \u003c 4 \ 3x + 1, & x \geq 4 \end{cases}. $$Find $$\lim_{x 	o 4^-} f(x)$$ and $$\lim_{x 	o 4^+} f(x).$$

Accepted Answer

$$\lim_{x 	o 4^-} f(x) = 28$$, $$\lim_{x 	o 4^+} f(x) = 13$$

Question 16

Given $$f(x) = (x + 6)/(x^2$ + 3)$$, evaluate $$\lim_{x 	o -3} f(x).$$

Accepted Answer

$$3$$

Question 17

Use the greatest integer function $$y = \lfloor x floor to $$find $$\lim_{x 	o 2^-} \frac{\lfloor x floor}{2}.$$

Accepted Answer

$$1$$

Question 18

Evaluate $$\lim_{x 	o 0} \frac{8 \sin \sqrt{9	heta}}{\sqrt{9	heta}}$$, which could represent the amplitude of a wave as $$	heta$$ approaches zero.

Accepted Answer

$$8$$

Question 19

Find $$\lim_{x 	o 0} \frac{x \csc 5x}{6 \cos 18x}$$ using the fact that $$\lim_{x 	o 0} \frac{\sin x}{x} = 1.$$

Accepted Answer

$$\frac{1}{5}$$

Question 20

Use the relation $$\lim_{	heta 	o 0} \frac{\sin 	heta}{	heta} = 1 to $$determine $$\lim_{	heta 	o 0} \frac{1 - \cos 	heta}{\sin 15	heta}.$$

Accepted Answer

$$0$$

Calculus Study Guide: Average Rate of Change, Limits, and Tangents

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