# 12.4 Derivatives  (Page 11/18)

 Page 11 / 18

## Finding Limits: Properties of Limits

For the following exercises, find the limits if $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}f\left(x\right)=-3\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\underset{x\to c}{\mathrm{lim}}g\left(x\right)=5.$

$\underset{x\to c}{\mathrm{lim}}\left(f\left(x\right)+g\left(x\right)\right)$

2

$\underset{x\to c}{\mathrm{lim}}\frac{f\left(x\right)}{g\left(x\right)}$

$\underset{x\to c}{\mathrm{lim}}\left(f\left(x\right)\cdot g\left(x\right)\right)$

$-15$

3

$\underset{x\to {3}^{+}}{\mathrm{lim}}\left(3x-\mathrm{〚x〛}\right)$

For the following exercises, evaluate the limits using algebraic techniques.

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{{\left(h+6\right)}^{2}-36}{h}\right)$

12

$\underset{x\to 25}{\mathrm{lim}}\left(\frac{{x}^{2}-625}{\sqrt{x}-5}\right)$

$\underset{x\to 1}{\mathrm{lim}}\left(\frac{-{x}^{2}-9x}{x}\right)$

$-10$

$\begin{array}{c}\mathrm{lim}\\ {}^{x\to 4}\end{array}\frac{7-\sqrt{12x+1}}{x-4}$

$\underset{x\to -3}{\mathrm{lim}}\left(\frac{\frac{1}{3}+\frac{1}{x}}{3+x}\right)$

$-\frac{1}{9}$

## Continuity

For the following exercises, use numerical evidence to determine whether the limit exists at $\text{\hspace{0.17em}}x=a.\text{\hspace{0.17em}}$ If not, describe the behavior of the graph of the function at $\text{\hspace{0.17em}}x=a.$

At $\text{\hspace{0.17em}}x=4,$ the function has a vertical asymptote.

removable discontinuity at $\text{\hspace{0.17em}}a=-\frac{5}{2}$

For the following exercises, determine where the given function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is continuous. Where it is not continuous, state which conditions fail, and classify any discontinuities.

$f\left(x\right)={x}^{2}-2x-15$

continuous on $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)$

$f\left(x\right)=\frac{{x}^{2}-2x-15}{x-5}$

$f\left(x\right)=\frac{{x}^{2}-2x}{{x}^{2}-4x+4}$

removable discontinuity at $\text{\hspace{0.17em}}x=2.$ $\text{\hspace{0.17em}}f\left(2\right)\text{\hspace{0.17em}}$ is not defined, but limits exist.

$f\left(x\right)=\frac{{x}^{3}-125}{2{x}^{2}-12x+10}$

$f\left(x\right)=\frac{{x}^{2}-\frac{1}{x}}{2-x}$

discontinuity at $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}x=2.\text{\hspace{0.17em}}$ Both $\text{\hspace{0.17em}}f\left(0\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}f\left(2\right)\text{\hspace{0.17em}}$ are not defined.

$f\left(x\right)=\frac{x+2}{{x}^{2}-3x-10}$

$f\left(x\right)=\frac{x+2}{{x}^{3}+8}$

removable discontinuity at is not defined.

## Derivatives

For the following exercises, find the average rate of change $\text{\hspace{0.17em}}\frac{f\left(x+h\right)-f\left(x\right)}{h}.$

$f\left(x\right)=3x+2$

$f\left(x\right)=5$

0

$f\left(x\right)=\frac{1}{x+1}$

$f\left(x\right)=\mathrm{ln}\left(x\right)$

$\frac{\mathrm{ln}\left(x+h\right)-\mathrm{ln}\left(x\right)}{h}$

$f\left(x\right)={e}^{2x}$

For the following exercises, find the derivative of the function.

$f\left(x\right)=4x-6$

$=4$

$f\left(x\right)=5{x}^{2}-3x$

Find the equation of the tangent line to the graph of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ at the indicated $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ value.
$f\left(x\right)=-{x}^{3}+4x$ ; $\text{\hspace{0.17em}}x=2.$

$y=-8x+16$

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

$f\left(x\right)=\frac{x}{|x|}$

Given that the volume of a right circular cone is $\text{\hspace{0.17em}}V=\frac{1}{3}\pi {r}^{2}h\text{\hspace{0.17em}}$ and that a given cone has a fixed height of 9 cm and variable radius length, find the instantaneous rate of change of volume with respect to radius length when the radius is 2 cm. Give an exact answer in terms of $\text{\hspace{0.17em}}\pi$

$12\pi$

## Practice test

For the following exercises, use the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ in [link] .

$f\left(1\right)$

3

$\underset{x\to {-1}^{+}}{\mathrm{lim}}f\left(x\right)$

$\underset{x\to {-1}^{-}}{\mathrm{lim}}f\left(x\right)$

0

$\underset{x\to -1}{\mathrm{lim}}f\left(x\right)$

$\underset{x\to -2}{\mathrm{lim}}f\left(x\right)$

$-1$

At what values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ discontinuous? What property of continuity is violated?

For the following exercises, with the use of a graphing utility, use numerical or graphical evidence to determine the left- and right-hand limits of the function given as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a.\text{\hspace{0.17em}}$ If the function has a limit as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches $\text{\hspace{0.17em}}a,$ state it. If not, discuss why there is no limit

$\underset{x\to {2}^{-}}{\mathrm{lim}}f\left(x\right)=-\frac{5}{2}a\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\underset{x\to {2}^{+}}{\mathrm{lim}}f\left(x\right)=9\text{\hspace{0.17em}}$ Thus, the limit of the function as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ approaches 2 does not exist.

For the following exercises, evaluate each limit using algebraic techniques.

$\underset{x\to -5}{\mathrm{lim}}\left(\frac{\frac{1}{5}+\frac{1}{x}}{10+2x}\right)$

$-\frac{1}{50}$

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{\sqrt{{h}^{2}+25}-5}{{h}^{2}}\right)$

$\underset{h\to 0}{\mathrm{lim}}\left(\frac{1}{h}-\frac{1}{{h}^{2}+h}\right)$

1

For the following exercises, determine whether or not the given function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is continuous. If it is continuous, show why. If it is not continuous, state which conditions fail.

$f\left(x\right)=\sqrt{{x}^{2}-4}$

$f\left(x\right)=\frac{{x}^{3}-4{x}^{2}-9x+36}{{x}^{3}-3{x}^{2}+2x-6}$

removable discontinuity at $\text{\hspace{0.17em}}x=3$

For the following exercises, use the definition of a derivative to find the derivative of the given function at $\text{\hspace{0.17em}}x=a.$

$f\left(x\right)=\frac{3}{5+2x}$

$f\left(x\right)=\frac{3}{\sqrt{x}}$

$f\text{'}\left(x\right)=-\frac{3}{2{a}^{\frac{3}{2}}}$

$f\left(x\right)=2{x}^{2}+9x$

For the graph in [link] , determine where the function is continuous/discontinuous and differentiable/not differentiable.

discontinuous at –2,0, not differentiable at –2,0, 2.

For the following exercises, with the aid of a graphing utility, explain why the function is not differentiable everywhere on its domain. Specify the points where the function is not differentiable.

$f\left(x\right)=|x-2|-|x+2|$

$f\left(x\right)=\frac{2}{1+{e}^{\frac{2}{x}}}$

not differentiable at $\text{\hspace{0.17em}}x=0\text{\hspace{0.17em}}$ (no limit)

For the following exercises, explain the notation in words when the height of a projectile in feet, $\text{\hspace{0.17em}}s,$ is a function of time $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ in seconds after launch and is given by the function $\text{\hspace{0.17em}}s\left(t\right).$

$s\left(0\right)$

$s\left(2\right)$

the height of the projectile at $\text{\hspace{0.17em}}t=2\text{\hspace{0.17em}}$ seconds

$s\text{'}\left(2\right)$

$\frac{s\left(2\right)-s\left(1\right)}{2-1}$

the average velocity from

$s\left(t\right)=0$

For the following exercises, use technology to evaluate the limit.

$\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}\left(x\right)}{3x}$

$\frac{1}{3}$

$\underset{x\to 0}{\mathrm{lim}}\frac{{\mathrm{tan}}^{2}\left(x\right)}{2x}$

$\underset{x\to 0}{\mathrm{lim}}\frac{\mathrm{sin}\left(x\right)\left(1-\mathrm{cos}\left(x\right)\right)}{2{x}^{2}}$

0

Evaluate the limit by hand.

At what value(s) of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ is the function below discontinuous?

$f\left(x\right)=\left\{\begin{array}{c}4x-7\text{\hspace{0.17em}}\text{\hspace{0.17em}}x\ne 1\\ {x}^{2}-4\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=1\end{array}$

For the following exercises, consider the function whose graph appears in [link] .

Find the average rate of change of the function from

2

Find all values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ at which $\text{\hspace{0.17em}}f\text{'}\left(x\right)=0.$

$x=1$

Find all values of $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ at which $\text{\hspace{0.17em}}f\text{'}\left(x\right)\text{\hspace{0.17em}}$ does not exist.

Find an equation of the tangent line to the graph of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ the indicated point:

$y=-14x-18$

For the following exercises, use the function $\text{\hspace{0.17em}}f\left(x\right)=x{\left(1-x\right)}^{\frac{2}{5}}$ .

Graph the function $\text{\hspace{0.17em}}f\left(x\right)=x{\left(1-x\right)}^{\frac{2}{5}}\text{\hspace{0.17em}}$ by entering $\text{\hspace{0.17em}}f\left(x\right)=x{\left({\left(1-x\right)}^{2}\right)}^{\frac{1}{5}}\text{\hspace{0.17em}}$ and then by entering $\text{\hspace{0.17em}}f\left(x\right)=x{\left({\left(1-x\right)}^{\frac{1}{5}}\right)}^{2}$ .

Explore the behavior of the graph of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ around $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ by graphing the function on the following domains, [0.9, 1.1], [0.99, 1.01], [0.999, 1.001], and [0.9999, 1.0001]. Use this information to determine whether the function appears to be differentiable at $\text{\hspace{0.17em}}x=1.$

The graph is not differentiable at $\text{\hspace{0.17em}}x=1\text{\hspace{0.17em}}$ (cusp).

For the following exercises, find the derivative of each of the functions using the definition: $\text{\hspace{0.17em}}\underset{h\to 0}{\mathrm{lim}}\frac{f\left(x+h\right)-f\left(x\right)}{h}$

$f\left(x\right)=2x-8$

$f\left(x\right)=4{x}^{2}-7$

${f}^{\text{'}}\left(x\right)=8x$

$f\left(x\right)=x-\frac{1}{2}{x}^{2}$

$f\left(x\right)=\frac{1}{x+2}$

${f}^{\text{'}}\left(x\right)=-\frac{1}{{\left(2+x\right)}^{2}}$

$f\left(x\right)=\frac{3}{x-1}$

$f\left(x\right)=-{x}^{3}+1$

${f}^{\text{'}}\left(x\right)=-3{x}^{2}$

$f\left(x\right)={x}^{2}+{x}^{3}$

$f\left(x\right)=\sqrt{x-1}$

$f\text{'}\left(x\right)=\frac{1}{2\sqrt{x-1}}$

#### Questions & Answers

So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and$2 for every 300 texts. Plan B: $25 per month and$0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's$26.50 monthly payment, you'll need 3,000 texts which will cost an additional \$10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply

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