Carlos ate 1/3 of his caramel pie. Carlos is having a friend come over and wants to give her 1/4 of the pie he has remaining. What fraction of the original pie will he give his friend?

(1)

(a) Use the fact that [tex]\sqrt{x^2} = |x|[/tex] for all [tex]x[/tex]. Since [tex]x\to+\infty[/tex], we have [tex]x>0[/tex] and [tex]|x| = x[/tex].

[tex]\displaystyle \lim_{x\to\infty} \frac{\sqrt{9x^2 - 2}}{x + 4} = \lim_{x\to\infty} \frac{\sqrt{x^2} \sqrt{9 - \frac2{x^2}}}{x + 4} \\\\= \lim_{x\to\infty} \frac{ x \sqrt{9 - \frac2{x^2}}}{x+4} \\\\= \lim_{x\to\infty} \frac{\sqrt{9 - \frac2{x^2}}}{1 + \frac4x} \\\\= \frac{\sqrt9}1 = \boxed{3}[/tex]

(b) This time [tex]x\to-\infty[/tex], so [tex]x < 0[/tex] and [tex]|x| = -x[/tex].

[tex]\displaystyle \lim_{x\to\infty} \frac{\sqrt{9x^2 - 2}}{x + 4} = \lim_{x\to\infty} \frac{\sqrt{x^2} \sqrt{9 - \frac2{x^2}}}{x + 4} \\\\= \lim_{x\to\infty} \frac{ \boxed{-x} \sqrt{9 - \frac2{x^2}}}{x+4} \\\\= (-1) \times \lim_{x\to\infty} \frac{\sqrt{9 - \frac2{x^2}}}{1 + \frac4x} \\\\= -\frac{\sqrt9}1 = \boxed{-3}[/tex]

(c) We immediately have

[tex]\displaystyle \lim_{x\to\infty} (x - \sqrt x) = \boxed{\infty}[/tex]

since [tex]x > \sqrt x[/tex] for all [tex]x > 1[/tex].

(d) Introduce a difference of squares by factoring in the limand's conjugate. The rest mirrors what we did in (a)/(b).

[tex]\displaystyle \lim_{x\to\infty} \left(\sqrt{x^2 + 12x} - x\right) = \lim_{x\to\infty} \frac{\left(\sqrt{x^2+12x} - x\right) \left(\sqrt{x^2+12x} + x\right)}{\sqrt{x^2 + 12x} + x} \\\\ = \lim_{x\to\infty} \frac{\left(\sqrt{x^2+12x}\right)^2 - x^2}{\sqrt{x^2+12x}+x} \\\\ = \lim_{x\to\infty} \frac{12x}{\sqrt{x^2+12x}+x} \\\\ = \lim_{x\to\infty} \frac{12}{\sqrt{1 + \frac{12}x} + 1} = \frac{12}{\sqrt1 + 1} = \boxed{6}[/tex]

(e) Divide through by the highest-degree exponential term.

[tex]\displaystyle \lim_{x\to\infty} \frac{12e^{2x} - 3e^{3x}}{2e^{2x} + 4e^{3x}} = \lim_{x\to\infty} \frac{12e^{-x} - 3}{2e^{-x} + 4} = \frac{0 - 3}{0 + 4} = \boxed{-\frac34}[/tex]

(2) By definition of the derivative, we have

[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{f(x+h) - f(x)}h[/tex]

For [tex]f(x) = \sqrt{x^2+1}[/tex], the limit becomes

[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\sqrt{(x+h)^2+1} - \sqrt{x^2+1}}h[/tex]

Factor in the conjugate.

[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\left(\sqrt{(x+h)^2+1} - \sqrt{x^2+1}\right) \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]

[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\left(\sqrt{(x+h)^2+1}\right)^2 - \left(\sqrt{x^2+1}\right)^2}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]

[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{\bigg((x+h)^2+1\bigg) - (x^2+1)}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]

[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{2xh + h^2}{h \left(\sqrt{(x+h)^2+1} + \sqrt{x^2+1}\right)}[/tex]

[tex]f'(x) = \displaystyle \lim_{h\to0} \frac{2x + h}{\sqrt{(x+h)^2+1} + \sqrt{x^2+1}}[/tex]

[tex]\implies \boxed{f'(x) = \displaystyle \lim_{h\to0} \frac{x}{\sqrt{x^2+1}}}[/tex]

(3) The tangent line to

[tex]y = \frac1{x^2+1}[/tex]

at the point (2, 1/5) has slope equal to the derivative [tex]\frac{dy}{dx}[/tex] when [tex]x = 2[/tex]. Compute the derivative; since [tex]y = \frac1{f(x)^2}[/tex] where [tex]f(x)[/tex] is the function from the previous problem, using the chain rule gives

[tex]y = \dfrac1{f(x)^2} \implies \dfrac{dy}{dx} = -\dfrac{2f'(x)}{f(x)^3} = -\dfrac{2 \times \frac{x}{\sqrt{x^2+1}}}{\left(\sqrt{x^2+1}\right)^3} \\\\ \implies \dfrac{dy}{dx} = -\dfrac{2x}{(x^2+1)^2}[/tex]

The tangent line at (2, 1/5) then has slope

[tex]\dfrac{dy}{dx}\bigg|_{x=2} = -\dfrac{2\times2}{(2^2+1)^2} = -\dfrac4{25}[/tex]

Using the point-slope formula, the equation of the tangent line is

[tex]y - \dfrac15 = -\dfrac4{25} (x - 2) \implies \boxed{y = -\dfrac{4x - 13}{25}}[/tex]

Answer:

hope i helped...if anythng not understood please write in the comments

Answer:

12 passengers can travel in Rs. 900.

Step-by-step explanation:

see the above attachment.

if you face any problem comment me, I am there for help.

hope this helps you.

Answer:

-9

Step-by-step explanation:

(-4-2*-1/2)3

(-4+1)3

-3*3

-9

Answer:

-27

Step-by-step explanation:

Hi there!

[tex](a-2b)^3[/tex]

Plug in the given information:

[tex](-4-2(\displaystyle -\frac{1}{2} ))^3\\\\(-4-(\displaystyle -\frac{2}{2} ))^3\\\\(-4+1)^3\\\\(-3)^3\\\\-27[/tex]

I hope this helps!

Answer:

7.083

Step-by-step explanation:

well there's the ones place, tenths, hundredths, and thousandths. easy math, unless i'm stoopid and this is wrong :D

Answer: 4

pls give brainliest

9514 1404 393

Answer:

Step-by-step explanation:

Jarvis is paid for 1 1/2 times the number of hours he worked on Saturday. So, for the 5 hours he worked, he is paid for ...

  5 × 1 1/2 = 7 1/2 hours . . . . paid on Saturday

Jarvis is paid for 1 3/4 times the number of hours he worked on Sunday, So, for the 2 hours he worked, he is paid for ...

  2 × 1 3/4 = 3 1/2 hours . . . . paid on Sunday

The total of hours for which Jarvis is paid is ...

  7 1/2 + 3 1/2 = 11 hours . . . . paid for the weekend

At $9 per hour, the amount Jarvis is paid for the weekend is ...

  $9 × 11 = $99.00 . . . . weekend pay

Answer:

C. 11 customers

Step-by-step explanation:

The pattern is that it rises or goes up by 2 in the y axis, it runs or goes to the right once in the x-axis.

Answer:

?

Step-by-step explanation:

y=mx+b

-3=3(3)+b

-3=9+b

-12=b

Answer:

Any expression that sums up to 96 is equivalent to 12+84.

Step-by-step explanation:

Since, you haven't provided any options.

12+84=96

Any expression that sums up to 96 is equivalent to 12+84.

Answer:

Banks play an important role in the economy by offering a service for people wishing to save. Banks also play an important role in offering finance to businesses that wish to invest and expand. These loans and business investment are important for enabling economic growth.

The financial services sector is the primary driver of a nation's economy. It provides the free flow of capital and liquidity in the marketplace. When the sector is strong, the economy grows, and companies in this industry are better able to manage risk.

Step-by-step explanation:

Answer:

Define:

Domain is the set of values that you can input in a function

Range is the set of values that corresponds to the output of a function, the set of values corresponding to its results

The vertical line test is a simple test you can do with a graph to know if this graph correspond to a function or not: you have to draw a vertical line, parallel to the y-axis and if that line touch the graph in just one point, it is a function

What is the domain and range of each function?

Domain = {3, 5, 7, 8, 11}

Range = {6, 7, 9, 14}

Is each relation a function?

Yes

Answer:

my butt

Step-by-step explanation:

why well because when were dinosaurs created in 1957 and when they were alive they drove cars to school to learn about extra chromosomes and and then the meatore hit and created my butt.

I hope that helps if you need anymore help let me know :)

Answer: a

Step-by-step explanation:

Answer:

A robot can complete tasks in

hour. Each task takes the same amount of time.

a. How long does it take the robot to complete one task?

It takes one hour to complete one task.

b. How many tasks can the robot complete in one hour?

It complete only one task in one hour.

Answer:

[tex]y = \frac{1}{2} x - 1[/tex]

Explanation:

The first pic shows the line graphed, while the second pic shows how to turn the equation into proper slope-intercept form.

The only steps the second pic is missing however, is flipping the sides around to be better understood (optional), and more importantly subtracting 2 on both sides in order to move it to the other side, to get 2y = x - 2. And dividing the 2 on both sides to get y by its self, to get:

y = (x-2)/2 , or y = 1/2 x - 2/2 , which if you notice 2/2 makes it equal to one whole so your answer is y = 1/2 x - 1 .

If you like this answer can you please mark me brainliest? :)

Answer:-16

Step-by-step explanation:

add all the temperature together =-112

then you divide by the number of temperatures given which is 7 which gives you -16

Answer:

-16 degrees C.

Step-by-step explanation:

When they say mean, they mean the average. You add the numbers together and divide them on how many numbers there are.

Answer:

1/10

Step-by-step explanation:

The default placeholder value for tenths in math is 0.1, or 1/10. the first digit after the decimal point.

can you help me this 4^x x ^8^x^+^1 = 2⁴

Answer:

The answer is 86° Fahrenheit.

Step-by-step explanation:

The equation for converting between Fahrenheit and Celsius is °F = 1.8 x °C + 32. So you would just plug 30 in for the °C.

°F = 1.8 x 30° + 32.

°F = 54 + 32

°F = 86°

Answer:

30ºC = 86ºF

Step-by-step explanation:

The temperature T in degrees Fahrenheit (ºF) is equal to 30 degrees Celsius (ºC) times 9/5 plus 32:

T(ºF) = 30ºC × 9/5 + 32 = 86ºF

Answer:

1/64

Step-by-step explanation:

P(pick 3 consecutive spades with replacement of drawn cards) =

(13/52)(13/52)(13/52) =

(1/4)(1/4)(1/4) =

1/64

9514 1404 393

Answer:

  D: 21

Step-by-step explanation:

The first term is (a1). The third term is (a1)r². The product of these terms is ...

  (a1)(a1)r² = ((a1)r)², the square of the second term.

The missing term is √(7·63) = √441 = 21.

_____

Additional comment

In consideration of the above, you can also see that the common ratio can be found from ...

  a3/a1 = ((a1)r²)/(a1) = r² = 63/7 = 9

Then the common ratio is r = √9 = 3, and the second term is 7·3 = 21.

Answer:

Anupam ask me where l live

hope it's help you (^0^)

Answer:

-12

Step-by-step explanation:

-3 + -9 = 12

hope this helps

Answer:

-12

Step-by-step explanation:

-3 + -9

turns into

-3 - 9

Which means on the number line, we start at -3 and move to the left 9 units.

When we move to left on the number line by 9, we are at -12.

Which means -3 - 9 (-3 + -9) = -12

Answer:

5. yes It still has the same density because it is the same thing.

6. a. float   less than one

b. Float less than one

C. Sink   greater than one

7. 1652.4 This is because the density equation is mass/volume. We have the density and mass but no volume. do get it we need to re arrange the equation. mass x density = volume.

Step-by-step explanation:

Answer:

80 I believe

Step-by-step explanation:

Mark as brainliest :)