Mathematics College

Let f(X)=1/x+1

Use the limit definition of the derivative to find:

i) f(-4)
ii) f(-3)
iii) f(1)
iv) f(3)

Answers

Answer 1
[tex]\bf f(x)=\cfrac{1}{x+1}\qquad \qquad \stackrel{d e f in i tion~of~a~derivative}{\lim\limits_{h\to 0}~\cfrac{f(t+h)-f(t)}{h}} \\\\\\ \lim\limits_{h\to 0}~\cfrac{\frac{1}{x+h+1}-\frac{1}{x+1}}{h}\implies \cfrac{\frac{(x+1)~-~(x+h+1)}{(x+h+1)(x+1)}}{h} \\\\\\ \cfrac{\frac{x+1-x-h-1}{(x+h+1)(x+1)}}{h}\implies \cfrac{\frac{\underline{x+1}\underline{-x}-h\underline{-1}}{(x+h+1)(x+1)}}{h}\implies \cfrac{\frac{-h}{(x+h+1)(x+1)}}{h} [/tex]

[tex]\bf \cfrac{\frac{-h}{x^2+xh+2x+h+1}}{h}\implies \cfrac{-h}{x^2+xh+2x+h+1}\cdot \cfrac{1}{h} \\\\\\ \lim\limits_{h\to 0}~\cfrac{-1}{x^2+xh+2x+h+1}\implies \cfrac{-1}{x^2+x(0)+2x+0+1} \\\\\\ \lim\limits_{h\to 0}~\cfrac{-1}{x^2+2x+1}[/tex]

Related Questions

Stacy wishes to make a hamburger which is lower in fat without having to give up the great taste. She purchases 5 pounds of ground beef which is 15% fat and combines it with 7 pounds of ground turkey which is only 3% fat. What is the percentage of fat in the mixture?

Please help, 15 pts.

Answers

Ground beef:
Amount = 5 lb
Fat = 15% of 5 lb = 0.15*5 = 0.75 lb fat

Turkey:
Amount = 7 lb
Fat = 3% of 7 lb = 0.03*7 = 0.21 lb fat

Total mass = 5 + 7 = 12 lb
Total fat = 0.75 + 0021 = 0.96 lb
Percent fat = 100*(0.96/12) =  8%

Answer: 8%
15% fat in 5 pounds & 3% fat in 7 pounds,

So, 15 x 5 + 3 x 7
      = 75 + 21
      = 96
After mixture, Total weight is 12 pounds,
So, 96 ÷ 12 = 8%

Therefore, 8% is the correct answer.

Write the following term as a cube. 64z^9
​(Type your answer using exponential​ notation.)

Answers

64 is 4³
z^9 is (z³)³
so 64z^9 can be written into (4z³)³

0.52 reapeating as a fraction

Answers

the idea being, hmmm... you'd multiply the value by some value of 10 raised at some power, that way the repeating part, moves over to the left-side of the decimal..... so let's do so, in this case the repeating part is 52, so we need to move two fellows over, so, we'll use 100, 10 at the second power then.

[tex]\bf 0.5252\overline{52}\qquad \boxed{x=0.5252\overline{52}} \\\\\\ \begin{array}{llll} 100\cdot x&=&52.5252\overline{52}\\ &&52+0.5252\overline{52}\\ &&52+\boxed{x}\\ 100x&=&52+x \end{array} \\\\\\ 100x-x=52\implies 99x=52\implies x=\cfrac{52}{99}[/tex]

can someone help me please
show your work

Answers

Answer:

a.

[tex]l = 2w - 3[/tex]

b. Yesterday, Gina again mixed red and white paint and made the same amount of paint, but she used one more pint of red paint than she used last week. Predict how the new paint color will compare to the paint she mixed last week

Answers

The paint will be tinted more red.

Can 3.65909090909 be expressed as a fraction whose denominator is a power of 10? Explain.

Answers

[tex]\bf 3.659\textit{ can also be written as }\cfrac{3659}{1000}\textit{ therefore }3.6590909\overline{09}\\\\ \textit{can be written as }\cfrac{3659.0909\overline{09}}{1000}[/tex]

notice above, all we did, was isolate the "recurring part" to the right of the decimal point, so the repeating 09, ended up on the right of it.

now, let's say, "x" is a variable whose value is the recurring part, therefore then

[tex]\bf \cfrac{3659.0909\overline{09}}{1000}\qquad \boxed{x=0.0909\overline{09}} \qquad \cfrac{3659+0.0909\overline{09}}{1000}\implies \cfrac{3659+x}{1000}[/tex]

now, the idea behind the recurring part is that, we then, once we have it all to the right of the dot, we multiply it by some power of 10, so that it moves it "once" to the left of it, well, the recurring part is 09, is two digits, so let's multiply it by 100 then, 

[tex]\bf \begin{array}{llllllll} 100x&=&09.0909\overline{09}\\ &&9+0.0909\overline{09}\\ &&9+x \end{array}\quad \implies 100x=9+x\implies 99x=9 \\\\\\ x=\cfrac{9}{99}\implies \boxed{x=\cfrac{1}{11}}\\\\ -------------------------------\\\\ \cfrac{3659.0909\overline{09}}{1000}\qquad \boxed{x=0.0909\overline{09}} \quad \cfrac{3659+0.0909\overline{09}}{1000}\implies \cfrac{3659+x}{1000} \\\\\\ \cfrac{3659+\frac{1}{11}}{1000}[/tex]

and you can check that in your calculator.

Which of the functions sketched below could be a probability density function?

Answers

We need a picture to solve it
Right functions or neather functions

use cramers rule to solve each system 6x +2y= -78 7x - 4y= -15

Answers

[tex]\bf \begin{cases} 6x+2y=-78\\ 7x-4y=-15 \end{cases}\qquad \begin{bmatrix} 6&2\\7&-4 \end{bmatrix}\implies (6\cdot -4)-(7\cdot 2)\implies \stackrel{D}{\boxed{-38}}\\\\ -------------------------------\\\\ x=\cfrac{D_x}{D}\implies \cfrac{ \begin{bmatrix} -78&2\\-15&-4 \end{bmatrix}}{-38}\implies \cfrac{(-78\cdot -14)-(-15\cdot 2)}{-38} \\\\\\ \cfrac{312+30}{-38}\implies\cfrac{342}{-38} \implies -9\\\\ -------------------------------[/tex]

[tex]\bf y=\cfrac{D_y}{D}\implies \cfrac{ \begin{bmatrix} 6&-78\\7&-15 \end{bmatrix}}{-38}\implies \cfrac{(6\cdot -15)-(-7\cdot 78)}{-38} \\\\\\ \cfrac{-80+546}{-38}\implies \cfrac{456}{-38}\implies -12[/tex]

Find the equation of the parabola y = ax^2 + bx + c that is tangent to the line y = x - 1 at (0,1)

Answers

We need to find the x value for which the slope of the tangent line to y = ax^2 + bx + c is 1 at the point on the parabola (0,1).  Realize that y = x -1 is the actual equation of the tangent line.  From this we see that the slope of the tangent line is m = 1.

Differentiate y = ax^2 + bx + c with respect to x:  dy/dx = 2ax + b.  This is the slope of the tangent line to the parabola at (0,1)  But we already know that the line y = x - 1 is tangent to the parabola at (0,1).  Thus, 2ax + b = 1 (because the slope of y = x - 1) is 1).  At (0,1), 2a(0) + b = 1, so b=1.

Thus, the equation of the parabola becomes more evident:  
a) since (0,1) satisfies this equation, 1 = a(0)^2 + 1(0) + c, so c=1

Then y = ax^2 + 1x + 1, and we need only find the value of a.

Since the graph of y = ax^2 + 1x + 1 passes thru (0,1), 

y = 1 = a(0)^2 + 1(0) + 1, or      1 = a(0)^2 + 1.  1 = 1, yes, but this does not tell us the value of a!


Is it possible that a could have any value?

Let a = 1.  Then y = 1x^2 + 1x + 1.  What is the slope of the tan. line to this curve?  Differentiating, dy/dx = 2x + 1.  At x = 0, the slope of the tan. line is 1.  This agrees with the given y= 1x - 1.

Thus, for a = 1 at least, y = 1x^2 + 1x + 1 is the equation of the parabola.

Graphing this equation and the equation of the given line y = x -1 shows that y = x - 1 is tangent to the graph of y = x^2 + x + 1.


A tree has a height of 18 feet and casts a shadow that is 6 feet in length. A boy, standing at the base of the tree’s shadow, has a height of 4 feet. How long is the shadow cast by the boy?

Answers

The two triangles of this statement are similar, so:

height of the three / its shadow = height of the boy / it shadow

18 / 6 = 4 / x => x = 4*6 / 18 = 24 / 18 = 4 / 3

Convert to mixed number: 4 / 3 = 1 and 1/3.

So, the answer is the option A. [tex]1 \frac{1}{3} feet[/tex]

If someone sleeps 8 hours a day for 75 years how many hours did that person sleep in his lifetime?

Answers

The person sleeps 8 hours a day.

There are 365 days in a year and 366 days in a leap year.

A leap year occurs every 4 years.

If we divide 75 by 4, we get 18.75. This number represents how many leap years occurred in 75 years. So, the person experienced approximately 19 leap years.

19 leap years has 6954 days.

The other 56 years, which have 365 days each year, have a total of 20,440 days.

That's a total of 27,394 days.

27,394 × 8 = 219,152 hours

This is approximately 24 years or 9,131 days.

Find the slope of the line passing through the points , (−7,2) and , (9,6) .

Answers

6-2 = 4

9- -7 =16

4/16 = 1/4

slope = 1/4

slope of the line is 

y₂ - y₁
------- = slope
x₂ - x₁

choose one as your "2" & the other as your "1" Plug them in
6 - 2
------
9 - (-7)

4/16
simplify the fraction
4/16 = 1/4

the slope is 1/4

hope this helps

A store plans to mark down all dresses by 60%. Julie wants to purchase a dress that is currently priced at $75. What amount of money will Julie save if she buys the dress after the store marks it down?

Answers

0.6 × 75$ = 45$ saved

Answer:45

Step-by-step explanation:For Julie's situation, the percent equation becomes:

money saved = percent of markdown × current price

Since the markdown is 60% off $75,

money saved====percent of markdown × current price60%×$750.6×7545

So, Julie will save $45.

WHOEVER ANSWER THIS AND HAS THE SMARTEST ANSWER I WILL MARK YOU AS BRAINLIEST. 15 out of 20 students own hand-held games; 105 out of 160 students own hand-held games. Determine if each pair of ratios or rates are equivalent. Show ALL your work.

Answers

The first ratio would be 3:4 (dividing by 5).
The second ratio would be 21:32 (also dividing by five).
To determine if they are equivalent rates, you must somehow make them even. Multiplying 3:4 by 7 would give you 21:28. Thus proving they are not equivalant.

I'm struggling on this!!!!!!!!!
In a photograph, a sculpture is 4.2 cm tall and 2.5 cm wide. The scale factor is 1 to 16. Find the size of the actual sculpture.

Answers

That means that the sculpture is going to be 16 times the size in the photo. 4.2*16=67.2 cm, 2.5*16=40 cm wide. The actual sculpture is 67.2 cm tall and 40 cm wide! If you need the answer in anything other than centimeters, just tell me!

Step-by-step explanation:

67.2 Length And 40cm Wide

A class of 40 students elected a class president. There were 12 votes for Candidate A, and 18 votes for Candidate B. The remaining students in the class did not vote. Which expression gives the percent of voters who voted for Candidate A?

Answers

Hey You!

12/40 = Candidate A.

18/40 = Candidate B.

12 + 18 = 30

40 - 30 = 10

10/40 = Those who didn't vote.

ANSWER:

12 ÷ 40 = .3 (30%) So 30% voted for Candidate A.

18 ÷ 40 = .45 (45%) So 45% voted for Candidate B.

10 ÷ 40 = .25 (25%) So 25% didn't vote at all.

Those Are Your Answers! ^^^

What is the expression in simplest form ?

(3x - 2) - (4x + 1)
a. x-3
b. -x - 3
c. -x + 1
d. x + 1

Answers

Answer:

  b.  -x - 3

Step-by-step explanation:

(3x -2) -(4x +1) . . . given

3x -2 -4x -1 . . . . . . distribute the minus sign; eliminate parentheses

(3x -4x) + (-2 -1) . . . group like terms

-x -3 . . . . . . . . . . . . simplify

Express each rate as a unit rate.
8 tomatoes for $2

Answers

For every 8 tomatoes you sell you receive $2

Each tomato is 0.25 cents a piece

What is the area under the normal curve between z = 0.0 and z = 1.79?

Answers

0.4633.

I remember some of the questions to this assignments.

Final answer:

The area under the normal curve between z = 0.0 and z = 1.79 is 0.4641.

Explanation:

The area under the normal curve between z = 0.0 and z = 1.79 can be found by subtracting the area to the left of z = 0.0 from the area to the left of z = 1.79.

The area to the left of z = 0.0 is 0.5000, and the area to the left of z = 1.79 is 0.9641 according to the Z-table.

Subtracting the two values, we get an area of 0.4641 between z = 0.0 and z = 1.79.

Learn more about normal distribution here:

https://brainly.com/question/30390016

#SPJ3

The product of 1.3×10^-4 and a number n results in 2.6×10^12 , what is the value of n?

Answers

n = 20,000,000,000,000,000 or 2.0×10^16

Answer:

2x10^16

Step-by-step explanation:

ik im late but in case anyone still needs it

also tell me if u want an explanation

in 3/4 pound of a spice mix there is 5/6 cup of cinnamon. how much cinnamon dose the spice mix contain per pound

Answers

[tex]\bf \begin{array}{ccll} \stackrel{lbs}{spice}&\stackrel{cups}{cinnamon}\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ \frac{3}{4}&\frac{5}{6}\\\\ 1&c \end{array}\implies \cfrac{\frac{3}{4}}{1}=\cfrac{\frac{5}{6}}{c}\implies \cfrac{\frac{3}{4}}{\frac{1}{1}}=\cfrac{\frac{5}{6}}{\frac{c}{1}}\implies \cfrac{3}{4}\cdot \cfrac{1}{1}=\cfrac{5}{6}\cdot \cfrac{1}{c} \\\\\\ [/tex]

[tex]\bf \cfrac{3}{4}=\cfrac{5}{6c}\implies 18c=20\implies c=\cfrac{20}{18}\implies c=\cfrac{10}{9}\implies c=1\frac{1}{9} [/tex]

The ratio of blue marbles to white marbles in a bag is 4 to 5 at this rate how many blue marbles are there if there are 15 white marbles

Answers

12 blue marbles for 15 white marbles because you multiply 5*3 so multiply 4*3 to get answer

What is 2,384 divided by 23 and what is 378 divided by 56 pls help me i will give the brainliest answer to someone who answers first

Answers

1st 103.652173913

2nd 6.75
he first one is 103.652173913 and the second one is 6.75

Nadine is constructing a line perpendicular to KE←→ . She has constructed the arcs as shown. What is the next step in her construction?

Use a straight edge to draw a line through D and M.

Use a straight edge to draw a line through M and K.

Use a straight edge to draw a line through M and E.

The image shows a horizontal line with points K, D, and E from left to right. Point D looks halfway between points K and E. An arc is shown at point K turning inwards towards point D. An arc is shown at point E turning inwards towards point D. Above point D is point M. Two arcs go through point M. One arc faces southwest towards point K and one points southeast towards point E. A line m and an arc. An arc is drawn on line m through center P. Center P lies above the line m. The arc cuts the line on two points labeled as X and Y. Two rays share a common endpoint at R. One ray is horizontal, pointing to the left. The second ray is vertical in an upward direction. A horizontal line segment M N with endpoints labeled as M and N. A horizontal line j and a point A are drawn. The point A is below the horizontal line j.

Answers

The correct answer is "Draw a straight line from D to M". In order to draw a perpendicular line to line KDE you need to create two acrs that intersect above or below KDE. This is illustrated by the point M. This is the point that will create a perpendicular line and intersect KDE at a 90 degree angle. 

A punch recipe says to mix 4 cups pineapple juice, 8 cups orange juice, and 12 cups seltzer in order to make 18 servings of punch. How many cups of each ingredient do you need to make 108 cups of punch? How many servings can be made from 108 cups of punch? For every cup of seltzer you use, how much orange juice do you use?

Answers

108/18 = 6

We know how many servings there are in 108 cups.

4 × 6 = 24
8 × 6 = 48
12 × 6 = 72

You need 24 cups of pineapple juice, 48 cups of orange juice, and 72 cups of seltzer to make 108 cups of punch.

I hope that helped! :)

You and two friends got jobs at stores in the mall. You earn
$117.60
$117.60
for
14
14
hours of work. Ty earns
$148.50
$148.50
in
18
18
hours. Kim earns
$137.60
$137.60
in
16
16
hours. Who has the highest pay rate in dollars per hour?



Answers

Divide each amount of money by its hours.

$117.60/14 = $8.40/hour

$148.50/18 = $8.25/hour

$137.60/16 = $8.60/hour

Kim earns the highest pay per hour.

What is 3.659090909090909 as a fraction?

Answers

There is not much that can be done to figure out how to write 3.659090909090909 as a fraction, except to literally use what the decimal portion of your number, the .659090909090909 , means.

Since there are 16 digits in 659090909090909 , the very last digit is the "10000000000000000th" decimal place.

So we can just say that .659090909090909 is the same as 659090909090909 /10000000000000000.

The fraction fmtterm: Can't handle [-2147483648/-2147483648] 3? is not reduced to lowest terms. We can reduce this fraction to lowest
terms by dividing both the numerator and denominator by -2.14748e+09.

Why divide by -2.14748e+09? -2.14748e+09 is the Greatest Common Divisor (GCD)
or Greatest Common Factor (GCF) of the numbers 6.59091e+14 and 1e+16.
So, this fraction reduced to lowest terms is

So your final answer is: 3.659090909090909 can be written as the fraction

The potential due to a point charge q at the origin may be written as v=q4πϵ0r=q4πϵ0x2+y2+z2√ part a calculate ex using equation ex=−∂v∂x.

Answers

The electric potential due to a point charge q at the origin is given by
[tex]v= \frac{q}{4 \pi \epsilon _{0} r} = \frac{q}{4 \pi \epsilon _{0} \sqrt{x^{2}+y^{2}+z^{2}} } [/tex]

The x-component of electric field is
[tex]e_{x} = - \frac{\partial v}{\partial x} = -\frac{q}{4 \pi \epsilon {0}}(- \frac{1}{2}) \frac{2x}{(x^{2}+y^{2}+z^{2})^{3/2}} = \frac{qx}{4 \pi \epsilon _{0} (x^{2}+y^{2}+z^{2})^{3/2}} [/tex]

Answer:
[tex]e_{x} = \frac{qx}{4 \pi \epsilon _{0} (x^{2}+y^{2}+z^{2})^{3/2}} [/tex]

Eₓ = [(qx) / [(4πϵ₀r)√(x² + y² + z²)³]

Further explanation

In this problem, we will solve the derivatives of the composite function. The formula used is the chain rule as follows.

[tex]\boxed{ \ If \ f(x) = h(g(x)) \ then \ f'(x) = h'(g(x)) \times g'(x) \ }[/tex]

In words: differentiate the outer function, then multiply it by the derivative of the innermost function.

Given:

[tex]\boxed{ \ V = \frac{q}{4 \pi \epsilon_0 r} \rightarrow V = \frac{q}{4 \pi \epsilon_0 r \sqrt{x^2 + y^2 + z^2}} \ }[/tex]

Question:

Calculate the x-component of electric field by using [tex]\boxed{ \ E_x = - \frac{\delta V}{\delta x} \ }[/tex]

The Process:

This case is a derivative application for electrostatics in physics.

The potential due to a point charge q at the origin may be written as

[tex]\boxed{ \ V = \frac{q}{4 \pi \epsilon_0 r \sqrt{x^2 + y^2 + z^2}} \rightarrow V = \frac{q}{4 \pi \epsilon_0 r}(x^2 + y^2 + z^2)^{-\frac{1}{2}} \ }[/tex]

The outside function is [tex] \boxed{ \ h(x) = \frac{q}{4 \pi \epsilon_0 r}(g(x))^{-\frac{1}{2}} \ } [/tex]The inside function is [tex]\boxed{ \ g(x) = x^2 + y^2 + z^2 \ }[/tex]

Let's determine the derivative and run the composite function rule.

The outside function: [tex] \boxed{ \ h'(x) = -\frac{1}{2} \frac{q}{4 \pi \epsilon_0 r}(g(x))^{-\frac{3}{2}} \ } [/tex]The inside function: [tex]\boxed{ \ g'(x) = 2x \ }[/tex]

Let us calculate the x-component of electric field by using

[tex]\boxed{ \ E_x = - \frac{\delta V}{\delta x} \ }[/tex]

The composite  function rule tells us that

[tex]\boxed{ \ V(x) = h(g(x)) \rightarrow V'(x) = h'(g(x)) \times g'(x) \ }[/tex]

Therefore, [tex]\boxed{ \ E_x = - V'(x) \rightarrow -[h'(g(x)) \times g'(x)] \ }[/tex]

[tex]\boxed{ \ E_x = -[-\frac{1}{2} \frac{q}{4 \pi \epsilon_0 r}(x^2 + y^2 + z^2)^{-\frac{3}{2}} \times 2x] \ }[/tex]

Thus, the result is [tex]\boxed{\boxed{ \ E_x = \frac{qx}{4 \pi \epsilon_0 r}(x^2 + y^2 + z^2)^{-\frac{3}{2}} \ }}[/tex]

[tex]\boxed{\boxed{ \ E_x = \frac{qx}{4 \pi \epsilon_0 r \sqrt{(x^2 + y^2 + z^2)^3} }} \ }}[/tex]

Learn moreUsing the product rule  https://brainly.com/question/1578252  The characteristics of electromagnetic waves https://brainly.com/question/727976 Determine the density of our sun at the end of its lifetime https://brainly.com/question/5189537

Keywords: calculus, differential, the composite function rule, the potential due to a point charge q, at the origin, physics, application, the chain rule

HELP PLZ Which math expression represents the phrase? 9 times the sum of a number and 7 A. 9 · n + 7 B. 9 · (n + 7) C. 9 · (n – 7) D. 9 · (n · 7)

Answers

equals what tho??????????????????
"The sum of a number and 7"
Sum implies addition.
We would write this part as n + 7.

Then we would write the "9 times" part in like this:
9 · (n + 7)
We can't write just 9 · n + 7 without the parentheses, because then we would be multiplying just 9 and n instead of 9 and the sum of n and 7. (order of operations says we must do multiplication first)

ok i have calculated this and i srill havent got the right answer

Answers

66 x 0.3 = 19.8
66-19.8 = 46.2
66 x 0.05 = 3.3
46.2 + 3.3 = 49.5

B
1. The original price of the shoes is 100% of the price of the shoes.
If you discount them by 30%, you will only pay 100% - 30% = 70% of their original price.

70% of $66 =

= 70% * $66

= 0.7 * $66

= $46.2

2. Now you have an item that costs $46.20, and you need to pay 5% tax on it.

$46.20 is 100% of the price of the shoes.
If you add 5% to the price, you need to pay 100% + 5% = 105% of the price.

105% of $46.20 =

= 105% * $46.20

= 1.05 * $46.20

= $48.51

Answer: $48.51
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