find a curve that passes through the point (1,-2 ) and has an arc length on the interval 2 6 given by 1 144 x^-6

Answer:

[tex]f(x) = \frac{6}{x^2} -8[/tex] or [tex]f(x) = -\frac{6}{x^2} + 4[/tex]

Step-by-step explanation:

Given

[tex](x,y) = (1,-2)[/tex] --- Point

[tex]\int\limits^6_2 {(1 + 144x^{-6})} \, dx[/tex]

The arc length of a function on interval [a,b]:  [tex]\int\limits^b_a {(1 + f'(x^2))} \, dx[/tex]

By comparison:

[tex]f'(x)^2 = 144x^{-6}[/tex]

[tex]f'(x)^2 = \frac{144}{x^6}[/tex]

Take square root of both sides

[tex]f'(x) =\± \sqrt{\frac{144}{x^6}}[/tex]

[tex]f'(x) = \±\frac{12}{x^3}[/tex]

Split:

[tex]f'(x) = \frac{12}{x^3}[/tex] or [tex]f'(x) = -\frac{12}{x^3}[/tex]

To solve fo f(x), we make use of:

[tex]f(x) = \int {f'(x) } \, dx[/tex]

For: [tex]f'(x) = \frac{12}{x^3}[/tex]

[tex]f(x) = \int {\frac{12}{x^3} } \, dx[/tex]

Integrate:

[tex]f(x) = \frac{12}{2x^2} + c[/tex]

[tex]f(x) = \frac{6}{x^2} + c[/tex]

We understand that it passes through [tex](x,y) = (1,-2)[/tex].

So, we have:

[tex]-2 = \frac{6}{1^2} + c[/tex]

[tex]-2 = \frac{6}{1} + c[/tex]

[tex]-2 = 6 + c[/tex]

Make c the subject

[tex]c = -2-6[/tex]

[tex]c = -8[/tex]

[tex]f(x) = \frac{6}{x^2} + c[/tex] becomes

[tex]f(x) = \frac{6}{x^2} -8[/tex]

For: [tex]f'(x) = -\frac{12}{x^3}[/tex]

[tex]f(x) = \int {-\frac{12}{x^3} } \, dx[/tex]

Integrate:

[tex]f(x) = -\frac{12}{2x^2} + c[/tex]

[tex]f(x) = -\frac{6}{x^2} + c[/tex]

We understand that it passes through [tex](x,y) = (1,-2)[/tex].

So, we have:

[tex]-2 = -\frac{6}{1^2} + c[/tex]

[tex]-2 = -\frac{6}{1} + c[/tex]

[tex]-2 = -6 + c[/tex]

Make c the subject

[tex]c = -2+6[/tex]

[tex]c = 4[/tex]

[tex]f(x) = -\frac{6}{x^2} + c[/tex] becomes

[tex]f(x) = -\frac{6}{x^2} + 4[/tex]

You can use the formula for finding the arc length on specified interval on x axis.
The curves whose arc length on the given interval is described are

[tex]f(x) = 6x^{-2} -8[/tex]

and

[tex]f(x) = -6x^{-2} + 4[/tex]

If the function is differentiable in the given interval, then we have:

[tex]s = \int_a^b\sqrt{(1 + (f'(x))^2)}\:dx[/tex]

where s denotes the length of the arc of the given function from x = a to x = b

Using the above formula, as we're already given the arc length, thus,

[tex]\int_a^b\sqrt{(1 + (f'(x))^2)}\:dx = \int_2^6\sqrt{(1 +144x^{-6})}\:dx[/tex]

This gives us

[tex]f'(x) = \pm \sqrt{144x^{-6}} = \pm 12x^{-3}[/tex]

Integrating both sides with respect to x, we get:

[tex]f(x) = \pm \int 12x^{-3}\\\\f(x) = \pm 6x^{-2} + c[/tex]

where c is constant of integration.

Since the curve passes through (1,-2), thus, putting f(x) = -2, x = 1, we get:

[tex]f(x) = \pm 6x^{-2} + c\\-2 = \pm 6 + c\\c = -2 \mp 6 = -8, or ,4[/tex]

c = -8 when we have [tex]f(x) = 6x^{-2} + c[/tex]

c = + 4 when we have [tex]f(x) = -6x^{-2} + c[/tex]

The curves whose arc length on the given interval is described are

[tex]f(x) = 6x^{-2} -8[/tex]

and

[tex]f(x) = -6x^{-2} + 4[/tex]

Learn more about length of the arc of a curve here:

https://brainly.com/question/14319881

Please help me, thank you!​

Which triangle most likely has the angles 56°, 48°, and 76°?
A. triangle A
B. triangle B
C. triangle C
D. triangle D

Use the figure 25m circle find the circumference. Round your answer to the nearest hundredth use 3.14 for *pi sign*

Yo someone help i'm lost

constant proportanalities in the equation y=rx 15,5 25,8 1/3 33,11

This graph shows the value of Jessica's vehicle at different years after she purchased it.

Which answer is the best estimate for the average rate of change between year 0 and year 3

A. -$6300/year
B. -16000/year
C. -$5300/years
D. -$4000/year​

Lauren has a purse full of dimes and quarters. She has 25 less dimes than quarters. If she has a total of $10.45, which system of equations would she write to find the number of dimes, d and quarters, q?​

Convert the mixed number into a improper fraction 5 1/3

i need help on this im strugling​

7. Fred had 85 marbles. After a few games, he had 120% of his original number. How many
marbles does he have now?

HELp pls il give brainliest

Andy works at a grocery store. The manager of the store would like Andy to set up a display of apples. Part of the display will include bags of apples. If there are 39 pounds of apples in the back of the store, how many bags of apples can Andy make for the display?

Is 18.4 rational or irrational ?

don’t know how to label the sides as “opposite, adjacent, hypotenuse” im kinda understand it but need bit support

Dennis is getting balloons for his grandmother's birthday party. He wants each balloon string to be 9 feet long. At the party store, string is sold by the yard. If Dennis wants to get 56 balloons, how many yards of string will he need?
1 foot = 1/3 yard

help if you can please :)

What is 2 1/3 divided by 1/2?

Also, I have the answer but is it 7/6 or 4 2/3?

PLEASE HELP I WILL GIVE BRAINLEST!!!

a bag contains 3 blue and 5 red marbles find the probability of drawing 2 blue marbles in a row without replacing the first marble write as a fraction

Can some please explain how to solve this? Thank you. 1/2(3/2y+1/3)=-1/2(y-5/2)

Hi can you help me pls I would very much appreciate it!!!

B2. Write any four alchemies of nature mentioned in the passage:
(iv)​

14 apartments on 2 floors =
apartments on 1 fioor

pls answer, show work A jacket at a clothing store is being discounted by 10%. If the jacket was originally priced at x dollars, which of the following expressions does NOT represent the final price of the jacket before sales tax is added?

A.
x – 0.10x

B.
x(1 – 0.10)

C.
x – 1.10x

D.
0.90x

What are the domain and the range of this function? f(x) = log(x + 3) - 2

A. domain: (-3, ∞); range: (∞, 2)
B. domain: (3, ∞); range: (∞, 2)
C. domain: (3, ∞); range: (∞, ∞)
D. domain: (-3, ∞); range: (∞, ∞)

Please help me with this question!!

If n<5, then n<3 true or false pls help me?!?

2+2
3x5
44+5
pels help meeeeeeee

What is the value of x in the figure? Enter your answer in the box. 145 degrees and (2x + 15) degrees

You are estimating the cost of optical sensors based on the resolution of the sensor. You decide to calculate the coefficient of determination (R2) as part of determining the goodness of fit of an equation. Using the preliminary calculations below, calculate the R2 and determine its meaning.
Summation of (Yi - Ybar)2 = 147172
Summation of (Yhat - Ybar)2 = 130542
Summation of (Y - Yhat)2 = 16630
a. 88.70% of the variation in the cost is being explained by the resolution.
b. 88.70% of the variation in the resolution is being explained by the cost.
c. 11.30% of the variation in the resolution is being explained by the cost.
d. 11.30% of the variation in the cost is being explained by the resolution.