Mathematics, 30.09.2019 06:00 dhhdndnd

Find a point on the curve y= x^2 that is closest to the point (18, find a point on the curve y= x^2 that is closest to the point (18, 0). after taking the derivative of this one, what should i do? plug in different values into the derivative?

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Answer from: brockmerino

You're trying to minimize the distance between the point (18,0)

and an arbitrary point on the curve, (x,y)=(x,x^2)

.

The distance between two such points is given by the function

$d(x)=\sqrt{(x-18)^2+(x^2-0)^2}=\sqrt{x^4+x^2-36x+324}$

so this is the function whose derivative you should check.

But before you do that, it's helpful to know that d(x)

is minimized at the same point as the modified distance function d^*(x)=d^2(x)=x^4+x^2-36x+324

.

Differentiating, you have

$\frac{\mathrm d}{\mathrm dx}d^*(x)=4x^3+2x-36$

Set this to zero and solve for the critical points.

4x^3+2x-36=2(2x^3+x-18)=0

You can use the rational root theorem to find some potential candidates for roots to the cubic. The constant term has factors $\pm1,\pm2,\pm3,\pm6,\pm9,\pm18$ , while the leading coefficient has factors $\pm1,\pm2$ . The only candidates for rational roots are $\pm1,\pm\dfrac12,\pm2,\pm3,\pm\dfrac32,\pm6,\pm9,\pm\dfrac92,\pm18$ . The only one of these that works is

, so

is a root to the cubic above.

Polynomial division reveals that we can factor the cubic as

2(x-2)(2x^2+4x+9)=0

which has only one real root at x=2

. Checking the value of the derivative of d^*

to the left and right of this point confirms that a minimum occurs here.

Therefore the closest point on the curve to (18,0)

Answer from: Quest

All you have to do is multiply

Answer from: Quest

your exact question is kind of difficult to understand, so maybe an example will explain what you're supposed to do? i'll work through the first problem on the worksheet.

suppose you have a tray of 652 brownies, and you have 5 friends that you want to feed without picking favorites. how many brownies can you dole out to each of your friends in equal amounts? how many will be left over for you to eat?

what you're doing here is finding a way to arrange the number 652 as the sum of integer multiple of 5 and some remainder: $652=5q+r$

here $q$ ("q" for quotient) is the number of brownies each of your friends get, and $r$ is the remainder. that's the basic idea behind division.

we break up the number 652 into parts so that the leading digit stands on its own:

652 = 600 + 52

first we check if it's possible to hand out exactly 100 brownies to each friend. with 5 friends, this means we want to know if we can take out 500 brownies from the tray. we can do this 1 time because we have 600 brownies to take from, and when we do that we're left with 100 + 52 brownies to divvy up. so we know that

652 = 1*500 + 152

next we want to know if it's possible to hand out exactly 10 brownies per friend. if we can, we have to remove 50 brownies from the tray each time until we can't. we have 152 brownies left, and we know that 3(50) = 150, so

152 = 3*50 + 2

now we have 2 brownies left in the tray. is it possible to hand 1 out to each of your 5 friends? no, you'd need a minimum of 5 brownies to do that. so the last 2 are yours.

putting everything together, we found that

652 = 1*500 + 3*50 + 2

and we can rearrange this to write

652 = 5*100 + 5*30 + 2

652 = 5(100 + 30) + 2

652 = 5*130 + 2

so $q=130$ and $r=2$ .

you might be more accustomed to the table depicting the algorithm. that's just a compact way of organizing what we've done above. you can consult the image i've attached below.

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