Finding the distance from a point on a curve to another

[blackroserei]

Hi, sorry to ask for help again but there are two questions which are very similar that I cannot figure out. The first is

"The point B (a, b) is on the curve f(x)=x^2 such that B is the point which is closest to A (0,6). Calculate the value of a."

I had no idea of where to start so I used the distance formula and plugged in some of the expressions and got d= ? (a-6)^2 + (b-0)^2. I distributed and then got, d= ? a^2-12a + 36 + b^2.

Then I got stuck there and I'm not sure if what I've done up to this point is even correct.

The second question is

"A point P(x, x^2) lies on the curve y=x^2. Calculate the minimum distance from the point A[2, -.5) to the point P."

Again, I used the distance formula ?(x-2)^2 + (x^2 - .5). I distributed everything and got d= ?x^2 - 4x + 4 +x^4 -x^2 + .25. Like the last problem I got stuck here and again, I'm not sure if what I've done so far is correct.

 

[mmm4444bot]

… The point B (a, b) is on the curve f(x)=x^2 such that B is the point which is closest to A (0,6). Calculate the value of a …

… I used the distance formula … and got d= ? [(a-6)^2 + (b-0)^2] …

You've made two errors.

1) You've got b in your equation; I don't think that they want you to solve for a in terms of b. I think that they want the numerical value of a that minimizes the distance between points A and B.

We know that f(x) = x^2.

We know that b = f(a).

So write b in terms of a, first. Use this expression instead of b in the distance formula.

2) You transposed the x and y values. The distance formula uses the difference between the two x values and also the difference between the two y values.

You took the difference between x and y.

After you fix these mistakes, you'll have an expression for the distance between point A and point B for any value of a.

Now it's time to take the derivative, in order to find the value of a that minimizes this distance.

Please show your work up to this stage, and we'll go from there.

 

[mmm4444bot]

… I used the distance formula ?[(x-2)^2 + (x^2 - .5)]. I distributed everything and got d= ?[x^2 - 4x + 4 +x^4 -x^2 + .25] …

You set up the equation properly, and your initial result in simplifying the radicand is good.

Do you know?

x^2 - x^2 is zero.

4 + 0.25 is 4.25.

If you continue by making these additional simplifications, then you'll find that taking the derivative is less work.

 

[blackroserei]

For the first problem, I did actually plug in x^2, sorry I thought I had typed it.

when I did that, I got, d= ?[ x^2 - 12x + 36 + x^4]. I'm not sure if x^4 is correct because your plugging in (x^2) ^2. Is this the correct process?

Then I found the derivative and got, ?4x^3 + 2x - 12

For the second question, I'm not sure how I transposed the x and y values. For the first point, the coordinates are (2, -.5) and the second is (x, x^2). The distance formula is ?(x(2) - x(1))^2 + (y(2)- y(1))^2 (sorry for the 2 and the 1 in parentheses, I don't have a function on my mac to do subscript). I was positive that I had substituted in everything correctly.

I then simplified everything and got ?[x^4 -4x + 4.25]. Then I found the derivative, and ended up with ?[4x^3 -4]. From there, I set the equation to zero 0= ?[4x^3 - 4] and squared both sides to get rid of the radical. I added 4 to both sides, divided each side by 1 and then took the 3? of 1 and got 1. Again, not sure if I did this correctly.

 

[mmm4444bot]

For the first problem, I did actually plug in x^2 …

This seems problematic to me because now you've eliminated the symbol a from your distance function. Do you have a strategy for finding the value of a in some other way? (I'm open to alternate approaches, but I need to know what you're thinking, if you want my help with another method.)

For the method that I have in mind, the suggestion that I provided to you in my first response deals with the symbol b, not the symbol a.

You initially posted a distance formula for the first exercise that contains both the symbols a and b. If you want to find the numerical value of a, then you need an equation that contains only one symbol -- the symbol a.

In other words, you want to define the distance function (I'll name it d) where a is its independent variable, and the definition of function d contains no other symbols.

d(a) = sqrt(some expression that contains only the symbol a)

We can replace the symbol b in your work with a new expression that contains the symbol a, instead, because we know that the y-coordinate of point B is f(a).

Tell me. What is f(a)?

(Use the given definition for function f to answer this question.)

If you think you've got it, then substitute it for b.

Also, don't forget to fix your transposition error in the distance formula.

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For the second question, I'm not sure how I transposed the x and y values.

You did not transpose anything in your work for the second exercise. You transposed x and y values in your distance function in the first exercise.

My first response in this thread pertains only to your first exercise. This is clear because I quoted your typing.

My second response in this thread pertains only to your second exercise. This is clear because I quoted your typing.

(Many people who post on this site get confused after batching multiple exercises within a single discussion. If you are one of these people, then start a New Topic for each individual exercise. I actually prefer that everyone do this, anyway, lest I get confused, too.)

-

… I then simplified everything and got ?[x^4 - 4x + 4.25]. Then I found the derivative, and ended up with ?[4x^3 -4] …

That's a good simplification.

Regarding your derivative result: whoops! It looks to me like you forgot about the Chain Rule.

d(x) = ?[x^4 - 4x + 4.25]

This is a composite function. The inner function is the polynomial, and the outer function is the square root function.

We must use the Chain Rule when calculating the derivative of a composite function.

 

[PAULK]

Hi, sorry to ask for help again but there are two questions which are very similar that I cannot figure out. The first is

"The point B (a, b) is on the curve f(x)=x^2 such that B is the point which is closest to A (0,6). Calculate the value of a."

I had no idea of where to start so I used the distance formula and plugged in some of the expressions and got d= ? (a-6)^2 + (b-0)^2. I distributed and then got, d= ? a^2-12a + 36 + b^2.

Then I got stuck there and I'm not sure if what I've done up to this point is even correct.

What is going on here with all this 'I'm stuck on the algebra'? The algebra is nothing, unless you choose to make it a mess by doing things like using the distance formula in the form

D = sqrt(some stuff)

Instead, write:

D^2 = (x - x0)^2 + (y - y0)^2

and try to minimize THAT instead. (You really love working with square roots, right?)

(An old trick used by lazy mathematicians -- is there any other kind?)


Now if:

"The point B (a, b) is on the curve f(x)=x^2 such that B is the point which is closest to A (0,6). Calculate the value of a."

Let B = (a,a^2) << it's on y = x^2, right?

DD = a^2 + (6 - a^2)^2 << distance formula squared.

DD = a^2 + 36 - 12a^2 + a^4 << remove paren.

DD = 36 - 11a^2 + a^4 << simplify

DD' = -22a + 4a^3 << differentiate

-22a + 4a^3 = 0

2a(2a^2 - 11) = 0

a = 0 << one root.

a = +- sqrt(11/2) < other root(s)

DD'' = -22 + 12a^2 < second derivative test.

DD''(0) = -22, so this is a max.

DD''(+- sqrt(11/2)) = -22 + 12(11/2) = -22 + 66, so this is a min.