Uploaded by TheIntegralCALC on 13.04.2011

Transcript:

Linear Approximation in One Variable Example 3

Hi, everyone.

Welcome back to integralcalc.com.

Today we're going to be doing another linear approximation problem.

And in this one, we're given the function f of x equals sine of x

and we've been asked to evaluate the linear approximation around the point x equals zero.

So we're going to be using a linear approximation formula here, L of x equals f of a plus f prime of a times x minus a.

Keep in mind that when we're given you know,

Keep in mind that when we're given you know,

supplemental equations like this to our original function x equals zero,

x equals zero, that's also going to be equal to a in regards to the formula that we have here.

You see f of a, f prime of a, x minus a.

You see f of a, f prime of a, x minus a.

This x equals zero is also a equals zero.

So with linear approximation...

I'm just writing the linear approximation formula, we're going to be looking at finding three pieces of information.

We need to plug in our value for a to our original function f of x to get f of a and we're going to plug in our answer to...

for this portion of our formula.

Then we're going to take the derivative of our original equation so we get f prime of x.

We'll plug a into that function and we'll plug in our answer for this component of our formula.

Then we're going to plug in the original value we've given for a which in our case is zero,

for this a right here and we're going to leave x completely alone.

That’s going to remain in our formula.

So again, three pieces of information, once we get them, we can plug in to our formula.

We’ll simplify our linear approximation equation as much as we can and that will be our final answer.

So the first thing we'll do, let's plug in a equals zero to our original function.

So we'll be plugging in zero for x.

So our original equation f of x, let's plug in zero, we'll get sine of zero.

And you can either evaluate sine of zero on your calculator, you can do so using the unit circle but either way, sine of zero is going to be zero.

So that is now equal to f of a. f of a.

Now, so that's going to go in right here. Check.

Now we need f prime of a.

So in order to do that we'll need to first find the derivative of f of x.

So the derivative of f of x is going to be f prime of x

and the derivative of sine of x is cosine of x.

That is a formula that you should definitely memorize.

But most calculators these days are going to be able to find them for you, in case you don't know it.

But the derivative of sine of x is cosine x.

That's our derivative equation.

Now we need to plug in a equals zero.

So f prime of zero equals cosine of zero and cosine of zero again,

you can either use your calculator or the unit circle but cosine of zero is one.

So that is going to be f prime of a and that's going to go in right here. So check.

And then our third piece of information, finding a, we've already been provided with that information. It's zero.

So we can check that one off too.

So now that we've got all three pieces of information,

we can go ahead and plug these in to our linear approximation formula.

So our linear approximation formula will look like this.

L of x equals f of a, we've got zero.

So zero plus f prime of a, we got one so we'll plug that in.

times... remember we said we're going to leave x alone,

That remains in the equation and then we subtract from that a,

which we were provided in our original problem.

So x minus zero.

And now we're just going to simplify this as much as possible to get our final answer.

So L of x equals...

L of x equals one times x minus zero is just x so we just end up with x.

So our final answer for the linear approximation equation about the point x equals zero is L of x equals x.

And that's it.

That’s our final answer.

I hope that video helped you guys

and I'll see you in the next one.

Bye!

Hi, everyone.

Welcome back to integralcalc.com.

Today we're going to be doing another linear approximation problem.

And in this one, we're given the function f of x equals sine of x

and we've been asked to evaluate the linear approximation around the point x equals zero.

So we're going to be using a linear approximation formula here, L of x equals f of a plus f prime of a times x minus a.

Keep in mind that when we're given you know,

Keep in mind that when we're given you know,

supplemental equations like this to our original function x equals zero,

x equals zero, that's also going to be equal to a in regards to the formula that we have here.

You see f of a, f prime of a, x minus a.

You see f of a, f prime of a, x minus a.

This x equals zero is also a equals zero.

So with linear approximation...

I'm just writing the linear approximation formula, we're going to be looking at finding three pieces of information.

We need to plug in our value for a to our original function f of x to get f of a and we're going to plug in our answer to...

for this portion of our formula.

Then we're going to take the derivative of our original equation so we get f prime of x.

We'll plug a into that function and we'll plug in our answer for this component of our formula.

Then we're going to plug in the original value we've given for a which in our case is zero,

for this a right here and we're going to leave x completely alone.

That’s going to remain in our formula.

So again, three pieces of information, once we get them, we can plug in to our formula.

We’ll simplify our linear approximation equation as much as we can and that will be our final answer.

So the first thing we'll do, let's plug in a equals zero to our original function.

So we'll be plugging in zero for x.

So our original equation f of x, let's plug in zero, we'll get sine of zero.

And you can either evaluate sine of zero on your calculator, you can do so using the unit circle but either way, sine of zero is going to be zero.

So that is now equal to f of a. f of a.

Now, so that's going to go in right here. Check.

Now we need f prime of a.

So in order to do that we'll need to first find the derivative of f of x.

So the derivative of f of x is going to be f prime of x

and the derivative of sine of x is cosine of x.

That is a formula that you should definitely memorize.

But most calculators these days are going to be able to find them for you, in case you don't know it.

But the derivative of sine of x is cosine x.

That's our derivative equation.

Now we need to plug in a equals zero.

So f prime of zero equals cosine of zero and cosine of zero again,

you can either use your calculator or the unit circle but cosine of zero is one.

So that is going to be f prime of a and that's going to go in right here. So check.

And then our third piece of information, finding a, we've already been provided with that information. It's zero.

So we can check that one off too.

So now that we've got all three pieces of information,

we can go ahead and plug these in to our linear approximation formula.

So our linear approximation formula will look like this.

L of x equals f of a, we've got zero.

So zero plus f prime of a, we got one so we'll plug that in.

times... remember we said we're going to leave x alone,

That remains in the equation and then we subtract from that a,

which we were provided in our original problem.

So x minus zero.

And now we're just going to simplify this as much as possible to get our final answer.

So L of x equals...

L of x equals one times x minus zero is just x so we just end up with x.

So our final answer for the linear approximation equation about the point x equals zero is L of x equals x.

And that's it.

That’s our final answer.

I hope that video helped you guys

and I'll see you in the next one.

Bye!