We're asked to find the curvature
and radius of curvature of Y equals E
raised to the power of two X at X equals negative one.
The function is graphed in blue,
and this red point is the point of the function
when X equals negative one.
Again, the goal is to find the curvature at this point
and the radius of curvature.
The curvature at a point measures how sharply
the curve bends or how quickly it changes direction.
The curvature of a line is zero.
The curvature of a circle is equal
to the reciprocal of the radius.
When we find the curvature of a space curve
given by a vector function,
we use this formula here,
but because in our case we have a function of X,
which gives us a plane curve,
we can simplify this formula.
If Y equals F of X,
then the curve can be parameterized
using the vector function R of T
where the S component is T,
the Y component is F of T,
and the Z component is zero,
and X prime of T is equal to one.
X double prime of T is equal to zero.
So then if we apply this formula
using this vector function,
the curvature formula simplifies
to the absolute value of Y double prime of X
divided by the quantity
one plus the square of Y prime of X
raised to the three halves power.
Before we apply this though,
let's look at our animation.
The graph of our function is graphed in black.
As we animate a point along the curve,
we will see the circle of curvature.
So as we approach this bend in the curve,
the circle of curvature will get smaller.
Notice as we approach the curve,
we are beginning to see the blue circle,
the circle of curvature,
and now we can see the entire circle of curvature.
As we pass the bend in the curve,
the circle of curvature begins to enlarge again,
and our goal is to find the curvature
and radius of curvature at X equals negative one.
Let's move to that point.
Here's the point in our function
where X equals negative one.
Again, our goal is to find the curvature at this point
and the radius of curvature,
which are reciprocals of one another,
so going back to our problem,
let's find the curvature using this formula here
when we have a function of X.
So we're given that Y is equal to E
raised to the power of two X.
We need to find Y prime and Y double prime.
To find Y prime,
we must apply the chain rule.
The derivative of E to the two X
with respect to X is E to the two X times two,
or two, E to the two X,
and Y double prime of X is equal to two
times E to the two X times two,
or four, E to the two X.
Now we can find the curvature as a function of X,
which was equal to the absolute value
of Y double prime of X,
which is the absolute value of four, E to the two X,
divided by the quantity
one plus the square of Y prime of X,
which is the square of two, E to the two X.
Now this is raised to the three halves power.
Well, four, E to the two X is always positive,
so we can drop the absolute value in the numerator.
We have K of X equals
four, E to the two X
divided by one plus,
and we need to be careful when squaring here.
We square the two, which is four,
and then when we square E to the two X,
we multiply the exponents.
Two X times two is four X,
giving us four times E to the four X,
or the square of two, E to the two X.
This is still raised to the three halves power.
Now that we have the formula for the curvature,
we can determine the curvature at X equals negative one.
Let's do this on the next slide.
The curvature at X equals negative one
is equal to four times E
raised to the power of two times negative one,
divided by the quantity one plus four
times E raised to the power of four times negative one,
raised to the three halves power,
which is equal to four times E
raised to the power of negative two,
divided by the quantity one plus four
times E raised to the power of negative four
raised to the three halves power.
So this is the exact value of the curvature
at X equals negative one.
Let's also get a decimal approximation.
Open parenthesis, four,
second natural log brings up
E raised to the power of negative two,
right arrow, closed parenthesis,
divided by, open parenthesis,
one plus four.
Second natural log brings up
E raised to the power of negative four,
right arrow, closed parenthesis,
raised to the power of three divided by two,
and enter.
Curvature is approximately 0.48687.
So this is the first part of the question.
Now we need to find the radius of curvature,
and the relationship is if the curvature
is equal to the reciprocal of the radius of curvature.
So the curvature is equal
to one over the radius of curvature,
or the radius is equal to one over the curvature.
So the radius of curvature at X equals negative one
is equal to the reciprocal of this fraction here,
which is the quantity one plus four
times E raised to the power of negative four
raised to the three halves
divided by four times E raised to the power of negative two.
Let's also get a decimal approximation.
On the calculator, what we can do is just press the X
raised to the power of negative one key.
This will take the reciprocal of the previous answer,
and now if we press enter,
we can see it's approximately 2.05394.
So we have the exact and approximate values
for the curvature K and the radius of curvature R.
Before we go, let's go back to our graph.
We now know the radius of the circle of curvature
at X equals negative one
is approximately 2.05,
which means the curvature is approximately 0.49.
I hope you found this helpful.