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Use Theorem 10 to find the curvature.

$$\mathbf{r}(t)=t \mathbf{i}+t^{2} \mathbf{j}+e^{\prime} \mathbf{k}$$

$$

\kappa(t)=\frac{\sqrt{4 e^{2 t}(t-1)^{2}+e^{2 t}+4}}{\left(1+4 t^{2}+e^{t}\right)^{3 / 2}}

$$

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you want to use and defend the curvature of the curve Represented by the vector are tee. You should be written as t turns I plus t squared times J plus need of the power of t times. Okay, so, visitor each of the three component vectors and we can read that in our angle bracket notation as t t squared and eat of the power of T. So to be able to apply here in 10 we need to take the first and second derivative of our So First, let's look it Our prime of tea, which is going to be one to t and eat of the power of T is our three component functions. And let's take the second derivative. So are a double prime of tea which is going to be zero to and u to the power of teeth. So let's go ahead. And from the cross product of those two were getting closer to the numerator of our, uh, curvature function from theorem 10. So looking at our prime crossed with our double prime, which we can write as the determinant of a three by three matrix sooner. Second row, we're gonna have one to t and E to the power of T third rule, We're gonna have zero to you, The power of T. In that topper, we have the component vectors I, j and K. So be sure to review how to take the determinant of a three by three matrix as necessary. But we're gonna do it here. So we're gonna look at the items in the columns for J and K. So we're gonna have two t two neither the power of T and E to the power of t that determine it. Times component vector I We're going to subtract latitude by two determinant times vector J So we're gonna look in r I N K columns will have 10 need of the power of T and even power teeth and then we're going to add a two by two determinant times Vector k. We're going to look in columns I and J. So we'll have 10 to t and to. And then what if we go through and compute that will have to tee times e to the power of T minus two times eat of power of tee times Vector I minus e to the power of T minus zero. Inspector J plus tu minus zero times Vector K And if we want, we can write that in our angle bracket notation I'm gonna factor two times ive the power of t out from the first term which leaves us with two times e to the power of t times t minus one. We'll have negative each of the power of T for a second component function and to for our third component functions. And then the last stuff that we need to find the numerator of the curvature function is behind ing the norm of that cross product. So norm of our prime crossed with our double prime and that's going to give us the square of each of our component functions squared. So if I square that first component function, we're gonna have four times e to the power of to t times T minus one squared by square. The second component function will have positive you the power to t. And if I square that third component function, we're just gonna get plus for so there is the numerator for curvature function. Now let's go ahead and find the denominator to do that we need to find the norm of the first derivative of our and that is going to be the square root of each of our component functions for our prime square. And so that's written up here at the top. So we'll have one squared, which is just one plus two t quantity squared were 40 square plus eat of the power of two teeth. And then if we want to go ahead and write that in our curvature function from here, um, 10. What we're gonna have is in the numerator the square root of or of four times E to the power of to t that a little bit more clearly there times t minus one squared plus e to the power of to t plus for and then in the denominator, we're going to have the square root of one plus four t squared plus e to the power of to tea. But instead of writing that as the square root cubed, I'm goingto go ahead and bring it to the power of three halves just cause I think it looks a little bit cleaner. And that is the curvature function for the vector represented by or for the curve for presented by the vector are