The formula κ(x)=(|f^''(x)|)/([1+(f^'(x))^2]^3 / 2) derived in Exercise 5, expresses the curvatu

The formula κ(x)=(|f^''(x)|)/([1+(f^'(x))^2]^3 / 2) derived...

The formula $$ \kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}} $$ derived in Exercise $5,$ expresses the curvature $\kappa(x)$ of a twice-differentiable plane curve $y=f(x)$ as a function of $x$ . Find the curvature function of each of the curves. Then graph $f(x)$ together with $\kappa(x)$ over the given interval. You will find some surprises. $$ y=e^{x},-1 \leq x \leq 2 $$

The formula
$$
\kappa(x)=\frac{\left|f^{\prime \prime}(x)\right|}{\left[1+\left(f^{\prime}(x)\right)^{2}\right]^{3 / 2}}
$$
derived in Exercise $5,$ expresses the curvature $\kappa(x)$ of a twice-differentiable plane curve $y=f(x)$ as a function of $x$ . Find the curvature function of each of the curves. Then graph $f(x)$ together with $\kappa(x)$ over the given interval. You will find some surprises.
$$
y=e^{x},-1 \leq x \leq 2
$$

Key Concepts

Curvature of a Plane Curve

Curvature is a measure of how sharply a curve bends at a given point on the curve. In the context of differential geometry, the curvature of a plane curve defined by y = f(x) is calculated using both the first and second derivatives of the function. The formula ?(x) = |f''(x)| / [1 + (f'(x))^2]^(3/2) arises from considering the rate of change of the tangent vector with respect to the arc-length, thus providing an intrinsic measure of the curve’s bending independent of its parametrization.

Differentiation and Derivative Computation

Differentiation is the process of finding the derivative of a function, which measures how the function value changes as its input changes. In this context, computing the first derivative f'(x) gives the slope of the tangent to the curve, while the second derivative f''(x) indicates the rate of change of this slope—essential in determining the curvature of the curve. Mastery of differentiation techniques is critical in applying the curvature formula and understanding the behavior of continuously differentiable functions.

Graphical Analysis of Functions

Graphical analysis involves visualizing functions and their properties to better understand their behavior over a given interval. In problems that involve both a function and its curvature, plotting the original function alongside its curvature function can reveal insights such as changes in bending and the relative rates at which the curve twists. This dual visualization helps in interpreting theoretical results, validating analytical findings, and uncovering any unexpected behaviors in the curvature compared to the original function.

Transcript

00:01 We are asked to consider, oops see, we're asked to consider the formula i've written right here.

00:07 And this was derived in exercise five.

00:11 It expresses the curvature of kappa x of a twice differentiable plain curve.

00:20 Or y equals f of x as a function of x.

00:23 We're asked to find the curvature function of each of the curves.

00:27 And then we're going to graph each of them together over the given.

00:30 Interval and here is what we're graphing.

00:36 So again, let's get started on this.

00:41 What we're graphing on this one is e of x and then we'll differentiate the first, differentiate the first time, and we get e of x for everything.

00:59 So then here's our equation again and substituting our values, we're going to get e of x over 1 plus e to the x.

01:14 Squared 3 over 2 and then e over x over 1 plus e to the 2 x 3 over 2.

01:30 And here is my equation i'm looking for.

01:35 Next let's do our curves and i've already drawn my graph here.

01:39 1, 2, 3, 4, 5, 1, 2, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5, 5, 3, 4.

01:52 5, 1, 2, 3, 4, 5.

01:57 So this will be 0, then i have 0 .5, 1 .0, 1 .0, 2 .0, and 2 .5.

02:11 Okay, so now let's go pick a couple colors with which to do this.

02:16 Let's make our k, our cap of x, blue...