Question

The curvature κ of a function h(x) at a point is the reciprocal of the radius of the circle that best approximates the shape of the function at that point. It is given by the following formula: κ=∣h′′(x)∣​/(1+[h′(x)]2)³/²Find the curvature, in exact and approximate form, of the following functions at the given points. (Remember that you can use Rational (3,2) to get the exact value of 2/3​. Just dividing 3 by 2 gives a floating point.) (a) h(x)=x²+3x+5 at x=2 (b) h(x)=tan(x) at x=π​/3 (c) h(x)=7x−1 at x=5 (d) h(x)=√25−x² at x=1 (e) In a print statement, give a geometric relationship between the answers to (c) and (d) and characteristics of those curves.

Answer 1

The curvature of the function h(x) is found using its first and second derivatives. For h(x)=x^2+3x+5 at x=2, the exact curvature is 2/√(50³).

The curvature κ of a function h(x) at a point is defined as κ=|h''(x)|/(1+[h'(x)]²)^{3/2}. Determining the curvature involves calculating the first and second derivatives of the function. Let's consider the function h(x)=x^2+3x+5.

To find the curvature at x=2, we calculate:

1. The first derivative h'(x)=2x+3.
2. Then evaluate h'(2)=2(2)+3=7.
3. The second derivative h''(x)=2.
4. Using the formula, κ=|2|/(1+7^2)^{3/2}=κ=2/√(1+49)³=2/√(50³).

The result gives the curvature in exact form. To find the approximate form, perform the arithmetic under the radical and compute accordingly.