Question

3. The curvature ? 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: \kappa = \frac{|h''(x)|}{(1 + [h'(x)]^2)^{3/2}} (1) Find 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 $\frac{3}{2}$. Just dividing 3 by 2 gives a floating point.) (a) $h(x) = x^2 + 3x + 7$ at $x = 3$ (b) $h(x) = \sin(x)$ at $x = \frac{\pi}{3}$ (c) $h(x) = 2x + 4$ at $x = 1$ (d) $h(x) = \sqrt{36 - x^2}$ at $x = 2$ (e) Give a geometric explanation for why your answers to (c) and (d) make sense.

          3. The curvature ? 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:
\kappa = \frac{|h''(x)|}{(1 + [h'(x)]^2)^{3/2}}
(1)
Find 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 $\frac{3}{2}$. Just dividing 3 by 2 gives a floating point.)
(a) $h(x) = x^2 + 3x + 7$ at $x = 3$
(b) $h(x) = \sin(x)$ at $x = \frac{\pi}{3}$
(c) $h(x) = 2x + 4$ at $x = 1$
(d) $h(x) = \sqrt{36 - x^2}$ at $x = 2$
(e) Give a geometric explanation for why your answers to (c) and (d) make sense.

3. The curvature ? 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)^3/2)
(1)
Find 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 (3)/(2). Just dividing 3 by 2 gives a floating point.)
(a) h(x) = x^2 + 3x + 7 at x = 3
(b) h(x) = sin(x) at x = (π)/(3)
(c) h(x) = 2x + 4 at x = 1
(d) h(x) = √(36 - x^2) at x = 2
(e) Give a geometric explanation for why your answers to (c) and (d) make sense.

Added by Chelsea A.

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