Chapter 11: Q. 46 (page 890)

Show that the curvature on the parabola defined by $y = x^{2}$ is greatest at the origin.

Short Answer

Expert verified

It is proved that the curvature on the parabola defined by $y = x^{2}$ is greatest at the origin.

Step by step solution

Step 1. Given Information.

The given parabola is $y = x^{2} .$

Step 2. Showing.

To show that the curvature on the parabola defined by $y = x^{2}$ is greatest at the origin, we will use the formula for the curvature in the plane.

The curvature in the plane is $k = \frac{|f'' (x)|}{{(1 + {(f' (x))}^{2})}^{\frac{3}{2}}} .$

Now, $y = f (x) = x^{2} .$

So,

$f (x) = x^{2} f' (x) = 2 x f'' (x) = 2 Now, the curvature of the graph of f is k = \frac{|f'' (x)|}{{(1 + {(f' (x))}^{2})}^{\frac{3}{2}}} k = \frac{|2|}{{(1 + {(2 x)}^{2})}^{\frac{3}{2}}} k = \frac{2}{{(1 + 4 x^{2})}^{\frac{3}{2}}}$

Step 3. Use the first derivative test.

Now, to find the point on the parabola where the curvature is greatest, we will apply the derivative test.

So,

$Let g (x) = \frac{2}{{(1 + 4 x^{2})}^{\frac{3}{2}}} g' (x) = 2 [- \frac{3}{2} {(1 + 4 x^{2})}^{- \frac{5}{2}} (8 x)] g' (x) = \frac{- 24 x}{{(1 + 4 x^{2})}^{\frac{5}{2}}} Put g' (x) = 0 0 = \frac{- 24 x}{{(1 + 4 x^{2})}^{\frac{5}{2}}} 0 = - 24 x x = 0$

Substitute $x = 0 in the equation y = x^{2}$ to find the value,

$y = 0$

Thus, at the origin $(0, 0),$ the curvature is greatest.

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Show that the curvature on the parabola defined by $y = x^{2}$ is greatest at the origin.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Showing.

Step 3. Use the first derivative test.

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Show that the curvature on the parabola defined byy=x2 is greatest at the origin.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Showing.

Step 3. Use the first derivative test.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Calculus

Applied Mathematics

Pure Maths

Theoretical and Mathematical Physics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Show that the curvature on the parabola defined by $y = x^{2}$ is greatest at the origin.