Chapter 11: Q. 16 (page 889)

Show that the curvature of the function of $y = \sqrt{1 - x^{2}}, x \in (- 1, 1)$ , is constant, but its second derivative varies with x.

Short Answer

Expert verified

The value of the second derivative depends on x.

The curvature of the function is 1 which is constant.

Step by step solution

Step 1. Given information.

We have to show that the curvature of the function of $y = \sqrt{1 - x^{2}}, x \in (- 1, 1)$ , is constant, but its second derivative varies with x.

Step 2. To show that the second derivative varies with x

if y=f(x) is a twice-differentiable function then the curvature of f is given by

$k = \frac{|f^{''} (x)|}{{(1 + {(f^{'} (x))}^{2})}^{\frac{3}{2}}}$

Since,

$\begin{aligned} y = f (x) \\ = \sqrt{1 - x^{2}} \\ f^{'} (x) = \frac{1}{2 \sqrt{1 - x^{2}}} (- 2 x) \\ \Rightarrow f^{'} (x) = \frac{- x}{\sqrt{1 - x^{2}}} \end{aligned}$

Use the quotient rule to find $f^{''} (x)$ :

$\begin{aligned} f^{''} (x) = \frac{\sqrt{1 - x^{2}} (- 1) + x \frac{1}{2 \sqrt{1 - x^{2}}} (- 2 x)}{1 - x^{2}} \\ f^{''} (x) = \frac{- (1 - x^{2}) - x^{2}}{{(1 - x^{2})}^{\frac{3}{2}}} \\ f^{''} (x) = \frac{- 1}{{(1 - x^{2})}^{\frac{3}{2}}} \end{aligned}$

The value of the second derivative depends on x.

Step 3. To show that the curvature of the function is constant

Substituting $f^{'} and f^{''}$ values in the formula for k we get :

$\begin{aligned} k = \frac{|\frac{- 1}{{(1 - x^{2})}^{\frac{3}{2}}}|}{{(1 + {(\frac{- x}{\sqrt{1 - x^{2}}})}^{2})}^{\frac{3}{2}}} \\ k = \frac{|{(1 - x^{2})}^{\frac{3}{2}}|}{{(1 + \frac{x^{2}}{1 - x^{2}})}^{\frac{3}{2}}} \\ = \frac{\frac{- 1}{{(1 - x^{2})}^{\frac{3}{2}}}}{{(\frac{1 - x^{2} + x^{2}}{1 - x^{2}})}^{\frac{3}{2}}} \\ k = \frac{1}{{(1 - x^{2})}^{\frac{3}{2}}} \cdot \frac{{(1 - x^{2})}^{\frac{3}{2}}}{1} \\ k = 1 \end{aligned}$

The curvature of the function is 1 which is constant.

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Show that the curvature of the function of $y = \sqrt{1 - x^{2}}, x \in (- 1, 1)$ , is constant, but its second derivative varies with x.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. To show that the second derivative varies with x

Step 3. To show that the curvature of the function is constant

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Show that the curvature of the function of y=1−x2,x∈(−1,1), is constant, but its second derivative varies with x.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. To show that the second derivative varies with x

Step 3. To show that the curvature of the function is constant

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Show that the curvature of the function of $y = \sqrt{1 - x^{2}}, x \in (- 1, 1)$ , is constant, but its second derivative varies with x.