Chapter 11: Q. 31 (page 889)

In Exercises 31–35 find the curvature of the given function at the indicated value of x. Then sketch the curve and the osculating circle at the indicated point.
$y = e^{x}, x = 0$

Short Answer

Expert verified

The curvature at $x = 0 is k = \frac{\sqrt{2}}{4}$ .

The graph is

Step by step solution

Step 1. Given information.

The given function and point is $y = e^{x}, x = 0$ .

Step 2. Curvature at the point.

The curvature is given by,

$k = \frac{|f^{'} (x)|}{{(1 + {(f^{'} (x))}^{2})}^{\frac{3}{2}}} f^{'} (x) = e^{x} f^{''} (x) = e^{x} k = \frac{|e^{x}|}{{(1 + {(e^{x})}^{2})}^{\frac{3}{2}}} k = \frac{e^{x}}{{(1 + e^{2 x})}^{\frac{3}{2}}} At x = 0, k = \frac{e^{0}}{{(1 + e^{2 (0)})}^{\frac{3}{2}}} k = \frac{1}{2^{\frac{3}{2}}} k = \frac{2^{\frac{1}{2}}}{2^{\frac{3}{2}} \cdot 2^{\frac{1}{2}}} k = \frac{\sqrt{2}}{2^{2}} k = \frac{\sqrt{2}}{4}$

Step 3. Graph.

The graph will be,

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In Exercises 31–35 find the curvature of the given function at the indicated value of x. Then sketch the curve and the osculating circle at the indicated point.
$y = e^{x}, x = 0$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Curvature at the point.

Step 3. Graph.

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In Exercises 31–35 find the curvature of the given function at the indicated value of x. Then sketch the curve and the osculating circle at the indicated point.y=ex,x=0

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Curvature at the point.

Step 3. Graph.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Geometry

Theoretical and Mathematical Physics

Decision Maths

Pure Maths

Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 31–35 find the curvature of the given function at the indicated value of x. Then sketch the curve and the osculating circle at the indicated point.
$y = e^{x}, x = 0$