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Chapter 11: Q. 40 (page 902)

Find the equation of the osculating circle to the given scalar function at the specified point.
$f (x) = \ln x, (1, 0)$

Short Answer

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Step 1. Given information.

given,

$f (x) = \ln x, (1, 0)$

Step 2.

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Most popular questions from this chapter

Evaluate and simplify the indicated quantities in Exercises 35–41.
$5 (\cos t, \sin t)$

Let $r (t) = <x (t), y (t), z (t)>$ be a differentiable vector function on some interval $I ⊑ R$ such that the derivative of the unit tangent vector $T (t_{0}) = 0$ , where $t_{0} \in I$ . Prove that the binormal vector $B (t_{0}) = T (t_{0}) \times N (t_{0})$

(a) $B (t_{0})$ is a unit vector;

(b) $B (t_{0})$ is orthogonal to both $T (t_{0})$ and $N (t_{0})$ .

Also, prove that $T (t_{0}), N (t_{0})$ , and $B (t_{0})$ form a right-handed coordinate system.

Evaluate and simplify the indicated quantities in Exercises 35–41.

$(1, t, t^{2}) \times (1, t^{2}, t^{3})$

Let $f (t)$ be a differentiable scalar function and $r (t)$ be a differentiable vector function. Prove that $\frac{d}{d t} (f (t) r (t)) = f' (t) r (t) + f (t) r' (t)$ . (This is Theorem 11.11 (b).)

Using the definitions of the normal plane and rectifying plane in Exercises 20 and 21, respectively, find the equations of these planes at the specified points for the vector functions in Exercises 40–42. Note: These are the same functions as in Exercises 35, 37, and 39.

$r (t) = <e^{t}, e^{- t}, \sqrt{2} t) at t = 0$

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Find the equation of the osculating circle to the given scalar function at the specified point.
$f (x) = \ln x, (1, 0)$

Short Answer

Step by step solution

Step 1. Given information.

Step 2.

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Find the equation of the osculating circle to the given scalar function at the specified point. f(x)=ln⁡x,(1,0)

Short Answer

Step by step solution

Step 1. Given information.

Step 2.

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Calculus

Geometry

Theoretical and Mathematical Physics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Find the equation of the osculating circle to the given scalar function at the specified point.
$f (x) = \ln x, (1, 0)$