Chapter 11: Q. 4 (page 871)

State what it means for a vector function $r (t) = <x (t), y (t), z (t)>$ to be differentiable.

Short Answer

Expert verified

Proved that the given function is said to be differentiable

Step by step solution

Step 1. Given information

The given vector function $r (t) = <x (t), y (t), z (t)>$

Step 2. The object is to define the derivative of the vector-function r(t)

Defination of the derivative of the vector function

Let $r (t) = <x (t), y (t), z (t)>$ where $x (t), y (t), z (t)$ are three real-valued function which are differentiable at every point in the interval $I \subseteq R$ then the derivative of

$r (t) = <x (t), y (t), z (t)>$ is

$r' (t) = \frac{d r}{d t} r (t) = \frac{d}{d t} <x (t), y (t), z (t)> = <x' (t), y' (t), z' (t)> = x' (t) i + y' (t) j + z' (t) k$

Now the function is said to be differentiable

Step 3. Take an example

Let $r (t) = <t, t^{2}, t^{4}>$

$r' (t) = \frac{d r}{d t} = \frac{d}{d r} r (t) = \frac{d}{d r} <t, t^{2}, t^{4}> = <\frac{d t}{d t}, \frac{d}{d t} (t^{2}), \frac{d}{d t} (t^{4})> = <1, 2 t, 4 t^{3}>$

Thus , $r' (t) = <1, 2 t, 4 t^{3}>$

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State what it means for a vector function $r (t) = <x (t), y (t), z (t)>$ to be differentiable.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The object is to define the derivative of the vector-function r(t)

Step 3. Take an example

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State what it means for a vector function rt=xt,yt,ztto be differentiable.

Short Answer

Step by step solution

Step 1. Given information

Step 2. The object is to define the derivative of the vector-function r(t)

Step 3. Take an example

One App. One Place for Learning.

Most popular questions from this chapter

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State what it means for a vector function $r (t) = <x (t), y (t), z (t)>$ to be differentiable.