Chapter 11: Q. 23 (page 880)

For each of the vector-valued functions, find the unit tangent vector.
$r (t^{2}) = (t^{2} + 5, 5 t, 4 t^{3})$

Short Answer

Expert verified

$\frac{1}{\sqrt{144 t^{4} + 4 t^{2} + 25}} <2 t, 5, 12 t^{2}>$

Step by step solution

Step1. Given Information

$Consider r (t) = <t^{2} + 5, 5 t, 4 t^{3}> The objective is to find the unit tangent vector to r (t) The unit tangent vector The unit tangent vector of r (t) denoted by T (t) is defined as T (t) = \frac{r^{'} (t)}{∥r^{'} (t)∥} Consider r (t) = <t^{2} + 5, 5 t, 4 t^{3}> First we compute r^{'} (t) : \begin{array}{l} r^{'} (t) = \frac{d}{d t} <t^{2} + 5, 5 t, 4 t^{3}> \\ = <\frac{d}{d t} (t^{2} + 5), \frac{d}{d t} (5 t), \frac{d}{d t} (4 t^{3})> \\ = <2 t, 5, 12 t^{2}> \\ ∥r^{'} (t)∥ = ∥<2 t, 5, 12 t^{2}>∥ \\ = \sqrt{(2 t)^{2} + (5)^{2} + {(12 t^{2})}^{2}} \\ = \sqrt{4 t^{2} + 25 + 144 t^{4}} \end{array}$

Step2. Continue

$The unit tangent vector to r (t) is \begin{array}{l} T (t) = \frac{r^{'} (t)}{∥r^{''} (t)∥} \\ = \frac{<2 t, 5, 12 t^{2}>}{\sqrt{4 t^{2} + 25 + 144 t^{4}}} \\ = \frac{1}{\sqrt{144 t^{4} + 4 t^{2} + 25}} <2 t, 5, 12 t^{2}> \\ Thus the unit tangent vector to the given vector function is \\ \frac{1}{\sqrt{144 t^{4} + 4 t^{2} + 25}} <2 t, 5, 12 t^{2}> \end{array}$

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

For each of the vector-valued functions, find the unit tangent vector.
$r (t^{2}) = (t^{2} + 5, 5 t, 4 t^{3})$

Short Answer

Step by step solution

Step1. Given Information

Step2. Continue

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

Company

Product

Help

For each of the vector-valued functions, find the unit tangent vector.r(t2)=(t2+5,5t,4t3)

Short Answer

Step by step solution

Step1. Given Information

Step2. Continue

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Geometry

Calculus

Mechanics Maths

Study anywhere. Anytime. Across all devices.

For each of the vector-valued functions, find the unit tangent vector.
$r (t^{2}) = (t^{2} + 5, 5 t, 4 t^{3})$