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Chapter 11: Q. 39 (page 872)

In Exercises 35-39 a vector function $r (t)$ and scalar function $t = f (蟿)$ are given. Find $\frac{d r}{d 蟿}$ .
$39. r (t) = \cos 鈦� 3 t i + \sin 鈦� 4 t j + t k, t = 5 蟿鈭� 2$

Short Answer

Expert verified

$\frac{d r}{d 蟿} = 鉄� 鈭� 15 \sin 鈦� (15 蟿鈭� 6), 20 \cos 鈦� (20 蟿鈭� 8), 5 鉄�$

Step by step solution

01

Step 1. Given data

The given vector function is $r (t) = \cos 鈦� 3 t i + \sin 鈦� 4 t j + t k, t = 5 蟿鈭� 2$

We have to find $\frac{d r}{d 蟿}$

02

Step 2. Use chain rule

If $r (t)$ is a vector function and $t = f (蟿)$ , a scalar function. Then the chain rule states that $\frac{d r}{d 蟿} = \frac{d r}{d t} 鈰� \frac{d t}{d 蟿}$

We have,

$\begin{aligned} r (t) = \cos 鈦� 3 t i + \sin 鈦� 4 t j + t k \\ r (t) = 鉄� \cos 鈦� 3 t, \sin 鈦� 4 t, t 鉄� \\ \frac{d r}{d t} = 鉄� 鈭� 3 \sin 鈦� 3 t, 4 \cos 鈦� 3 t, 1 鉄� \end{aligned}$

It is given, $t = 5 蟿鈭� 2$

$\frac{d t}{d 蟿} = 5$

By the chain rule,

$\frac{d r}{d 蟿} = 鉄� 鈭� 3 \sin 鈦� 3 t, 4 \cos 鈦� 4 t, 1 鉄� 5 = 鉄� 鈭� 15 \sin 鈦� 3 t, 20 \cos 鈦� 4 t, 5 鉄� = 鉄� 鈭� 15 \sin 鈦� 3 (5 蟿鈭� 2), 20 \cos 鈦� 4 (5 蟿鈭� 2), 5 鉄� = 鉄� 鈭� 15 \sin 鈦� (15 蟿鈭� 6), 20 \cos 鈦� (20 蟿鈭� 8), 5 鉄�$

Thus $\frac{d r}{d 蟿} = 鉄� 鈭� 15 \sin 鈦� (15 蟿鈭� 6), 20 \cos 鈦� (20 蟿鈭� 8), 5 鉄�$

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Most popular questions from this chapter

$Let r (t) = <x (t), y (t), z (t)> . Provide a definition for the continuity of the vector function r at a point c in the domain of r .$

Given a vector-valued function r(t) with domain $鈩�,$ what is the relationship between the graph of r(t) and the graph of r(kt), where k > 1 is a scalar?

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鈥�42. Note: These are the same functions as in Exercises 35, 37, and 39.

$r (t) = 鉄� \sin 鈦� 2 t, \cos 鈦� 2 t, t 鉄� at t = \frac{蟺}{2}$

Evaluate the limits in Exercises 42鈥�45.

$\underset{t 鈫� 0}{l i m} (\sin 3 t, \cos 3 t)$

For each of the vector-valued functions, find the unit tangent vector.

$r (t^{2}) = (t^{2} + 5, 5 t, 4 t^{3})$

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