Chapter 11: Q. 22 (page 872)

In Exercises 21–23 you are given a vector function $r$ and a scalar function $t = f (τ)$ . Compute $\frac{d r}{d τ}$ in the following two ways:
(a) By using the chain rule $\frac{d r}{d τ} = \frac{d r}{d t} \frac{d t}{d τ}$ .
(b) By substituting $t = f (τ)$ into the formula for $r$ . Ensure that your two answers are consistent.

Short Answer

Expert verified

Part (a) The answer is $\frac{d r}{d t} = 3 <- T^{2} \sin T^{3}, T^{2} \cos T^{3}, T^{2}>$

Part (b) The value is $\frac{d r}{d t} = 3 <- T^{2} \sin T^{3}, T^{2} \cos T^{3}, T^{2}>$

The two answers are consistent, that is $\frac{d r}{d τ} = 3 <- τ^{2} \sin τ^{3}, τ^{2} \cos τ^{3}, τ^{2}>$

Step by step solution

Part (a) Step 1. Given data

The given vector function is $r (t) = ⟨ \cos t, \sin t, t ⟩, t = τ^{3}$

We have to find $\frac{d r}{d τ}$ in two ways,

Part (a) Step 2. Find drdτ

By using the chain rule, $\frac{d r}{d τ} = \frac{d r}{d t} \cdot \frac{d t}{d τ}$

$\begin{aligned} \frac{d r}{d τ} & = \frac{d}{d t} r (t) \cdot \frac{d}{d τ} (t) \\ = \frac{d}{d t} ⟨ \cos t, \sin t, t ⟩ \cdot \frac{d}{d τ} (τ^{3}) \\ = ⟨ - \sin t, \cos t, 1 ⟩ (3 τ^{2}) \end{aligned} \begin{aligned} = ⟨ - \sin τ^{3}, \cos τ^{3}, 1 ⟩ (3 τ^{2}) \\ = 3 (- τ^{2} \sin τ^{3}, τ^{2} \cos τ^{3}, τ^{2}) \end{aligned}$

Part (b) Step 1. Find drdτ by substituting t=f(τ)

By substituting $t = \sin τ$ in $r (t)$

$\begin{aligned} r (t) & = ⟨ c o s t, s i n t, t ⟩ \\ = ⟨ c o s τ^{3}, s i n τ^{3}, τ^{3} ⟩ \\ \frac{d r}{d τ} & = \frac{d}{d τ} r (t) \\ = \frac{d}{d τ} ⟨ c o s τ^{3}, s i n τ^{3}, τ^{3} ⟩ \\ = ⟨ - 3 τ^{2} s i n τ^{3}, 3 τ^{2} c o s τ^{2}, 3 τ^{2} ⟩ \\ = 3 ⟨ - τ^{2} s i n τ^{2}, τ^{2} c o s τ^{2}, τ^{2} ⟩ \end{aligned}$

Here, results obtained in part (a) and part (b) are equal.

Therefore, the two answers are consistent.

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Short Answer

Step by step solution

Part (a) Step 1. Given data

Part (a) Step 2. Find drdτ

Part (b) Step 1. Find drdτ by substituting t=f(τ)

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In Exercises 21–23 you are given a vector function rand a scalar function t=f(τ). Computedrdτin the following two ways:(a) By using the chain rule drdτ=drdtdtdτ.(b) By substituting t=f(τ)into the formula for r. Ensure that your two answers are consistent.

Short Answer

Step by step solution

Part (a) Step 1. Given data

Part (a) Step 2. Find drdτ

Part (b) Step 1. Find drdτ by substituting t=f(τ)

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Pure Maths

Decision Maths

Mechanics Maths

Discrete Mathematics

Calculus

Study anywhere. Anytime. Across all devices.