Q. 57 Let r1(t)聽and聽r2(t)聽 be conti... [FREE SOLUTION]

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Chapter 11: Q. 57 (page 862)

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with two components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617407427" $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (t)) = r_{1} (t_{0}) 鈰� r_{2} (t_{0})$ .

Short Answer

Expert verified

Ans: It is proved that $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (t)) = r_{1} (t_{0}) 鈰� r_{2} (t_{0})$

Step by step solution

Step 1. Given information.

given,

$lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (t)) = r_{1} (t_{0}) 鈰� r_{2} (t_{0})$

Step 2. Solution

Consider two continuous vector functions with two components. Let

$r_{1} (t) = <x_{1} (t), y_{1} (t)> and r_{2} (t) = <x_{2} (t), y_{2} (t)>$

The objective is to prove that

$lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (t)) = r_{1} (t_{0}) 鈰� r_{2} (t_{0})$

Where $t_{0}$ belongs to the domains of $r_{1} a n d r_{2}$ .

Since $r_{1} (t) a n d r_{2} (t)$ are continuous, their components $x_{1} (t), y_{1} (t), x_{2} (t) and y_{2} (t)$ are continuous.

Thus product and sum of the continuous functions are also continuous.

Step 3. Consider

$\begin{aligned} r_{1} (t) 鈰� r_{2} (t) = <x_{1} (t), y_{1} (t)> 鈰� <x_{2} (t), y_{2} (t)> \\ = x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t) \end{aligned}$

is continuous

Therefore, the function $r_{1} (t) . r_{2} (t)$ is continuous

Consider

$\begin{aligned} lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (t)) = lim_{t 鈫� t_{0}} 鈥� (x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t)) \\ = lim_{t 鈫� t_{0}} 鈥� x_{1} (t) lim_{t 鈫� t_{0}} 鈥� x_{2} (t) + lim_{t 鈫� t_{0}} 鈥� y_{1} (t) lim_{t 鈫� t_{0}} 鈥� y_{2} (t) \\ = x_{1} (t_{0}) x_{2} (t_{0}) + y_{1} (t_{0}) y_{2} (t_{0}) definition of continuous function \\ = r_{1} (t_{0}) 鈰� r_{2} (t_{0}) \\ Thus lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (k)) = r_{1} (t_{0}) 鈰� r_{2} (t_{0}) \end{aligned}$

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91影视

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with two components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617407427" $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (t)) = r_{1} (t_{0}) 鈰� r_{2} (t_{0})$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

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91影视

Let r1(t)andr2(t) be continuous vector functions with two components, and let t0be a point in the domains of both r1andr2. Prove that localid="1649617407427" limt鈫�t0鈥�r1(t)鈰�r2(t)=r1t0鈰�r2t0.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Probability and Statistics

Discrete Mathematics

Logic and Functions

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with two components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617407427" $lim_{t 鈫� t_{0}} 鈥� (r_{1} (t) 鈰� r_{2} (t)) = r_{1} (t_{0}) 鈰� r_{2} (t_{0})$ .