Chapter 11: Q. 54 (page 862)

Let $r (t) = ⟨ x (t), y (t) |$ be a vector-valued function defined on an open interval containing the point $t_{0}$ . Prove that r(t) is continuous at $t_{0}$ if and only if $x (t)$ and $y (t)$ are both continuous at $t_{0}$ .

Short Answer

Expert verified

Ans: It is proved that $r (t)$ is continuous at $t_{0}$ if and only if $x (t) a n d y (t)$ are continuous at $t_{0}$

Step by step solution

Step 1. Given information.

given,

$r (t) = ⟨ x (t), y (t) |$

Step 2. Consider a vector valued function r(t)=⟨x(t),y(t)⟩ defined on an interval containing t0.

The objective is to prove that r(t) is continuous at $t_{0}$ if and only if $x (t)$ and $y (t)$ are both continuous at $t_{0}$ .

Note: A vector function $r (t)$ is said to be continuous at $t_{0}$ if $lim_{t \to t_{0}} x (t) and lim_{t \to t_{0}} y (t)$ exists, then use say that x(t) and y(t) are both continuous at $t_{0}$ .

Let $lim_{t \to t_{0}} x (t) = x (t_{0}) and let lim_{t \to t_{0}} y (t) = y (t_{0})$

Step 3. Consider

$\begin{aligned} lim_{t \to t_{0}} r (t) = lim_{t \to t_{0}} ⟨ x (t), y (t) ⟩ \\ = lim_{t \to t_{0}} x (t) i + lim_{t \to t_{0}} y (t) j \\ = x (t_{0}) i + y (t_{0}) j \\ = <x (t_{0}), y (t_{0})> \\ = r (t_{0}) \end{aligned}$

$lim_{t \to t_{0}} r (t) = r (t_{0})$ happens only when role="math" localid="1649614127296" $x (t) a n d y (t)$ are continuous at $t_{0}$ .

Thus, $r (t)$ is continuous at $t_{0}$ if and only if $x (t) a n d y (t)$ are continuous at $t_{0}$ .

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Let $r (t) = ⟨ x (t), y (t) |$ be a vector-valued function defined on an open interval containing the point $t_{0}$ . Prove that r(t) is continuous at $t_{0}$ if and only if $x (t)$ and $y (t)$ are both continuous at $t_{0}$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider a vector valued function r(t)=⟨x(t),y(t)⟩ defined on an interval containing t0.

Step 3. Consider

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Let r(t)=⟨x(t),y(t)|be a vector-valued function defined on an open interval containing the point t0. Prove that r(t) is continuous at t0if and only if x(t)and y(t)are both continuous at t0.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Consider a vector valued function r(t)=⟨x(t),y(t)⟩ defined on an interval containing t0.

Step 3. Consider

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

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Let $r (t) = ⟨ x (t), y (t) |$ be a vector-valued function defined on an open interval containing the point $t_{0}$ . Prove that r(t) is continuous at $t_{0}$ if and only if $x (t)$ and $y (t)$ are both continuous at $t_{0}$ .