Chapter 11: Q. 56 (page 862)

Let $c_{1}$ and $c_{2}$ be scalars, $r_{1} (t)$ and $r_{2} (t)$ be continuous vector functions with two components, and $t_{0}$ be a point in the domains of both $r_{1}$ and $r_{2}$ . Prove that
$lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})$

Short Answer

Expert verified

Ans: It is proved that $lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})$ .

Step by step solution

Step 1. Given information.

To prove,

$lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})$

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

role="math" localid="1649615795278"

lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})

there

c_{1} a n d c_{2}

are two scales and to belong 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 $x_{1} (t), y_{1} (t), x_{2} (t), y_{2} (t)$ are continuous. The product and sum of the continuous functions are also continuous.

Step 3. Consider

$(c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})$

$\begin{aligned} = <c_{1} x_{1} (t), c_{1} y_{1} (t)> + <c_{2} x_{2} (t), c_{2} y_{2} (t)> \\ = <c_{1} x_{1} (t) + c_{2} x_{2} (t), c_{1} y_{1} (t) + c_{2} y_{2} (t)> is continuous. \end{aligned}$

Therefore, $c_{1} r_{1} (t) + c_{2} r_{2} (t)$ is a continuous function.

Consider

$\begin{aligned} lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = lim_{t \to t_{0}} <c_{1} x_{1} (t) + c_{2} x_{2} (t), c_{1} y_{1} (t) + c_{2} y_{2} (t)> \\ = lim_{t \to t_{0}} (c_{1} x_{1} (t) + c_{2} x_{2} (t)) i + lim_{t \to t_{0}} (c_{1} y_{1} (t) + c_{2} y_{2} (t)) j \\ = (c_{1} x_{1} (t_{0}) + c_{2} x_{2} (t_{0})) i + (c_{1} y_{1} (t_{0}) + c_{2} y_{2} (t_{0})) j \\ = (c_{1} x_{1} (t_{0}) i + c_{1} y_{1} (t_{0}) j) + (c_{2} x_{2} (t_{0}) i + c_{2} y_{2} (t_{0}) j) \\ = <c_{1} x_{1} (t_{0}), c_{1} y_{1} (t_{0})> + <c_{2} x_{2} (t_{0}), c_{2} y_{2} (t_{0})> \\ = c_{1} <x_{1} (t_{0}), y_{1} (t_{0})> + c_{1} <x_{2} (t_{0}), y_{2} (t_{0})> \\ = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0}) \end{aligned}$

Thus, $lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})$

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Let $c_{1}$ and $c_{2}$ be scalars, $r_{1} (t)$ and $r_{2} (t)$ be continuous vector functions with two components, and $t_{0}$ be a point in the domains of both $r_{1}$ and $r_{2}$ . Prove that
$lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

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Let c1and c2be scalars, r1(t)and r2(t)be continuous vector functions with two components, and t0be a point in the domains of both r1andr2. Prove thatlimt→t0 c1r1(t)+c2r2(t)=c1r1t0+c2r2t0

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

Let $c_{1}$ and $c_{2}$ be scalars, $r_{1} (t)$ and $r_{2} (t)$ be continuous vector functions with two components, and $t_{0}$ be a point in the domains of both $r_{1}$ and $r_{2}$ . Prove that
$lim_{t \to t_{0}} (c_{1} r_{1} (t) + c_{2} r_{2} (t)) = c_{1} r_{1} (t_{0}) + c_{2} r_{2} (t_{0})$