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Chapter 11: Q. 1 (page 901)

Sketching vector functions: Sketch the following vector functions.
$r (f) = \{t, t^{3}\}, t \in ℝ$

Short Answer

Expert verified

$r (f) = \{t, t^{3}\}, t \in ℝ$

Step by step solution

01

Step 1. Given information

$r (f) = \{t, t^{3}\}, t \in ℝ$

02

Step 2. Sketching the vector

On X-axis, x=t and Y-axis, y=t³

Plotting the graph,

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Most popular questions from this chapter

Explain why we do not need an “epsilon–delta” definition for the limit of a vector-valued function.

For constants $α, β$ , and $γ$ , the graph of a vector-valued function of the form

$r (t) = α e^{γ t} (\cos t) i + α e^{γ t} (\sin t) j + β e^{γ t} k, t \geq 0$

Let $r_{1} (f)$ and $r_{2} (t)$ be differentiable vector functions with three components each. Prove that

\frac{d}{d t} (r_{1} (t) - r_{2} (t)) = r_{1}^{'} (t) - r_{2} (t) + r_{1} (t) - r_{2}^{'} (t) ∣

(This is Theorem 11.11 (c).)

Prove that the cross product of two orthogonal unit vectors is a unit vector.

Let $x_{1} (t)$ , $x_{2} (t)$ , $y_{1} (t)$ , and $y_{2} (t)$ be differentiable scalar functions. Prove that the dot product of the vector-valued functions role="math" localid="1649579098744" $r_{1} (t) = <x_{1} (t), y_{1} (t)>$ and role="math" localid="1649579122624" $r_{2} (t) = <x_{2} (t), y_{2} (t)>$ is a differentiable scalar function.

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