Q. 16 Let r(t)=聽x(t),聽y(t),聽聽t鈭圼... [FREE SOLUTION]

Chapter 11: Q. 16 (page 860)

Let $r (t) = (x (t), y (t)), t 鈭� [a, 鈭�),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a vertical asymptote as t 鈫� 鈭�? Provide an example illustrating your answer.

Short Answer

Expert verified

The vector-valued function has a vertical asymptote if $\lim_{t 鈫� a^{+}} y (t) = 鈭�, \lim_{t 鈫� a^{+}} y (t) = - 鈭�$ conditions are satisfied.

Step by step solution

Step 1. Given Information.

The given vector-valued function is $r (t) = (x (t), y (t)), t 鈭� [a, 鈭�) .$

Step 2. Explanation.

The vector-valued function has a vertical asymptote if $\lim_{t 鈫� a^{+}} y (t) = 鈭�, \lim_{t 鈫� a^{+}} y (t) = - 鈭�$ conditions are satisfied.

So, let's take an example to illustrate the answer,

$Let r (t) = (\frac{1}{t}, \frac{t}{t + 1}), t 鈭� [a, 鈭�) So, y (t) = \frac{1}{t + 1} \lim_{t 鈫� - 1^{+}} y (t) = \lim_{t 鈫� - 1^{+}} \frac{t}{t + 1} \lim_{t 鈫� - 1^{+}} y (t) = \lim_{h 鈫� 0} \frac{- 1 + h}{- 1 + h + 1} \lim_{t 鈫� - 1^{+}} y (t) = \lim_{h 鈫� 0} \frac{- 1 + h}{h} \lim_{t 鈫� - 1^{+}} y (t) = \frac{- 1}{0} \lim_{t 鈫� - 1^{+}} y (t) = - 鈭�$

Hence proved.

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Let $r (t) = (x (t), y (t)), t 鈭� [a, 鈭�),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a vertical asymptote as t 鈫� 鈭�? Provide an example illustrating your answer.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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91影视

Let r(t)=x(t),y(t),t鈭�[a,鈭�), be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a vertical asymptote as t 鈫� 鈭�? Provide an example illustrating your answer.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Discrete Mathematics

Decision Maths

Pure Maths

Geometry

Calculus

Study anywhere. Anytime. Across all devices.

Let $r (t) = (x (t), y (t)), t 鈭� [a, 鈭�),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a vertical asymptote as t 鈫� 鈭�? Provide an example illustrating your answer.