Q. 24 Explain why the graph of every v... [FREE SOLUTION]

Chapter 11: Q. 24 (page 860)

Explain why the graph of every vector-valued function $r (t) = (\cos t, \sin t, f (t))$ lies on the surface of the cylinder $x^{2} + y^{2} = 1$ for every continuous functionf.

Short Answer

Expert verified

The graph of every vector-valued function $r (t) = (\cos t, \sin t, f (t))$ lies on the surface of the cylinder $x^{2} + y^{2} = 1$ for every continuous function f because every point of r(t)satisfies the surface of the cylinder.

Step by step solution

Step 1. Given Information.

The given vector-valued function is $r (t) = (\cos t, \sin t, f (t)),$ and the surface of the cylinder is $x^{2} + y^{2} = 1 .$

Step 2. Explanation.

To explain why the graph of every vector-valued function lies on the surface of the cylinder for every continuous function f,substitute $x = \cos t and y = \sin t$ in the given surface of the cylinder equation.

So,

$x^{2} + y^{2} = 1 \cos^{2} t + \sin^{2} t = 1 1 = 1$

It is true.

Thus, every point of r(t)satisfies the given equation of the surface of the cylinder.

Hence the graph of $r (t) = (\cos t, \sin t, f (t)),$ lies on the surface of the cylinder $x^{2} + y^{2} = 1$ for every continuous functionf.

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Explain why the graph of every vector-valued function $r (t) = (\cos t, \sin t, f (t))$ lies on the surface of the cylinder $x^{2} + y^{2} = 1$ for every continuous functionf.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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91影视

Explain why the graph of every vector-valued function r(t)=cost,sint,f(t)lies on the surface of the cylinder x2+y2=1for every continuous functionf.

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

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Decision Maths

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Explain why the graph of every vector-valued function $r (t) = (\cos t, \sin t, f (t))$ lies on the surface of the cylinder $x^{2} + y^{2} = 1$ for every continuous functionf.