Q. 32 For each of the vector-valued fu... [FREE SOLUTION]

Chapter 11: Q. 32 (page 880)

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.
$r (t) = (\cos^{3} (t), \sin^{3} (t)), t = \frac{蟺}{4}$

Short Answer

Expert verified

The unit tangent vector and principle unit normal vector are

$T (\frac{蟺}{4}) = (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) N (\frac{蟺}{4}) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Step by step solution

Step 1. Given information

Given $r (t) = (\cos^{3} (t), \sin^{3} (t))$

Step 2. Parta. Find unit tangent vector Partb. Find principle unit normal vector

$r (t) = (\cos^{3} (t), \sin^{3} (t)) r' (t) = (- 3 \cos^{2} (t) \sin (t), 3 \sin^{2} (t) \cos (t)) 鈭� r' (t) 鈭� = 鈭� (- 3 \cos^{2} (t) \sin (t), 3 \sin^{2} (t) \cos (t)) 鈭� = \sqrt{({(- 3 \cos^{2} (t) \sin (t))}^{2} + {(3 \sin^{2} (t) \cos (t))}^{2})} = 3 \cos (t) \sin (t) P a r t (a) T (t) = \frac{r' (t)}{鈭� r' (t) 鈭�} = \frac{(- 3 \cos^{2} (t) \sin (t), 3 \sin^{2} (t) \cos (t))}{3 \cos (t) \sin (t)} = (- \cos (t), \sin (t)) A t t = \frac{蟺}{4} T (\frac{蟺}{4}) = (- \cos (\frac{蟺}{4}), \sin (\frac{蟺}{4})) = (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) T' (t) = (\sin (t), \cos (t)) 鈭� T' (t) 鈭� = \sqrt{(\sin^{2} (t) + \cos^{2} (t))} = 1 P a r t (b) N (t) = \frac{T' (t)}{鈭� T' (t) 鈭�} = \frac{(\sin (t), \cos (t))}{1} = (\sin (t), \cos (t)) A t t = \frac{蟺}{4} N (\frac{蟺}{4}) = (\sin (\frac{蟺}{4}), \cos (\frac{蟺}{4})) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

Step 3. The solution

The unit tangent vector and the principle unit normal vector are

$T (\frac{蟺}{4}) = (- \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) N (\frac{蟺}{4}) = (\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2})$

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91影视

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.
$r (t) = (\cos^{3} (t), \sin^{3} (t)), t = \frac{蟺}{4}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Parta. Find unit tangent vector Partb. Find principle unit normal vector

Step 3. The solution

One App. One Place for Learning.

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91影视

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.rt=cos3t,sin3t,t=蟺4

Short Answer

Step by step solution

Step 1. Given information

Step 2. Parta. Find unit tangent vector Partb. Find principle unit normal vector

Step 3. The solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

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Theoretical and Mathematical Physics

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Study anywhere. Anytime. Across all devices.

For each of the vector-valued functions in Exercises ,find the unit tangent vector and the principal unit normal vector at the specified value of t.
$r (t) = (\cos^{3} (t), \sin^{3} (t)), t = \frac{蟺}{4}$