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Chapter 11: Q. 42 (page 860)

Evaluate the limits in Exercises 42–45.
$\underset{t \to 0}{l i m} (\sin 3 t, \cos 3 t)$

Short Answer

Expert verified

The evaluation of the limit $\underset{t \to 0}{l i m} (\sin 3 t, \cos 3 t)$ is $j$ .

Step by step solution

Step 1. Given Information.

The function:

$\underset{t \to 0}{l i m} (\sin 3 t, \cos 3 t)$

Step 2. Apply the limits.

By applying the limit and solving,

$\underset{t \to 0}{l i m} (\sin 3 t, \cos 3 t) = \underset{t \to 0}{l i m} (\sin 3 t) i + \underset{t \to 0}{l i m} (\cos 3 t) j = 0 i + 1 j = j$

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Most popular questions from this chapter

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.

Let $r (t)$ be a differentiable vector function such that $r (t) \cdot r^{'} (t) = 0$ for every value of $t$ . Prove that $‖ r (t) ‖$ is a constant.

Every description of the DNA molecule says that the strands of the helices run in opposite directions. This is meant as a statement about chemistry, not about the geometric shape of the double helix. Consider two helices

$h_{1} (t) = ⟨ \cos t, \sin t, α t ⟩ a n d h_{2} (t) = ⟨ \sin t, \cos t, α t ⟩$

(a) Sketch these two helices in the same coordinate system, and show that they run geometrically in different directions.

(b) Explain why it is impossible for these two helices to fail to intersect, and hence why they could not form a configuration for DNA.

Find the unit tangent vector and the principal unit normal vector at the specified value of t.

$r (t) = (a \sin h t, b \cos h t) W h e r e a a n d b a r e p o s i t i v e, t = 0$

Let $f (t)$ be a differentiable scalar function and $r (t)$ be a differentiable vector function. Prove that $\frac{d}{d t} (f (t) r (t)) = f' (t) r (t) + f (t) r' (t)$ . (This is Theorem 11.11 (b).)

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Evaluate the limits in Exercises 42–45.limt→0(sin3t,cos3t)

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Apply the limits.

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Most popular questions from this chapter

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Evaluate the limits in Exercises 42–45.
$\underset{t \to 0}{l i m} (\sin 3 t, \cos 3 t)$