Chapter 11: Q. 6 (page 871)

Given a differentiable vector-valued function $r (t),$ explain why $r' (t_{0})$ is tangent to the curve defined by $r (t)$ when the initial point of $r' (t_{0})$ is placed at the terminal point of $r' (t_{0})$

Short Answer

Expert verified

$r' (t_{0})$ is a tangent to the curve defined by $r (t)$ when the initial point of $r' (t_{0})$ is placed at the terminal of $r (t_{0})$

Step by step solution

Step 1. Given information

The given a differentiable vector - valued function $r (t)$

Step 2. The objective is to explain why r'(t0) is a tangent to the curve defined by r(t) when the initial point of r'(t0) is placed at the terminal point of r(t) .

For this the definition of the derivative of the vector - valued function should be considered first:
The Derivative of a vector - valued function:
Let

r (t)

be a differentiable vector function. Then the derivative of

r (t)

r' (t) = \lim_{h \to 0} \frac{r (t + h) - r (t)}{h} r' (t) = \lim_{h \to 0} \frac{r (t_{0} + h) - r (t)}{h}

Derivative of

r (t)

consists of

r (t + h) - r (t)

in the numerator, the difference of vector point from the terminal point of

r (t)

. If

r' (t_{0})

starts from the terminal point of

r (t)

it becomes tangent to the curve.
Thus

r' (t_{0})

is a tangent to the curve defined by

r (t)

when the initial point of

r' (t_{0})

is placed at the terminal of

r (t_{0})

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Given a differentiable vector-valued function $r (t),$ explain why $r' (t_{0})$ is tangent to the curve defined by $r (t)$ when the initial point of $r' (t_{0})$ is placed at the terminal point of $r' (t_{0})$

Short Answer

Step by step solution

Step 1. Given information

Step 2. The objective is to explain why r'(t0) is a tangent to the curve defined by r(t) when the initial point of r'(t0) is placed at the terminal point of r(t) .

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Given a differentiable vector-valued function r(t),explain why r'(t0) is tangent to the curve defined by r(t) when the initial point of r'(t0) is placed at the terminal point ofr'(t0)

Short Answer

Step by step solution

Step 1. Given information

Step 2. The objective is to explain why r'(t0) is a tangent to the curve defined by r(t) when the initial point of r'(t0) is placed at the terminal point of r(t) .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

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Study anywhere. Anytime. Across all devices.

Given a differentiable vector-valued function $r (t),$ explain why $r' (t_{0})$ is tangent to the curve defined by $r (t)$ when the initial point of $r' (t_{0})$ is placed at the terminal point of $r' (t_{0})$