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Chapter 11: Q. 1 (page 900)

$Fill in the blanks to complete each of the following theorem statements : The derivative of a vector function r (t) is given by r' (t) = \lim_{h \to 0}______$

Short Answer

Expert verified

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

Step by step solution

Step 1. Given information is:

A vector function r(t)

Step 2. Derivative of vector function

$Let r (t) be a differentiable vector function with either two or three components . The derivative of r (t) is given by : r' (t) = \lim_{h \to 0} \frac{r (t + h) - r (t)}{h}$

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Most popular questions from this chapter

Explain why the graph of every vector-valued function $r (t) = (\cos t, \sin t, \cos t)$ lies on the intersection of the two cylinders $x^{2} + y^{2} = 1 and y^{2} + z^{2} = 1 .$

As we saw in Example 1, the graph of the vector-valued function $r (t) = (\cos t, \sin t, t), for t \in [0, 2 π]$ is a circular helix that spirals counterclockwise around the z-axis and climbs as t increases. Find another parametrization for this helix so that the motion is downwards.

$Complete the following definition : If r (t) = <x (t), y (t), z (t)> is a differentiable position function, then the velocity vector v (t) is . . . .$

Find parametric equations for each of the vector-valued functions in Exercises 26–34, and sketch the graphs of the functions, indicating the direction for increasing values of t.

$r (t) = (t, t^{2}, t^{3}) f o r t \in [0, 2]$

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.

$r (t) = (3 \sin (t), 5 \cos (t), 4 \sin (t)), t = π$

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Fillintheblankstocompleteeachofthefollowingtheoremstatements:Thederivativeofavectorfunctionr(t)isgivenbyr'(t)=limh→0______

Short Answer

Step by step solution

Step 1. Given information is:

Step 2. Derivative of vector function

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$Fill in the blanks to complete each of the following theorem statements : The derivative of a vector function r (t) is given by r' (t) = \lim_{h \to 0}______$