Chapter 1. calc_tutorial_13_2_025

1.1 Problem Statement

{3,5,7}
{2,4,6}
{5,10}
pow($a,2)
$a*$c

Evaluate (d / dt)r(g(t)) using the Chain Rule.

1.2 Step 1

Question Sequence

Question 1.1

Choose how to apply the Chain Rule to a differentiable vector-valued function r(t) and a differentiable 52kSQxpMMRmRRfKqkkozIWSginhc30P0GScbzJ4MvSQ= function g(t).

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

Recall how to find the derivative of a vector-valued function. A vector-valued function r(t) = <x(t), y(t), z(t)> is differentiable if and only if each component is differentiable. The derivative is given as follows.

r'(t) = (d / dt)r(t) = <x'(t), y'(t), z'(t)>

Calculate r'(t) for

r'(t) =

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

Calculate g'(t) for g(t) = $at + $c.

g'(t) = nc1ItEz0kR4=

Correct.
Incorrect.

1.3 Step 2

Question Sequence

Question 1.2

Use the Chain Rule to evaluate (d / dt)r(g(t)).

(d / dt)r(g(t)) = g'(t)r'(g(t)) =

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

=

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
Correct.
Incorrect.