Chapter 11: Q. 12 (page 901)

Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = \{e^{t}, t, e^{- t}\}$

Short Answer

Expert verified

$r (t) = \{e^{t}, t, e^{- t}\}$ Ans: $v (t) = <e^{t}, 1, - e^{- t}> and a (t) = <e^{t}, 0, e^{- t}>$

Step by step solution

Step 1. Given information:

$r (t) = \{e^{t}, t, e^{- t}\}$

Step 2. Denoting the velocity and acceleration of the vector:

The velocity vector, $v (t)$ is given by

$v (t) = \frac{d}{d t} r (t) = <\frac{d}{d t} x (t), \frac{d}{d t} y (t), \frac{d}{d t} z (t)>$ .

That is, we take the derivative of each component function of $r (t)$ .

The acceleration vector, $a (t)$ is given by

$a (t) = \frac{d}{d t} v (t)$ .

That is, $a (t)$ is obtained by taking the derivative of each component function of $v (t)$ .

Step 3. Finding the velocity and acceleration of the vector:

Consider $r (t) = <e^{t}, t, e^{- t}>$

$v (t) = \frac{d}{d t} r (t) = \frac{d}{d t} <e^{t}, t, e^{- t}> = <\frac{d}{d t} (e^{t}), \frac{d}{d t} (t), \frac{d}{d t} (e^{- t})> = <e^{t}, 1, - e^{- t}> a (t) = \frac{d}{d t} (v (t)) = \frac{d}{d t} <e^{t}, 1, - e^{- t}> = <\frac{d}{d t} (e^{t}), \frac{d}{d t} (1), \frac{d}{d t} (- e^{- t})> = <e^{t}, 0, + e^{- t}>$

thus

$v (t) = <e^{t}, 1, - e^{- t}> and a (t) = <e^{t}, 0, e^{- t}>$

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Velocity and acceleration vectors: Find the velocity and acceleration vectors for the given vector functions.
$r (t) = \{e^{t}, t, e^{- t}\}$

Short Answer

Step by step solution

Step 1. Given information:

Step 2. Denoting the velocity and acceleration of the vector:

Step 3. Finding the velocity and acceleration of the vector:

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