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A particle travels along the
x-axis such that its velocity is given by
v(t)=(t^(0.3)+5)cos(2t). What is the average velocity of the particle on the interval
0 <= t <= 4 ? You may use a calculator and round your answer to the nearest thousandth.
Answer:

A particle travels along the $x$ -axis such that its velocity is given by $v(t)=\left(t^{0.3}+5\right) \cos (2 t)$ . What is the average velocity of the particle on the interval $0 \leq t \leq 4$ ? You may use a calculator and round your answer to the nearest thousandth. $\newline$ Answer:

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Relate position, velocity, speed, and acceleration using derivatives

Full solution

Q. A particle travels along the

x

-axis such that its velocity is given by

v(t)=\left(t^{0.3}+5\right) \cos (2 t)

. What is the average velocity of the particle on the interval

0 \leq t \leq 4

? You may use a calculator and round your answer to the nearest thousandth.

\newline

Answer:

Set up integral for average velocity: To find the average velocity of the particle over the interval from $t = 0$ to $t = 4$ , we need to integrate the velocity function $v(t)$ over this interval and then divide by the length of the interval. $\newline$ The average velocity formula is given by: $\newline$ Average velocity = $(1/(b-a)) \times \int_{a}^{b} v(t) \, dt$ $\newline$ Here, $a = 0$ , $b = 4$ , and $v(t) = (t^{0.3}+5)\cos(2t)$ .
Simplify the integral: First, we set up the integral for the average velocity: $\newline$ Average velocity = $(1/(4-0)) \times \int_{0}^{4} (t^{0.3}+5)\cos(2t) \, dt$ $\newline$ This simplifies to: $\newline$ Average velocity = $(1/4) \times \int_{0}^{4} (t^{0.3}+5)\cos(2t) \, dt$
Evaluate integral using calculator: Next, we use a calculator to evaluate the integral. Since the integral involves both a power function and a trigonometric function, it's not straightforward to integrate by hand. We rely on numerical methods or a calculator with integration capabilities.
Multiply integral result: After evaluating the integral from $0$ to $4$ of $(t^{0.3}+5)\cos(2t) \, dt$ using a calculator, we get a numerical value. Let's assume this value is $I$ (since we don't have an actual calculator to perform the computation).
Round to nearest thousandth: Now, we multiply the result of the integral $I$ by $(1/4)$ to find the average velocity: $\newline$ Average velocity = $(1/4) \times I$
Round to nearest thousandth: Now, we multiply the result of the integral $I$ by $(1/4)$ to find the average velocity: $\newline$ Average velocity = $(1/4) \times I$ Finally, we round the result to the nearest thousandth as instructed. Let's say the rounded value is $V$ .

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