Question

Displacement from velocity The following functions describe the velocity of a car (in $\mathrm{mi} / \mathrm{hr}$ ) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval $[0, t]$, where $0 \leq t \leq 3$ (Check your book to see figure) $$v(t)=\left\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 2 \\50 & \text { if } 2< t \leq 2.5 \\44 & \text { if } 2.5< t \leq 3\end{array}\right.$$

Name: Problem 75, Chapter 5: Calculus: Early Transcendentals
Uploaded: 2020-05-22T15:58:01-08:00
Duration: 2 min 37 s
Channel: Jacob Steele
Description: Displacement from velocity The following functions describe the velocity of a car (in mi / hr ) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval [0, t], where 0 ≤t ≤3 (Check your book to see figure) v(t)={ 30 if 0 ≤ t ≤ 2 50 if 2< t ≤ 2.5 44 if 2.5< t ≤ 3 .

   Displacement from velocity The following functions describe the velocity of a car (in $\mathrm{mi} / \mathrm{hr}$ ) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval $[0, t]$, where $0 \leq t \leq 3$
(Check your book to see figure)
$$v(t)=\left\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 2 \\50 & \text { if } 2< t \leq 2.5 \\44 & \text { if } 2.5< t \leq 3\end{array}\right.$$

Calculus: Early Transcendentals

William Briggs, Lyle… 3rd Edition

Chapter 5, Problem 75

Instant Answer

Solved by Expert Jacob Steele

05/22/2020

Step 1

Since the velocity function is a piecewise function, we will integrate each piece separately over its respective interval. Show more…

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Displacement from velocity The following functions describe the velocity of a car (in $\mathrm{mi} / \mathrm{hr}$ ) moving along a straight highway for a 3-hr interval. In each case, find the function that gives the displacement of the car over the interval $[0, t]$, where $0 \leq t \leq 3$ (Check your book to see figure) $$v(t)=\left\{\begin{array}{ll}30 & \text { if } 0 \leq t \leq 2 \\50 & \text { if } 2< t \leq 2.5 \\44 & \text { if } 2.5< t \leq 3\end{array}\right.$$

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Key Concepts

Relationship between velocity and displacement

Displacement is determined by the integral of velocity with respect to time. This relationship comes from the fundamental concept that integrating a rate of change (velocity) gives the total accumulated change (displacement) over an interval.

Definite Integration

Definite integration is the process of calculating the cumulative sum of a function's values over a specific interval. It is used to find the net change in position by evaluating the area under the velocity-time curve between two points in time.

Piecewise Functions

Piecewise functions are defined by different expressions over different subintervals of the domain. In problems involving piecewise-defined velocity functions, integration must be performed separately over each interval where the function has a distinct expression before summing the results to obtain the overall displacement.

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