Piecewise velocity The velocity of a (fast) automobile on a...

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function $$v(t)=\left\{\begin{array}{ll} 3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45 \end{array}\right.$$, where $t$ is measured in seconds and $v$ has units of meters/second. a. Graph the velocity function, for $0 \leq t \leq 70 .$ When is the velocity a maximum? When is the velocity zero? b. What is the distance traveled by the automobile in the first 30 s? c. What is the distance traveled by the automobile in the first $60 \mathrm{s} ?$ d. What is the position of the automobile when $t=75 ?$

Piecewise velocity The velocity of a (fast) automobile on a straight highway is given by the function $$v(t)=\left\{\begin{array}{ll} 3 t & \text { if } 0 \leq t<20 \\ 60 & \text { if } 20 \leq t<45 \\ 240-4 t & \text { if } t \geq 45 \end{array}\right.$$, where $t$ is measured in seconds and $v$ has units of meters/second.
a. Graph the velocity function, for $0 \leq t \leq 70 .$ When is the velocity a maximum? When is the velocity zero?
b. What is the distance traveled by the automobile in the first 30 s?
c. What is the distance traveled by the automobile in the first $60 \mathrm{s} ?$
d. What is the position of the automobile when $t=75 ?$

Key Concepts

Piecewise Functions

A piecewise function is defined by different expressions over different intervals of its domain. This concept is essential when modeling situations where the rule changes at specific points, requiring careful consideration of each segment when analyzing or graphing the function.

Graphing Piecewise Functions

Graphing piecewise functions involves plotting each individual segment on its relevant interval and ensuring continuity or indicated breaks at the transition points. This skill is crucial for visualizing how a function behaves over time or across different conditions.

Definite Integration

Definite integration is used to calculate the accumulated change of a quantity over an interval, such as distance traveled from a velocity function. It involves summing the contributions from each segment of a function—especially important when the function is defined piecewise.

Relationship Between Velocity and Position

The velocity of an object is the derivative of its position with respect to time, so finding the position involves integrating the velocity. This relationship allows one to determine displacement and analyze motion by accumulating area under the velocity curve.

Analyzing Function Behavior

Determining key features of a function, such as maximum values or zeros, involves analyzing each piece of the function and identifying where these conditions occur. This includes solving for when the function’s output reaches a peak or when it crosses zero, which is fundamental in understanding the dynamics of the modeled system.

Transcript

00:01 Okay, so we have our problem here.

00:04 We're given a car, and it's driving at meters with a unit of meters per second, and we're using this piecewise function for velocity.

00:14 So 3t between 0 and 20 seconds, 60 between 20 and 45 seconds, and then 240 minus 4t above 45 seconds.

00:23 So if we do that, if we were to graph that, between 0 and 20 we get this.

00:39 So it goes up to 60.

00:41 Then it's going to stay 60.

00:46 Then it stays 60 until 45.

00:49 Then it drops down back to zero when it gets to the end.

00:53 Because of the 240 minus 4t.

00:55 So we have maximums at 20 to 20 and to 45.

01:05 And we have minimums or zeros at, t equals 0 and t equals 60.

01:15 From there, what we're going to do is we're going to move on.

01:20 We're going to move on to the next part.

01:22 The next part asks us how far the automobile has driven in the first 30 seconds.

01:32 So to figure that out, we're going to figure out how far it goes in 30 seconds.

01:36 So to do that, we're going to integrate.

01:38 So we're going to find the integral of 3t from 0 to 20, which is equal to, 3t squared over 2, which is equal to 320 squared over 2 minus 3 .0 squared over 2, which eventually becomes 600 meters.

02:08 And we can do the same thing for 60.

02:12 So we're going to integrate 60 from 20 to 30.

02:19 So we get 60 times 30.

02:24 Well, this becomes 60t.

02:26 So we get 60 times 30 minus 60 times 20, which is equal to 600 meters...

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