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Calculus for Team-Based Inquiry Learning 2025 Edition

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Section 6.1 Average Value (AI1)

Subsection 6.1.1 Activities

Activity 6.1.1.

Suppose a car drives due east at 70 miles per hour for 2 hours, and then slows down to 40 miles per hour for an additional hour.
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(a)
How far did the car travel in these 3 hours?
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  1. \(110\) miles
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  2. \(150\) miles
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  3. \(180\) miles
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  4. \(220\) miles
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(b)
What was its average velocity over these 3 hours?
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  1. \(55\) miles per hour
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  2. \(60\) miles per hour
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  3. \(70\) miles per hour
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  4. \(75\) miles per hour
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Activity 6.1.2.

Suppose instead the car starts with a velocity of \(30\) miles per hour, and increases velocity linearly according to the function \(v(t)=30+20t\) so its velocity after three hours is \(90\) miles per hour.
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(a)
How can we model the car’s distance traveled using calculus?
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  1. Integrate velocity, because position is the rate of change of velocity.
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  2. Integrate velocity, because velocity is the rate of change of position.
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  3. Differentiate velocity, because position is the rate of change of velocity.
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  4. Differentiate velocity, because velocity is the rate of change of position.
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(b)
Then, which of these expressions is a mathematical model for the car’s distance traveled after 3 hours?
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  1. \(\displaystyle \int (30+20t)\,dt\)
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  2. \(\displaystyle \int (30t+10t^2)\,dt\)
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  3. \(\displaystyle \int_0^3 (30+20t)\,dt\)
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  4. \(\displaystyle \int_0^3 (30t+10t^2)\,dt\)
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(c)
How far did the car travel in these 3 hours?
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  1. \(110\) miles
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  2. \(150\) miles
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  3. \(180\) miles
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  4. \(220\) miles
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(d)
Thus, what was its average velocity over three hours?
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  1. \(55\) miles per hour
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  2. \(60\) miles per hour
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  3. \(70\) miles per hour
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  4. \(75\) miles per hour
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Observation 6.1.3.

To obtain the average velocity of an object traveling with velocity \(v(t)\) for \(a\leq t\leq b\text{,}\) we may find its distance traveled by calculating \(\int_a^b v(t)\text{.}\) Thus, the average velocity is obtained by dividing by the time \(b-a\) elapsed:
\begin{equation*} \frac{1}{b-a}\int_a^b v(t)\,dt\text{.} \end{equation*}
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For example, the following calculuation confirms the previous activity:
\begin{equation*} \frac{1}{3-0}\int_0^3 (30+20t)\,dt\text{.} \end{equation*}
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Definition 6.1.4.

Given a function \(f(x)\) defined on \([a,b]\text{,}\) it’s average value is defined to be
\begin{equation*} \frac{1}{b-a}\int_a^b f(x) \,dx\text{.} \end{equation*}
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Activity 6.1.5.

(a)
Which of the following expressions represent the average value of \(f(x)=-12 \, x^{2} + 8 \, x + 4\) over the interval \([-1, 2]\text{?}\)
  1. \(\displaystyle \displaystyle \frac{1}{3}\int_{-1}^{2}\left(-12 \,x^{2} + 8 \, x + 4 \right) dx\)
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  2. \(\displaystyle \displaystyle \frac{-1}{1}\int_{1}^{2}\left(-12 \,x^{2} + 8 \, x + 4 \right) dx\)
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  3. \(\displaystyle \displaystyle \frac{1}{2}\int_{1}^{2}\left(-12 \,x^{2} + 8 \, x + 4 \right) dx\)
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  4. \(\displaystyle \displaystyle \frac{-1}{4}\int_{-1}^{2}\left(-12 \,x^{2} + 8 \, x + 4 \right) dx\)
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(b)
Show that the average value of \(f(x)=-12 \, x^{2} + 8 \, x + 4\) over the interval \([-1, 2]\) is \(-4\text{.}\)
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Activity 6.1.6.

(a)
Which of the following expressions represent the average value of \(f(x)=x\cos(x^2)+x\) on the interval \([\pi, 4\pi]\text{?}\)
  1. \(\displaystyle \displaystyle \frac{1}{3\pi}\int_{0}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)
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  2. \(\displaystyle \displaystyle \frac{1}{4\pi}\int_{0}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)
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  3. \(\displaystyle \displaystyle \frac{1}{3\pi}\int_{\pi}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)
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  4. \(\displaystyle \displaystyle \frac{1}{4\pi}\int_{\pi}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)
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(b)
Find the average value of \(f(x)=x\cos(x^2)+x\) on the interval \([\pi, 4\pi]\) using the chosen expression.
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Activity 6.1.7.

Find the average value of \(\displaystyle g(t)=\frac{t}{t^2+1}\) on the interval \([0, 4]\text{.}\)
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Activity 6.1.8.

A shot of a drug is administered to a patient and the quantity of the drug in the bloodstream over time is \(q(t)=3te^{-0.25t}\text{,}\) where \(t\) is measured in hours and \(q\) is measured in milligrams. What is the average quantity of this drug in the patient’s bloodstream over the first 6 hours after injection?
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Activity 6.1.9.

Which of the following is the average value of \(f(x)\) over the interval \([0,8]\text{?}\)
A plot of f(x).
Figure 117. Plot of \(f(x)\text{.}\)
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Note \(f(x)=\begin{cases} 1, & 0\leq x\leq 3 \\ 4, & 3 < x \leq 6 \\ 2, & 6 < x \leq 8 \end{cases}\text{.}\)
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  1. \(\displaystyle 4\)
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  2. \(\displaystyle 2\)
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  3. \(\displaystyle \displaystyle \frac{7}{3}\)
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  4. \(\displaystyle 19\)
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  5. \(\displaystyle 2.375\)
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Subsection 6.1.2 Videos

Figure 118. Video: Compute the average value of a function on an interval
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Subsection 6.1.3 Exercises

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