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Section 14.1 Average Value (AI1)

Subsection 14.1.1 Activities

Activity 14.1.

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.
(a)
How far did the car travel in these 3 hours?
  1. \(110\) miles
  2. \(150\) miles
  3. \(180\) miles
  4. \(220\) miles
(b)
What was its average velocity over these 3 hours?
  1. \(55\) miles per hour
  2. \(60\) miles per hour
  3. \(70\) miles per hour
  4. \(75\) miles per hour

Activity 14.2.

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

Observation 14.3.

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*}
For example, the following calculuation confirms the previous activity:
\begin{equation*} \frac{1}{3-0}\int_0^3 (30+20t)\,dt\text{.} \end{equation*}

Definition 14.4.

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*}

Activity 14.5.

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

Activity 14.6.

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

Activity 14.7.

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

Activity 14.8.

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?

Activity 14.9.

Which of the following is the average value of \(f(x)\) over the interval \([0,8]\text{?}\)
A plot of f(x).
Figure 173. Plot of \(f(x)\text{.}\)
Note \(f(x)=\begin{cases} 1, & 0\leq x\leq 3 \\ 4, & 3 < x \leq 6 \\ 2, & 6 < x \leq 8 \end{cases}\text{.}\)
  1. \(\displaystyle 4\)
  2. \(\displaystyle 2\)
  3. \(\displaystyle \displaystyle \frac{7}{3}\)
  4. \(\displaystyle 19\)
  5. \(\displaystyle 2.375\)

Subsection 14.1.2 Videos

Figure 174. Video: Compute the average value of a function on an interval

Subsection 14.1.3 Exercises

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