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Section 6.1 Average Value (AI1)

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.

(a)

How far did the car travel in these 3 hours?

(b)

What was their average velocity over these 3 hours?

Activity 6.1.2.

Suppose a car drives due east at a velocity of \(v(t)=40+\sqrt{300t}\) miles per hour, where \(t\) is the number of hours after they start driving.

(a)

Which of the following expressions measures how far the car travels in these 3 hours?

  1. \(\displaystyle \displaystyle \int_0^2 70x \,dx + \int_2^3 40x \,dx\)

  2. \(\displaystyle \displaystyle \int_0^2 70 \,dx + \int_2^3 40 \,dx\)

  3. \(\displaystyle \displaystyle \int_0^2 70x \,dx + \int_0^1 40x \,dx\)

  4. \(\displaystyle \displaystyle \int_0^2 70 \,dx + \int_0^1 40 \,dx\)

  5. \(\displaystyle 70(2)+40(1)\)

(b)

What was their average velocity over these 3 hours?

  1. \(180\) mi

  2. \(180\) mph

  3. \(60\) mi

  4. \(60\) mph

Activity 6.1.3.

Which of the following is the average value of \(f(x)\) over the interval \([0,8]\text{?}\)

A plot of f(x).
Figure 88. 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. \(\displaystyle 2.375\)

Activity 6.1.4.

Consider a function \(f(x)\) which has average value \(v\) on the interval \([a,b]\text{.}\) Which of the following is most likely always true about \(v\text{?}\)

  1. \(v\) is the average of \(a\) and \(b\text{.}\)

  2. \(v\) is the average of \(f(a)\) and \(f(b)\text{.}\)

  3. \(v\) is the average of \(\min\{f(x)\}\) and \(\max\{f(x)\}\) on the interval \([a,b]\text{.}\)

  4. \(\displaystyle \displaystyle \int_a^b f(x)dx=\int_a^b vdx\)

Definition 6.1.5.

Given a function \(f(x)\) defined on \([a,b]\text{,}\) it's average value \(v\) is

\begin{equation*} v=\frac{1}{b-a}\int_a^b f(x) dx\text{.} \end{equation*}

Activity 6.1.6.

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. \(\displaystyle \int_{0}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\text{.}\)

  2. \(\displaystyle \displaystyle \frac{1}{3\pi}\int_{0}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)

  3. \(\displaystyle \displaystyle \frac{1}{4\pi}\int_{0}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)

  4. \(\displaystyle \displaystyle \int_{\pi}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)

  5. \(\displaystyle \displaystyle \frac{1}{3\pi}\int_{\pi}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)

  6. \(\displaystyle \displaystyle \frac{1}{4\pi}\int_{\pi}^{4\pi}\Big(x\cos(x^2)+x\Big) dx\)

Activity 6.1.7.

Find the average value of \(f(x)=x\cos(x^2)+x\) on the interval \([\pi, 4\pi]\) using the chosen expression from Activity 6.1.6.

Activity 6.1.8.

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

Activity 6.1.9.

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?

Subsection 6.1.1 Videos

Figure 89. Video: Compute the average value of a function on an interval