Stewart 8th §6.5: Average Value of a Function

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Stewart 8th §6.5: Average Value of a Function 0/29
1 Average Value · Level 2
Find the average value of the function on the given interval. \(f(x) = 3x^2 + 8x\), \([-1, 2]\)
2 Average Value · Level 2
\(f(x) = \sqrt{x}\), \([0, 4]\)
3 Average Value · Level 2
\(g(x) = 3 \cos x\), \([-\dfrac{\pi}{2}, \dfrac{\pi}{2}]\)
4 Average Value · Level 3
\(g(t) = \dfrac{t}{\sqrt{3 + t^2}}\), \([1, 3]\)
5 Average Value · Level 3
\(f(t) = e^{\sin t} \cos t\), \([0, \dfrac{\pi}{2}]\)
6 Average Value · Level 3
\(f(x) = \dfrac{x^2}{(x^3 + 3)^2}\), \([-1, 1]\)
7 Average Value · Level 3
\(h(x) = \cos^4 x \sin x\), \([0, \pi]\)
8 Average Value · Level 3
\(h(u) = \dfrac{\ln u}{u}\), \([1, 5]\)
9 Average Value - MVT · Level 3
(a) Find the average value of \(f\) on the given interval.
(b) Find \(c\) such that \(f_{\text{ave}} = f(c)\).
(c) Sketch the graph of \(f\) and a rectangle whose area is the same as the area under the graph of \(f\). \(f(x) = (x - 3)^2\), \([2, 5]\)

Enter your answer directly below each part above.

10 Average Value - MVT · Level 3
\(f(x) = \dfrac{1}{x}\), \([1, 3]\)
11 Average Value - MVT · Level 3
\(f(x) = 2 \sin x - \sin 2x\), \([0, \pi]\)
12 Average Value - MVT · Level 3
\(f(x) = 2x e^{-x^2}\), \([0, 2]\)
13 Average Value · Level 3
If \(f\) is continuous and \(\displaystyle\int_{1}^{3} f(x) d x = 8\), show that \(f\) takes on the value 4 at least once on the interval \([1, 3]\).
14 Average Value · Level 3
Find the numbers \(b\) such that the average value of \(f(x) = 2 + 6x - 3x^2\) on the interval \([0, b]\) is equal to 3.
15 Average Value · Level 3
Find the average value of \(f\) on \([0, 8]\). [Figure: step function with values shown in a graph]
16 Average Value · Level 3
The velocity graph of an accelerating car is shown.
(a) Use the Midpoint Rule to estimate the average velocity of the car during the first 12 seconds.
(b) At what time was the instantaneous velocity equal to the average velocity?

Enter your answer directly below each part above.

17 Average Value · Level 3
In a certain city the temperature (in \(^{\circ}\)F) \(t\) hours after 9 AM was modeled by the function \(T(t) = 50 + 14 \sin \dfrac{\pi t}{12}\). Find the average temperature during the period from 9 AM to 9 PM.
18 Average Value · Level 4
The velocity \(v\) of blood that flows in a blood vessel with radius \(R\) and length \(l\) at a distance \(r\) from the central axis is \(v(r) = \dfrac{P}{4 \eta l}(R^2 - r^2)\) where \(P\) is the pressure difference between the ends of the vessel and \(\eta\) is the viscosity of the blood. Find the average velocity (with respect to \(r\)) over the interval \(0 \leq r \leq R\). Compare the average velocity with the maximum velocity.
19 Average Value · Level 3
The linear density in a rod 8 m long is \(\dfrac{12}{\sqrt{x + 1}}\) kg/m, where \(x\) is measured in meters from one end of the rod. Find the average density of the rod.
20 Average Value · Level 4
(a) A cup of coffee has temperature 95\(^{\circ}\)C and takes 30 minutes to cool to 61\(^{\circ}\)C in a room with temperature 20\(^{\circ}\)C. Use Newton's Law of Cooling (Section 3.
8) to show that the temperature of the coffee after \(t\) minutes is \(T(t) = 20 + 75 e^{-k t}\) where \(k \approx 0.02\).
(b) What is the average temperature of the coffee during the first half hour?

Enter your answer directly below each part above.

21 Average Value · Level 4
In Example 3.8.1 we modeled the world population in the second half of the 20th century by the equation \(P(t) = 2560 e^{0.017185 t}\). Use this equation to estimate the average world population during this time period.
22 Average Value · Level 4
If a freely falling body starts from rest, then its displacement is given by \(s = \dfrac{1}{2} g t^2\). Let the velocity after a time \(T\) be \(v_T\). Show that if we compute the average of the velocities with respect to \(t\) we get \(v_{\text{ave}} = \dfrac{1}{2} v_T\), but if we compute the average of the velocities with respect to \(s\) we get \(v_{\text{ave}} = \dfrac{2}{3} v_T\).
23 Average Value · Level 3
Use the result of Exercise 5.5.83 to compute the average volume of inhaled air in the lungs in one respiratory cycle.
24 Average Value · Level 4
Use the diagram to show that if \(f\) is concave upward on \([a, b]\), then \(f_{\text{ave}} > f\left(\dfrac{a + b}{2}\right)\).
25 Average Value - MVT · Level 4
Prove the Mean Value Theorem for Integrals by applying the Mean Value Theorem for derivatives (see Section 4.
2) to the function \(F(x) = \displaystyle\int_{a}^{x} f(t) d t\).
26 Average Value · Level 4
If \(f_{\text{ave}}[a, b]\) denotes the average value of \(f\) on the interval \([a, b]\) and \(a < c < b\), show that \(f_{\text{ave}}[a, b] = \dfrac{c - a}{b - a} f_{\text{ave}}[a, c] + \dfrac{b - c}{b - a} f_{\text{ave}}[c, b]\).
27 Average Value · Level 2
Find the average value of the function \(f(x) = 1 + x^2\) on the interval \([-1, 2]\).
28 Average Value - MVT for Integrals · Level 3
Since \(f(x) = 1 + x^2\) is continuous on \([-1, 2]\), the Mean Value Theorem for Integrals says there is a number \(c\) in \([-1, 2]\) such that \(\displaystyle\int_{-1}^2 (1 + x^2) d x = f(c)[2 - (-1)]\). Find \(c\).
29 Average Value - Velocity · Level 2
Show that the average velocity of a car over a time interval \([t_1, t_2]\) is the same as the average of its velocities during the trip.

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