Stewart 8th §6.5: Average Value of a Function

29 questions

--:--
0 / 29
Stewart 8th §6.5: Average Value of a Function 0/29
1 Integration and Accumulation of Change · Level 2
Find the average value of the function on the given interval. \(f(x) = 3x^2 + 8x\), \([-1, 2]\)
2 Integration and Accumulation of Change · Level 2
\(f(x) = \sqrt{x}\), \([0, 4]\)
3 Integration and Accumulation of Change · Level 2
\(g(x) = 3 \cos x\), \([-\dfrac{\pi}{2}, \dfrac{\pi}{2}]\)
4 Integration and Accumulation of Change · Level 3
\(g(t) = \dfrac{t}{\sqrt{3 + t^2}}\), \([1, 3]\)
5 Integration and Accumulation of Change · Level 3
\(f(t) = e^{\sin t} \cos t\), \([0, \dfrac{\pi}{2}]\)
6 Integration and Accumulation of Change · Level 3
\(f(x) = \dfrac{x^2}{(x^3 + 3)^2}\), \([-1, 1]\)
7 Integration and Accumulation of Change · Level 3
\(h(x) = \cos^4 x \sin x\), \([0, \pi]\)
8 Integration and Accumulation of Change · Level 3
\(h(u) = \dfrac{\ln u}{u}\), \([1, 5]\)
9 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · Level 3
\(f(x) = \dfrac{1}{x}\), \([1, 3]\)
11 Integration and Accumulation of Change · Level 3
\(f(x) = 2 \sin x - \sin 2x\), \([0, \pi]\)
12 Integration and Accumulation of Change · Level 3
\(f(x) = 2x e^{-x^2}\), \([0, 2]\)
13 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · Level 3
Find the average value of \(f\) on \([0, 8]\). [Figure: step function with values shown in a graph]
16 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · Level 2
Find the average value of the function \(f(x) = 1 + x^2\) on the interval \([-1, 2]\).
28 Integration and Accumulation of Change · 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 Integration and Accumulation of Change · 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.

Answered: 0 / 29