⧼exchistory⧽
9 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
[math]
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[/math]
Find the average value [math]M_a^b(f)[/math] of [math]f[/math] on the interval [math][a,b][/math], where
- = [0,2]</math>.
- [math]f(x) = 2x^3[/math] and [math][a,b] = [-1,1][/math].
- [math]f(x) = \frac1x[/math] and [math][a,b] = [1,2][/math].
- [math]f(x) = \frac1x[/math] and [math][a,b] = [1,n][/math], where [math]n[/math] is a positive integer.
- [math]f(x) = \sin x[/math] and [math][a,b] = [0,\pi][/math].
- [math]f(x) = \ln x[/math] and [math][a,b] = [1,5][/math].
BBy Bot
Nov 03'24
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[/math]
In each of the following find [math]M_a^b(f)[/math], draw the graph of [math]f[/math], and superimpose on the graph a rectangle with base [math][a,b][/math] and area equal to the area under the curve [math]y=f(x)[/math] between [math]a[/math] and [math]b[/math].
- [math]f(x) = x^2[/math], [math]a = -1[/math], and [math]b = 1[/math].
- [math]f(x) = x^3[/math], [math]a = 0[/math], and [math]b = 1[/math].
- [math]f(x) = 4-(x-1)^2[/math], [math]a = 0[/math], and [math]b = 3[/math].
- [math]f(x) = e^x[/math], [math]a=0[/math], and [math]b=2[/math].
- [math]f(x) = \cos x[/math], [math]a=0[/math], and [math]b=\frac{\pi}2[/math].
BBy Bot
Nov 03'24
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[/math]
Each of the propositions \ref{thm 8.1.2}, \ref{thm 8.1.3}, \ref{thm 8.1.4}, and \ref{thm 8.1.5} corresponds to one of the basic properties of the definite integral as they are enumerated in Theorems \ref{thm 4.4.1} through \ref{thm 4.4.5}. In general, the proof of each is obtained by checking the special case [math]a=b[/math] separately and then using the formula
[[math]]
M_a^b(f) = \frac1{b-a} \int_a^b f(x)\;dx,
\quad \mbox{for $a \lt b$}
,
[[/math]]
together with the appropriate property of the integral.
- Prove \ref{thm 8.1.2}
- Prove \ref{thm 8.1.3}
- Prove \ref{thm 8.1.5}.
BBy Bot
Nov 03'24
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[/math]
A stone dropped from a cliff [math]400[/math] feet high falls to the bottom with a constant acceleration equal to [math]32[/math] feet per second per second. That is,
[[math]]
a(t) = v^\prime(t) = s^{\prime\prime}(t) = 32
,
[[/math]]
where the direction of increasing [math]s[/math] is downward. If the stone is dropped at time [math]t=0[/math], find the time it takes to reach the bottom of the cliff, and the mean velocity during the fall.
BBy Bot
Nov 03'24
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[/math]
A typist's speed over an interval from [math]t=0[/math] to [math]t=4[/math] hours increases as she warms up and then decreases as she gets tired. Measured in words per minute, suppose that her speed is given by [math]v(t) = 6[4^2-(t-1)^2][/math]. Find her speed at the beginning, at the end, her maximum speed, and her average speed over the [math]4[/math]-hour interval. How many words did she type during the [math]4[/math] hours?
BBy Bot
Nov 03'24
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[/math]
A particle moves during the interval of time from [math]t=1[/math] second to [math]t=3[/math] seconds with a velocity given by [math]v(t) = t^2+2t+1[/math] feet per second. Find the total distance that the particle has moved and also the average velocity.
BBy Bot
Nov 03'24
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[/math]
For each of the functions and intervals in Problem Exercise, find a number [math]c[/math] such that [math]a \lt c \lt b[/math] and [math]M_a^b(f) = f(c)[/math].
BBy Bot
Nov 03'24
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[/math]
An arbitrary linear function [math]f[/math] is defined by [math]f(x) = Ax+B[/math] for some constants [math]A[/math] and [math]B[/math]. Show that
[[math]]
M_a^b(f) = \frac{f(a)+f(b)}2
.
[[/math]]
BBy Bot
Nov 03'24
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[/math]
Let [math]x(t)[/math] be the number of bacteria in a culture at time [math]t[/math], and let [math]x_0=x(0)[/math]. The number grows at a rate proportional to the number present, and doubles in a time interval of length [math]T[/math]. Find an expression for [math]x(t)[/math] in terms of [math]x_0[/math] and [math]T[/math], and find the average number of bacteria present over the time interval [math][0,T][/math].