Find the average value [math]M_a^b(f)[/math] of [math]f[/math] on the interval [math][a,b][/math], where

• [math]f(x)=x^2-2x+1[/math] and [math][a,b]= [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].

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].

Each of the propositions proposition, proposition, proposition, and proposition corresponds to one of the basic properties of the definite integral as they are enumerated in Theorems theorem through theorem. In general, the proof of each is obtained by checking the special case [math]a=b[/math] separately and then using the formula

together with the appropriate property of the integral.

• Prove proposition
• Prove proposition
• Prove proposition

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,

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.

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?

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.

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].

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

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].