"(a) Determine the average value of the following functions on the interval [1,2]. Give your answers first in exact form, and then in decimal form with at least three decimal digits.
y = x
, Average Value = (exact)
Average Value ? (decimal)
y = x2
, Average Value = (exact)
Average Value ? (decimal)
y = x3
, Average Value = (exact)
Average Value ? (decimal)
y = x4
, Average Value = (exact)
Average Value ? (decimal)
(b) Determine the average value of the following functions on the interval [0,?/2]. Give your answers first in exact form, and then in decimal form with at least three decimal digits.
y = cos x
, Average Value = (exact)
Average Value ? (decimal)
y = cos2x
, Average Value = (exact)
Average Value ? (decimal)
y = cos3x
, Average Value = (exact)
Average Value ? (decimal)
y = cos4x
, Average Value = (exact)
Average Value ? (decimal)
(c) The average values in part (a) increased, while the average values in part (b) decreased. The reason for this is that
(i) the interval [1,2] is shorter than the interval [0,?/2].
(ii) each function in part (a) is an increasing function of x in [1,2], while each function in part (b) is a decreasing function of x in [0,?/2].
(iii)When y > 1, we have y < y2 < y3 < y4, while when 0 < y < 1, we have y > y2 > y3 > y4."
