1.)If m ? f ( x ) ? M for a ? x ? b , where m is the absolute minimum and M is t
ID: 2886161 • Letter: 1
Question
1.)If m ? f(x) ? M for a ? x ? b, where m is the absolute minimum and M is the absolute maximum of f on the interval [a, b], then
m(b ? a) ?
? M(b ? a).
Use this property to estimate the value of the integral.
?/12
smaller value?
larger value?
2.)Find
if
f(x) =
2.)a) Use the definition to find an expression for the area under the curve
y = x3
from 0 to 1 as a limit.
lim n??
(b) The following formula for the sum of the cubes of the first n integers is proved in Appendix E. Use it to evaluate the limit in part (a).
.
b f(x) dx aExplanation / Answer
1)
?/12<x<?/9
=>?/4<3x<?/3
=>tan(?/4)<tan(3x)<tan(?/3)
=>1<tan(3x)<?3
=>5<5tan(3x)<5?3
=>5(?/9 -?/12)??[?/12 to ?/9]5tan(3x) dx?5?3(?/9 -?/12)
=>(5?/36)??[?/12 to ?/9]5tan(3x) dx?(5?3 ?/36)
smaller value=(5?/36)
larger value=(5?3 ?/36)
================================================
2)
?[0 to 8]f(x) dx =?[0 to 6]f(x) dx+?[6 to 8]f(x) dx
=>?[0 to 8]f(x) dx =?[0 to 6]6 dx+?[6 to 8] x dx
=>?[0 to 8]f(x) dx =|[0 to 6]6x+|[6 to 8] (1/2)x2
=>?[0 to 8]f(x) dx =6(6-0)+ (1/2)(82-62)
=>?[0 to 8]f(x) dx =36+ (1/2)(64-36)
=>?[0 to 8]f(x) dx =36+14
=>?[0 to 8]f(x) dx =50
============================================
2)
a)
a=0 ,b=1 ,?x=[b-a]/n=(1-0)/n=1/n
xi=a+i?x
=>xi=0+i(1/n)
=>xi=(i/n)
area = lim[n->?] ?[i=1 to n] (xi)3?x
area = lim[n->?] ?[i=1 to n] (i/n)3(1/n)
area = lim[n->?] ?[i=1 to n] (i3/n4)
b)
area = lim[n->?] (n2(n+1)2/22n4)
area = lim[n->?] (1/4)(1+(1/n))2
area = (1/4)(1+0)2
area = 1/4
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