Chapter 5.2, Problem 46E

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516

Chapter
Section

### Calculus: Early Transcendental Fun...

7th Edition
Ron Larson + 1 other
ISBN: 9781337552516
Textbook Problem

# Numerical Reasoning Consider a trapezoid of area 4 bounded by the graphs of y = x , y = 0 , x = 1 and x = 3 .(a) Sketch the region.(b) Divide the interval [1, 3] into n subintervals of equal width and show that the endpoints are 1 < 1 + 1 ( 2 n ) < ⋯ < 1 + ( n − 1 ) ( 2 n ) < 1 + n ( 2 n ) .(c) S how that S ( n ) = ∑ i = 1 n [ 1 + ( i − 1 ) ( 2 n ) ] ( 2 n ) .(d) Show that S ( n ) = ∑ i = 1 n [ 1 + i ( 2 n ) ] ( 2 n ) .(e) Find s ( n ) and S ( n ) for n = 5 ,10,50, and 100.(f) Show that lim n → ∞ s ( n ) = lim n → ∞ S ( n ) = 4 .

(a)

To determine

To graph: The trapezoidal region made by binding of graphs of y=x,y=0,x=3 and x=1

Explanation

Given:

There is a trapezoidal region formed by lines y=x,y=0,x=3Â andÂ x=1

Graph:

The graph of y=x is a diagonal line passing through the origin

(b)

To determine

To prove: The endpoints of the graph made in part (a) are 0<1(2n)<...<(n1)(2n)<n(2n)

(c)

To determine

To prove: The relation s(n)=i=1n[1+(i1)(2n)](2n)

(d)

To determine

To prove: The relation S(n)=i=1n[1+(i)(2n)](2n)

(e)

To determine

The values for s(n) and S(n) for n=5,10,50,100 if s(n)=i=1n[(i1)(2n)](2n) and S(n)=i=1n[(i)(2n)](2n)

(f)

To determine

To prove: That limns(n)=limnS(n)=4

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