Let I = where f is the function whose graph is shown. (a) Use the graph to find

Question

Let

I =

where f is the function whose graph is shown.

(a) Use the graph to find L2, R2 and M2.

(b) Are these underestimates or overestimates of I?

L2 is an underestimate.L2 is an overestimate.

R2 is an underestimate.R2 is an overestimate.

M2 is an underestimate.M2 is an overestimate.

(c) Use the graph to find T2.
T2 =  

How does it compare with I?
T2  ? < >

(d) For any value of n, list the numbers Ln, Rn, Mn, Tn and I in increasing order. (Enter your answers as a comma-separated list. Enter your answer using the variables rather than numerical values.)

L, M, T, I, R

Explanation / Answer

a) Divide the region into 2 reactangles

L2 = f(0)*2 +f(2)*2 = 1*2 +5*2 = 12

R2 = f(2)*2 +f(4)*2 = 5*2 + 7*2 = 24

M2 = f(1)*2 +f(3)*2 = 3*2 +6.4*2 =18.8

b) Imagine the rectangles used to get L2,R2 and M2 , we have

L2 is a underestimate as it doesn't take much area into consideration

R2 is a overestimate as R2 takes extra area into consideration

M2 is a overestimate as it takes extra area into consideration

c) T2 = area of trapeziums = (1+5)*2/2 +(7+5)*2/2 = 19

it underestimates as it doesn't include some area in the second trapezium

d) we need to give the order according to the estimated errors we get

L<T <I <M <R

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