Question

You want to estimate the area underneath the graph of a positive function by using four rectangles of equal width. The rectangles that must give the best estimate of this are those with height obtained from the:

- Left endpoints
- Midpoints
- Right endpoints
- This can’t be determined.

write down which option you think gives **the best
estimate.**

Your statements can be about the velocity of the particle but could also be about its position or its acceleration.

Answer #1

**Generally the best and closest approximation is given by
the mid-points, since it will average out the values of left and
right hand point, which will provide closer
approximation.**

**For increasing function, left approximation will be
lower than right one. For decreasing function, left approximation
will be higher than the right one.**

**The mid-points generally provide the best approximate by
averaging the value of left and right hand estimate in both types
of function (i.e. increasing and decreasing)**

**Note - Post any doubts/queries in comments
section.**

Estimate the area under the graph of f(x)=1/x from x=1
to x=2 using four approximating rectangles and left endpoints.

30) Estimate the area under the graph of f(x)= 1/x+4 over the
interval [1,3] using five approximating rectangles and
right endpoints.
Rn=
Repeat the approximation using left endpoints.
Ln=

Estimate the area under the graph of f(x)=1/(x+2) over the
interval [0,3]using eight approximating rectangles and
right endpoints.
Rn=
Repeat the approximation using left endpoints.
Ln=

Estimate the area under the graph of f(x)=x2+3x from x=1 to x=4
using 33 approximating rectangles and left endpoints.
Approximation =

Estimate the area under the graph of f(x)=x2+4x from x=2 to x=10
using 4 approximating rectangles and left endpoints. Approximation
=

Use a finite approximation to estimate the area under the graph
of the given function on the stated interval as instructed. 1) f(x)
= x 2 between x = 3 and x = 7 using a left sum with four rectangles
of equal width.

Estimate the area under the graph of f(x)=f(x)=3x^2+6x+7 over
the interval [0,5] using ten approximating rectangles and
right endpoints.
Repeat using left endpoints.

Estimate the area under the graph of f(x)=1/(x+3) over the
interval [−2,1] using ten approximating rectangles and
right endpoints. a=-2,b=1,n=10
Rn=?
Repeat the approximation using left endpoints.
Ln?
Accurate to 4 places.

You are given the function f(x) = 4 - x^2. Using four
approximating rectangles and left endpoints, estimate the area
under the graph of f(x) from x=0 to x=2. Estimated area
= Repeat the above calculation using right endpoints.
Estimated area = Your answers should have two digits
after the decimal.

Estimate the area under the graph of
f(x)=1x+1f(x)=1x+1 over the interval [2,7][2,7] using eight
approximating rectangles and right endpoints.
Rn=Rn=
Repeat the approximation using left
endpoints.
Ln=Ln=
Round answers to 4
places. Remember not to round too early in your
calculations.

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