Question 4.2.7: A Sum of Function Values at Equally Spaced x ’s Sum the valu...
A Sum of Function Values at Equally Spaced x ’s
Sum the values of f(x)=3 x^2-4 x+2 evaluated at x = 1.05, x = 1.15, x = 1.25, . . . , x = 2.95.
You will need to think carefully about the x’s. The distance between successive x-values is 0.1, and there are 20 such values. (Be sure to count these for yourself.) Notice that we can write the x’s in the form 0.95 + 0.1i , for i = 1, 2, . . . , 20.
We now have
\sum_{i=1}^{20} f(0.95+0.1 i)=\sum_{i=1}^{20}\left[3(0.95+0.1 i)^2-4(0.95+0.1 i)+2\right]=\sum_{i=1}^{20}\left(0.03 i^2+0.17 i+0.9075\right) Multiply out terms.
=0.03 \sum_{i=1}^{20} i^2+0.17 \sum_{i=1}^{20} i+\sum_{i=1}^{20} 0.9075 From Theorem 2.2.
=0.03 \frac{20(21)(41)}{6}+0.17 \frac{20(21)}{2}+0.9075(20) From Theorem 2.1 (i), (ii) and (iii).
= 139.95.
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