The function is continuous on the interval [10, 20] with some of its values given in the table above. Estimate the average value of the function with a Trapezoidal Sum Approximation, using the intervals between those given points.

Question

x 10;12;15;19;20
f(x) -2;-5;-9;-12;-16

A) -8.750
B) -7.000
C) -8.400
D) -5.500

Answer 1

DRAW IT !!
get area of trapezoids
(12-10)(-5-2)/2+(15-12)(-9-5)/2+(19-15)(-12-9)/2+(20-19)(-15-12)/2
divide by the length to get average height
20-10 = 10 is length

(2)(-7)/2 + (3)(-14)/2 + ...

Answer 2

Using the trapezoidal rule, the area A is approximated by
4
∑(x_k+1-x_k)*(f(x_k+1)+f(x_k))/2
k=1

= 2(-7/2) + 3(-7) + 4(-21/2) + 1(-14) = -84

The average value is A/(20-10) = -8.4