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.
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
Using the trapezoidal rule, the area A is approximated by
4
∑(xk+1-xk)*(f(xk+1)+f(xk))/2
k=1
= 2(-7/2) + 3(-7) + 4(-21/2) + 1(-14) = -84
The average value is A/(20-10) = -8.4
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 + ...
Well, I sure hope this function isn't a knock-knock joke, because I don't think it's going to be very funny. Let's give it a shot anyway!
To find the Trapezoidal Sum Approximation, we'll first need to calculate the sum of the areas of the trapezoids formed by the intervals between the given points.
The formula for the area of a trapezoid is (base1 + base2) * height / 2. In this case, the bases will be the function values at each point, and the height will be the difference between the x-values.
Here are the calculations for each trapezoid:
Trapezoid 1:
Base1 = -2, Base2 = -5, Height = 12 - 10 = 2
Area1 = (-2 + -5) * 2 / 2 = -7
Trapezoid 2:
Base1 = -5, Base2 = -9, Height = 15 - 12 = 3
Area2 = (-5 + -9) * 3 / 2 = -21
Trapezoid 3:
Base1 = -9, Base2 = -12, Height = 19 - 15 = 4
Area3 = (-9 + -12) * 4 / 2 = -42
Trapezoid 4:
Base1 = -12, Base2 = -16, Height = 20 - 19 = 1
Area4 = (-12 + -16) * 1 / 2 = -14
Now, let's add up all the areas:
Total area = Area1 + Area2 + Area3 + Area4 = -7 + -21 + -42 + -14 = -84
Finally, to find the average value of the function over the given interval, we divide the total area by the width of the interval:
Average value = -84 / (20 - 10) = -84 / 10 = -8.4
So, to answer your question, the average value of the function, as estimated by the Trapezoidal Sum Approximation, is -8.4. Looks like option C) -8.400 is the correct answer. Well done!
To estimate the average value of the function using Trapezoidal Sum Approximation, we need to calculate the average of the function values at the given points.
The formula for the Trapezoidal Sum Approximation is:
Average value = (f(a) + 2f(x1) + 2f(x2) + ... + 2f(xn-1) + f(b)) / (2n)
where f(a) and f(b) are the function values at the endpoints, n is the number of intervals, and f(xi) are the function values at the intermediate points.
In this case, the endpoints are a = 10 and b = 20, and the given points are:
x: 10; 12; 15; 19; 20
f(x): -2; -5; -9; -12; -16
To calculate the average value, we need to calculate the sum of the function values at the given points:
Sum = -2 + 2*(-5) + 2*(-9) + 2*(-12) + (-16)
= -2 - 10 - 18 - 24 - 16
= -70
The number of intervals, n, is equal to the number of given points minus 1:
n = 5 - 1 = 4
Now we can substitute the values into the formula:
Average value = (-2 + 2*(-5) + 2*(-9) + 2*(-12) + (-16)) / (2*4)
= -70 / 8
= -8.75
Therefore, the average value of the function with a Trapezoidal Sum Approximation is approximately -8.75.
So the correct answer is A) -8.750.
To estimate the average value of the function using a Trapezoidal Sum Approximation, follow these steps:
1. Calculate the width of each interval:
The width of each interval is the difference between consecutive x-values. In this case, the intervals are [10,12], [12,15], [15,19], and [19,20]. The widths are 2, 3, 4, and 1, respectively.
2. Calculate the average of the function values at the endpoints of each interval:
For each interval, add the values of f(x) at the endpoints and divide by 2. In this case, the averages for the intervals are:
(f(10) + f(12)) / 2 = (-2 + -5) / 2 = -7/2 = -3.5
(f(12) + f(15)) / 2 = (-5 + -9) / 2 = -7/2 = -3.5
(f(15) + f(19)) / 2 = (-9 + -12) / 2 = -21/2 = -10.5
(f(19) + f(20)) / 2 = (-12 + -16) / 2 = -14/2 = -7
3. Calculate the sum of the averages multiplied by the widths:
Multiply each average by its corresponding width and sum them up. In this case:
(-3.5 * 2) + (-3.5 * 3) + (-10.5 * 4) + (-7 * 1) = -7 - 10.5 - 42 + (-7) = -51.5
4. Divide the sum by the total width of the interval:
Divide the sum from step 3 by the total width of the interval, which is the sum of the widths. In this case:
2 + 3 + 4 + 1 = 10
5. The result is the average value of the function:
Divide the result from step 4 by the total width of the interval. In this case:
-51.5 / 10 = -5.15
So, the estimated average value of the function, using Trapezoidal Sum Approximation, is approximately -5.15.
None of the provided answer options exactly match this value, but the closest option is D) -5.500.