Use the expressions for left and right sums and the table below.
right-hand sum = summation (n above and i =1 on the bottom) f(ti) * change in t = f(t1) * change in t + f(t2) * change in t + .... + f (tn) * change in t
left-hand sum = summation (n above and i = 0 on the bottom) f(ti) * change in t = f (t0) * change in t + f (t1) * change in t + ... = f (tn - 1) * change in t.
Table: t 0 4 8 12 16
f (t) 26 24 23 21 18
If the table messed up, the points are (0, 26), (4, 24) , (8, 23) , (12, 21), (16, 18)
a.) If n = 4, what is change in t? What are t0, t1, t2, t3, t4? What are f (t0), f (t1), f (t2), f (t3), f(t4)?
t0= f (t0) =
t1 = f (t1) =
t2 = f (t2) =
t3 = f (t3) =
t4 = f (t4) =
b.) Find the left and right sums using n = 4.
Left sum =
Right sum =
c.) If n = 2, what is change in t? What are t0, t1, t2? What are f (t0), f (t1), f(t2)?
t0 = f (t0) =
t1 = f (t1) =
t2 = f (t2) =
d.) Find the left and right sums using n = 2
Left sum =
Right sum =
Please show work and explain. I have no idea where to start from.
arghhhh cant figure it out
same here Anonymous
To find the values for each part, let's go step by step:
a) If n = 4, we have 4 intervals. The change in t can be calculated by taking the difference between two consecutive t values:
change in t = t1 - t0 = 4 - 0 = 4
The t values for each interval are as follows:
t0 = 0
t1 = 4
t2 = 8
t3 = 12
t4 = 16
The f(t) values corresponding to each t value can be read from the table:
f(t0) = 26
f(t1) = 24
f(t2) = 23
f(t3) = 21
f(t4) = 18
b) To calculate the left and right sums, we use the given expressions:
Left sum = summation (4 above and i = 0 on the bottom) f(ti) * change in t
= f(t0) * change in t + f(t1) * change in t + f(t2) * change in t + f(t3) * change in t
= 26 * 4 + 24 * 4 + 23 * 4 + 21 * 4
= 104 + 96 + 92 + 84
= 376
Right sum = summation (4 above and i = 1 on the bottom) f(ti) * change in t
= f(t1) * change in t + f(t2) * change in t + f(t3) * change in t + f(t4) * change in t
= 24 * 4 + 23 * 4 + 21 * 4 + 18 * 4
= 96 + 92 + 84 + 72
= 344
c) If n = 2, we have 2 intervals. The change in t can be calculated again by taking the difference between two consecutive t values:
change in t = t1 - t0 = 8 - 0 = 8
The t values for each interval are as follows:
t0 = 0
t1 = 8
t2 = 16
The f(t) values corresponding to each t value can be read from the table:
f(t0) = 26
f(t1) = 23
f(t2) = 18
d) To calculate the left and right sums, we use the given expressions:
Left sum = summation (2 above and i = 0 on the bottom) f(ti) * change in t
= f(t0) * change in t + f(t1) * change in t
= 26 * 8 + 23 * 8
= 208 + 184
= 392
Right sum = summation (2 above and i = 1 on the bottom) f(ti) * change in t
= f(t1) * change in t + f(t2) * change in t
= 23 * 8 + 18 * 8
= 184 + 144
= 328
In summary:
a) Change in t = 4
t0 = 0, t1 = 4, t2 = 8, t3 = 12, t4 = 16
f(t0) = 26, f(t1) = 24, f(t2) = 23, f(t3) = 21, f(t4) = 18
b) Left sum = 376
Right sum = 344
c) Change in t = 8
t0 = 0, t1 = 8, t2 = 16
f(t0) = 26, f(t1) = 23, f(t2) = 18
d) Left sum = 392
Right sum = 328
To answer these questions, we first need to understand the given expressions for the left and right sums:
Right-hand sum: It is the sum of the products of the function values (f(ti)) and the change in t for each interval, where i varies from 1 to n.
left-hand sum: It is the sum of the products of the function values (f(ti)) and the change in t for each interval, where i varies from 0 to n-1.
Using the given table and expressions, let's solve the questions step by step:
a) If n = 4, what is the change in t? What are t0, t1, t2, t3, t4? What are f(t0), f(t1), f(t2), f(t3), f(t4)?
Change in t can be determined by finding the difference between consecutive t-values. Here, the t-values are 0, 4, 8, 12, and 16. The change in t is the difference between any two consecutive t-values. In this case, the change in t is 4 since the consecutive t-values have a constant difference of 4.
t0 = 0
t1 = 4
t2 = 8
t3 = 12
t4 = 16
From the table, we can determine the corresponding f(ti) values:
f(t0) = 26
f(t1) = 24
f(t2) = 23
f(t3) = 21
f(t4) = 18
So:
t0 = 0, f(t0) = 26
t1 = 4, f(t1) = 24
t2 = 8, f(t2) = 23
t3 = 12, f(t3) = 21
t4 = 16, f(t4) = 18
b) Find the left and right sums using n = 4.
Left sum is the sum of the products of f(ti) and the change in t for each interval i, where i varies from 0 to n-1.
Left sum = f(t0) * change in t + f(t1) * change in t + f(t2) * change in t + f(t3) * change in t
= 26 * 4 + 24 * 4 + 23 * 4 + 21 * 4
Right sum is the sum of the products of f(ti) and the change in t for each interval i, where i varies from 1 to n.
Right sum = f(t1) * change in t + f(t2) * change in t + f(t3) * change in t + f(t4) * change in t
= 24 * 4 + 23 * 4 + 21 * 4 + 18 * 4
c) If n = 2, what is the change in t? What are t0, t1, t2? What are f(t0), f(t1), f(t2)?
Similar to the previous question, we need to find the change in t, t-values, and f(ti) values.
In this case, the t-values are 0, 4, and 8.
Change in t = 4
t0 = 0
t1 = 4
t2 = 8
From the table, we can determine the corresponding f(ti) values:
f(t0) = 26
f(t1) = 24
f(t2) = 23
So:
t0 = 0, f(t0) = 26
t1 = 4, f(t1) = 24
t2 = 8, f(t2) = 23
d) Find the left and right sums using n = 2.
Left sum = f(t0) * change in t + f(t1) * change in t
= 26 * 4 + 24 * 4
Right sum = f(t1) * change in t + f(t2) * change in t
= 24 * 4 + 23 * 4
To calculate the actual values, substitute the change in t and the corresponding f(ti) values into the expressions.