Calculus Problems and Solutions
= Let the integrating factor be In
Let U be and dV =
=du=2cos x sin2xdx , v = -cos x.
Hence = +
Let cosx be U. then we have
=du = -sinx dx
=cos2x + C
=(x2z +2z2 -5z)-1
 Approximate using (a) left endpoints, (b) right endpoints (c) midpoints and n= 1000 partitioning intervals. Explain which technology you used in the estimate. Fill out the table
Let’s get the interval of the points. That is
b=1 and a =0 n=1000
Then = 0.001
The interval is 0.001
And the width of each rectangle formed will be = 0.25
Left points = choose xi* =xi
= 0 0.001 + = 0.0001579.
Right points =[f(xi+1) f(xi+2)+ f(xi+3)
=0 + = 0.001887
Midpoint = reftp[oints +right points then you divide by 2.
=where m =| f’’(x)|
= = 1.3706 * 107
|1.3706 * 107|
Let f(x) =. Sketch an approximate graph of F on the interval [ 0, ] by filling out this table and then plotting the corresponding points.
Interval = = 0.01772.
Fill the table using n =100 intervals. Explain what technology you used. I Integrated the equation using normal formula with respect to dt. I went ahead to get the point intervals which helped me to get the widths.
 Use the error estimation formula to determine how many partitioning intervals is needed to ensure that the midpoint rule approximations for = are accurate within 0.01
= but we get m from | F’’(x)|
M= 0.000048 after integration and the limits be 2 and 3.. Hence the error = 0.02.
 (a) Graph y=
Y2 = 1-x2 the table below shows the points to plot
(b) Partitioning into how many intervals does insure that can be approximated using the midpoint rule to within 0.01?
( c) Then use the technology to compute an approximation of A = accurate to within 0.01
Get the left points = R4 =f(xi*)
1* 0.01+ 1*0.001 +1*0.001 +0 = 0.003
Right points = [f (xi+1) f(xi+2)+ f(xi+3)
= by integration method the right point is = 0.4563
Hence the midpoint = left + right and then divide by 2.
0.003+ 0.4563 = 0.22965
(d) Finally calculate the number P= ( A- ). What number does 6p approximate?
P =( A- ). Where by A = 0.22965.then we substitute
=(0.22965 ) = 0.099441366
P = 0.099441366
If p =0.099441366 what of 6p?