2) We approximate the area under the graph of y = f(x), above the x-axis, over the interval [0, 4] by using n rectangles and left-hand endpoints. (Leave n as a parameter.) a) What is the width, Ax, of each rectangle used in this approximation? Ax =. b) Find a formula for the gridpoint xg. (The only variables appearing in your expression should be k and n.) Xk = c) Write the Riemann sum for this area approximation in Sigma notation. Sigma notation:
2) We approximate the area under the graph of y = f(x), above the x-axis, over the interval [0, 4] by using n rectangles and left-hand endpoints. (Leave n as a parameter.) a) What is the width, Ax, of each rectangle used in this approximation? Ax =. b) Find a formula for the gridpoint xg. (The only variables appearing in your expression should be k and n.) Xk = c) Write the Riemann sum for this area approximation in Sigma notation. Sigma notation:
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
2a. 2b. 2c.
![For questions 1 and 2, consider the function f(x) =
+ x +1 on the interval [0, 4]
12
10
8
6
2
-3
-2
-1
2
3
1) We approximate the area under the graph of y = f(x), above the x-axis, over the interval [0, 4] by using n = 4 rectangles and
left-hand endpoints.
a)
What
s the width, Ax, of each rectangle used in this approximation?
人るl4= 2 (6i-)x= f(Xo) + flx) + flx)tfCX3)
c0,4) and n=4
Given the interval [aib] =
b-a
4-0
= |
4
こ
n
Ax
AX = 1
b)
List the gridpoints, xo, X1, X2, X3, X4, used in this approximation.
メi+l=
Xit AX
'os
X, = xo+ AX = 0+1 = 1
Xz= 2
ド2: Y」t Ax= Itl=2
X3 = X2+ AX= 2+1 = 3
X3 = 3
X4=4
X4 - x3+AX= 3+1= 4
Gridpoints: 0, 1, 2, 3, 4
c)
Illustrate this area approximation by sketching the corresponding rectangles in the graph above.
Math 1151
Written Homework 5
Autumn 2020
d)
Is this area approximation going to be an overestimate or an underestimate. Explain.
We hnow that wnen the grapn is con care down Gne second derivative is negative )
then me iine Will ve above +hne grapn and the
approx imati on is an overestimare, otherwise it is an
underestimare, cit the graph is con cOne up), From the grapn we
can observe that
the graph ot fU)< x?+ xtl,on [0,4] wnich is given above as concave
vp on c0,41. so the area approximation tor hrs grapn is goins to be an
overestimate .
e)
Calculate this area approximation.
f(XoJ = fca) = fl0)=/
+ (K J= fla) = flI)- 914
+CXZ) = f(as= PlZ) = 4
f(X3) = fla)= P(3)- a5)4
ニ](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7323f76-5a9c-46b6-9930-86cefe6607d1%2F35192932-bb65-4c52-af5f-95f80eaeba69%2Fy2o2rsh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:For questions 1 and 2, consider the function f(x) =
+ x +1 on the interval [0, 4]
12
10
8
6
2
-3
-2
-1
2
3
1) We approximate the area under the graph of y = f(x), above the x-axis, over the interval [0, 4] by using n = 4 rectangles and
left-hand endpoints.
a)
What
s the width, Ax, of each rectangle used in this approximation?
人るl4= 2 (6i-)x= f(Xo) + flx) + flx)tfCX3)
c0,4) and n=4
Given the interval [aib] =
b-a
4-0
= |
4
こ
n
Ax
AX = 1
b)
List the gridpoints, xo, X1, X2, X3, X4, used in this approximation.
メi+l=
Xit AX
'os
X, = xo+ AX = 0+1 = 1
Xz= 2
ド2: Y」t Ax= Itl=2
X3 = X2+ AX= 2+1 = 3
X3 = 3
X4=4
X4 - x3+AX= 3+1= 4
Gridpoints: 0, 1, 2, 3, 4
c)
Illustrate this area approximation by sketching the corresponding rectangles in the graph above.
Math 1151
Written Homework 5
Autumn 2020
d)
Is this area approximation going to be an overestimate or an underestimate. Explain.
We hnow that wnen the grapn is con care down Gne second derivative is negative )
then me iine Will ve above +hne grapn and the
approx imati on is an overestimare, otherwise it is an
underestimare, cit the graph is con cOne up), From the grapn we
can observe that
the graph ot fU)< x?+ xtl,on [0,4] wnich is given above as concave
vp on c0,41. so the area approximation tor hrs grapn is goins to be an
overestimate .
e)
Calculate this area approximation.
f(XoJ = fca) = fl0)=/
+ (K J= fla) = flI)- 914
+CXZ) = f(as= PlZ) = 4
f(X3) = fla)= P(3)- a5)4
ニ
![2) We approximate the area under the graph of y = f(x), above the x-axis, over the interval [0, 4] by using n rectangles and
left-hand endpoints. (Leave n as a parameter.)
a)
What is the width, Ax, of each rectangle used in this approximation?
Ax =.
b)
Find a formula for the gridpoint x. (The only variables appearing in your expression should be k and n.)
X =
c)
Write the Riemann sum for this area approximation in Sigma notation.
Sigma notation:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fd7323f76-5a9c-46b6-9930-86cefe6607d1%2F35192932-bb65-4c52-af5f-95f80eaeba69%2Fyuna9gyv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2) We approximate the area under the graph of y = f(x), above the x-axis, over the interval [0, 4] by using n rectangles and
left-hand endpoints. (Leave n as a parameter.)
a)
What is the width, Ax, of each rectangle used in this approximation?
Ax =.
b)
Find a formula for the gridpoint x. (The only variables appearing in your expression should be k and n.)
X =
c)
Write the Riemann sum for this area approximation in Sigma notation.
Sigma notation:
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