For this process we'll use the function f(x) = -x² + 6x - 7. You might find the following links useful: Link to a Desmos calculator for definite integrals which displays shaded area Link to a Desmos calculator which calculates rectangle approximations of areas • Sketch the graph of f (x), with area shaded under the curve and above the x-axis over the interval [2, 4]. • Write a definite integral which represents the area of the shaded region. Don't solve it yet! • Draw a left endpoint rectangle approximation of the area with n = 4 and describe what each of the following are in your illustration: o Ax; ox for i= 1, 2, 3, 4; o The area estimate given by this approximation. (If you use technology to compute this, explain what computations the technology is doing.) • Draw a left endpoint rectangle approximation of the area with n = 8. o How do Ax and the change? o If you start with only one of your eight rectangles, on the left side of your interval, and add them in one at a time, until all of the approximating rectangles are present, how much area do you add at each step? (Don't actually calculate this estimate for eight rectangles, just describe it in terms of the information you have, and explain how you would calculate it.) o How is this related to Part 1 of the Fundamental Theorem of Calculus? • Explain how each of the following are related to each other for this function, and how they are each related to the question of how much area is under the curve over this interval. Be sure to write what each of them are. o Definite integral; o Indefinite integral; o General antiderivative; o Particular antiderivative. • Use Part 2 of the Fundamental Theorem of Calculus to find the exact area. (You can compare this with the value given by the Desmos calculator linked above, but please compute it by hand.) • Explain how the ideas of "accumulation" and "net change" are at work when we use Part 2 of the FTC.
For this process we'll use the function f(x) = -x² + 6x - 7. You might find the following links useful: Link to a Desmos calculator for definite integrals which displays shaded area Link to a Desmos calculator which calculates rectangle approximations of areas • Sketch the graph of f (x), with area shaded under the curve and above the x-axis over the interval [2, 4]. • Write a definite integral which represents the area of the shaded region. Don't solve it yet! • Draw a left endpoint rectangle approximation of the area with n = 4 and describe what each of the following are in your illustration: o Ax; ox for i= 1, 2, 3, 4; o The area estimate given by this approximation. (If you use technology to compute this, explain what computations the technology is doing.) • Draw a left endpoint rectangle approximation of the area with n = 8. o How do Ax and the change? o If you start with only one of your eight rectangles, on the left side of your interval, and add them in one at a time, until all of the approximating rectangles are present, how much area do you add at each step? (Don't actually calculate this estimate for eight rectangles, just describe it in terms of the information you have, and explain how you would calculate it.) o How is this related to Part 1 of the Fundamental Theorem of Calculus? • Explain how each of the following are related to each other for this function, and how they are each related to the question of how much area is under the curve over this interval. Be sure to write what each of them are. o Definite integral; o Indefinite integral; o General antiderivative; o Particular antiderivative. • Use Part 2 of the Fundamental Theorem of Calculus to find the exact area. (You can compare this with the value given by the Desmos calculator linked above, but please compute it by hand.) • Explain how the ideas of "accumulation" and "net change" are at work when we use Part 2 of the FTC.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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sub parts have been solved
![For this process we'll use the function f(x)
Link to a Desmos calculator for definite integrals which displays shaded area ª
Link to a Desmos calculator which calculates rectangle approximations of areas
• Sketch the graph of f (x), with area shaded under the curve and above the x-axis over the interval [2, 4].
• Write a definite integral which represents the area of the shaded region. Don't solve it yet!
-
-x² + 6x - 7. You might find the following links useful:
O
x for i 1, 2, 3, 4;
=
• Draw a left endpoint rectangle approximation of the area with n = = 4 and describe what each of the following are in your illustration:
ο Δα;
• The area estimate given by this approximation. (If you use technology to compute this, explain what computations the technology is doing.)
• Draw a left endpoint rectangle approximation of the area with n = = 8.
o How do Ax and the change?
o If you start with only one of your eight rectangles, on the left side of your interval, and add them in one at a time, until all of the approximating
rectangles are present, how much area do you add at each step? (Don't actually calculate this estimate for eight rectangles, just describe it in
terms of the information you have, and explain how you would calculate it.)
o How is this related to Part 1 of the Fundamental Theorem of Calculus?
Explain how each of the following are related to each other for this function, and how they are each related to the question of how much area is
under the curve over this interval. Be sure to write what each of them are.
o Definite integral;
o Indefinite integral;
o General antiderivative;
o Particular antiderivative.
• Use Part 2 of the Fundamental Theorem of Calculus to find the exact area. (You can compare this with the value given by the Desmos calculator
linked above, but please compute it by hand.)
• Explain how the ideas of "accumulation" and "net change" are at work when we use Part 2 of the FTC.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F85721d2a-3ed1-464f-b67e-e1169344369a%2F4496ce65-9cbe-47bf-b000-b8cc227df242%2F1i99tpp_processed.png&w=3840&q=75)
Transcribed Image Text:For this process we'll use the function f(x)
Link to a Desmos calculator for definite integrals which displays shaded area ª
Link to a Desmos calculator which calculates rectangle approximations of areas
• Sketch the graph of f (x), with area shaded under the curve and above the x-axis over the interval [2, 4].
• Write a definite integral which represents the area of the shaded region. Don't solve it yet!
-
-x² + 6x - 7. You might find the following links useful:
O
x for i 1, 2, 3, 4;
=
• Draw a left endpoint rectangle approximation of the area with n = = 4 and describe what each of the following are in your illustration:
ο Δα;
• The area estimate given by this approximation. (If you use technology to compute this, explain what computations the technology is doing.)
• Draw a left endpoint rectangle approximation of the area with n = = 8.
o How do Ax and the change?
o If you start with only one of your eight rectangles, on the left side of your interval, and add them in one at a time, until all of the approximating
rectangles are present, how much area do you add at each step? (Don't actually calculate this estimate for eight rectangles, just describe it in
terms of the information you have, and explain how you would calculate it.)
o How is this related to Part 1 of the Fundamental Theorem of Calculus?
Explain how each of the following are related to each other for this function, and how they are each related to the question of how much area is
under the curve over this interval. Be sure to write what each of them are.
o Definite integral;
o Indefinite integral;
o General antiderivative;
o Particular antiderivative.
• Use Part 2 of the Fundamental Theorem of Calculus to find the exact area. (You can compare this with the value given by the Desmos calculator
linked above, but please compute it by hand.)
• Explain how the ideas of "accumulation" and "net change" are at work when we use Part 2 of the FTC.
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