=S₁²= (a) Use the Fundamental Theorem of Calculus to show that arctan(1) = Sketch of function with n = 4 rectangles: (b) Express the signed area between y = 1 and the x-axis on the interval [0, 1] using any Riemann Sum you wish with n rectangles of equal base lengths for some arbitrary whole number n. Note that the summation notation is supposed to contain the variable n in it since we would like a general formula we can easily modify for different values of n. Sketch the function and the rectangles for n = 4. Use desmos https://www.desmos.com/calculator/oceoomwdiy to help with visualization and checking your answer (note that you need to update the function within desmos). Hint: Helpful questions to ask yourself at the end: Does the answer seem reasonable? Did I check computations by typing it into the desmos link above? π -4 1 x² + 1 -dx. Sigma notation for arbitrary n rectangles: Σ k= (c) Plug in the above Riemann Sum with n = 1000 into Desmos, Wolfram Alpha, or your calculator to get an approximate value for arctan (1) = 4 up to 6 decimals. No need to show work here. Note that this is an approximation, so it will not match arctan(1) exactly. arctan(1) ≈
=S₁²= (a) Use the Fundamental Theorem of Calculus to show that arctan(1) = Sketch of function with n = 4 rectangles: (b) Express the signed area between y = 1 and the x-axis on the interval [0, 1] using any Riemann Sum you wish with n rectangles of equal base lengths for some arbitrary whole number n. Note that the summation notation is supposed to contain the variable n in it since we would like a general formula we can easily modify for different values of n. Sketch the function and the rectangles for n = 4. Use desmos https://www.desmos.com/calculator/oceoomwdiy to help with visualization and checking your answer (note that you need to update the function within desmos). Hint: Helpful questions to ask yourself at the end: Does the answer seem reasonable? Did I check computations by typing it into the desmos link above? π -4 1 x² + 1 -dx. Sigma notation for arbitrary n rectangles: Σ k= (c) Plug in the above Riemann Sum with n = 1000 into Desmos, Wolfram Alpha, or your calculator to get an approximate value for arctan (1) = 4 up to 6 decimals. No need to show work here. Note that this is an approximation, so it will not match arctan(1) exactly. arctan(1) ≈
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
![Criteria for Success: I can solve conceptual questions related to Riemann sums, definite integrals, and
the Fundamental Theorem of Calculus that lie on the top half of Bloom's Taxonomy (analyze, evaluate, and
create).
Question: The goal of this question is to approximate arctan(1)
Riemann Sums.
(a) Use the Fundamental Theorem of Calculus to show that arctan(1) =
Sketch of function with n= 4 rectangles:
(b) Express the signed area between y = and the x-axis on the interval [0, 1] using any Riemann Sum
you wish with n rectangles of equal base lengths for some arbitrary whole number n. Note that the
summation notation is supposed to contain the variable n in it since we would like a general formula we
can easily modify for different values of n. Sketch the function and the rectangles for n = 4. Use desmos
https://www.desmos.com/calculator/oceoomwdiy to help with visualization and checking your
answer (note that you need to update the function within desmos). Hint: Helpful questions to ask
yourself at the end: Does the answer seem reasonable? Did I check computations by typing it into the
desmos link above?
4
||
(and therefore also 7) using
So² 12² + 1d²
arctan(1) ~
-dx.
Sigma notation for arbitrary n rectangles:
Σ
k=
(c) Plug in the above Riemann Sum with n = 1000 into Desmos, Wolfram Alpha, or your calculator to get
an approximate value for arctan(1) = up to 6 decimals. No need to show work here. Note that this is
an approximation, so it will not match arctan(1) exactly.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F97b652fc-cbf7-4903-96da-3b4650a71f93%2Fa753428d-c743-4748-b291-78a146ba663d%2Fdn6l02_processed.png&w=3840&q=75)
Transcribed Image Text:Criteria for Success: I can solve conceptual questions related to Riemann sums, definite integrals, and
the Fundamental Theorem of Calculus that lie on the top half of Bloom's Taxonomy (analyze, evaluate, and
create).
Question: The goal of this question is to approximate arctan(1)
Riemann Sums.
(a) Use the Fundamental Theorem of Calculus to show that arctan(1) =
Sketch of function with n= 4 rectangles:
(b) Express the signed area between y = and the x-axis on the interval [0, 1] using any Riemann Sum
you wish with n rectangles of equal base lengths for some arbitrary whole number n. Note that the
summation notation is supposed to contain the variable n in it since we would like a general formula we
can easily modify for different values of n. Sketch the function and the rectangles for n = 4. Use desmos
https://www.desmos.com/calculator/oceoomwdiy to help with visualization and checking your
answer (note that you need to update the function within desmos). Hint: Helpful questions to ask
yourself at the end: Does the answer seem reasonable? Did I check computations by typing it into the
desmos link above?
4
||
(and therefore also 7) using
So² 12² + 1d²
arctan(1) ~
-dx.
Sigma notation for arbitrary n rectangles:
Σ
k=
(c) Plug in the above Riemann Sum with n = 1000 into Desmos, Wolfram Alpha, or your calculator to get
an approximate value for arctan(1) = up to 6 decimals. No need to show work here. Note that this is
an approximation, so it will not match arctan(1) exactly.
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