Question 3. In this question you will examine relationships between integrals of the form [ S² (1) Compute these integrals for p = 2, p = 1/2, and p = 1. What do you notice? of the form -1/P dr. TP da and integrals
Question 3. In this question you will examine relationships between integrals of the form [ S² (1) Compute these integrals for p = 2, p = 1/2, and p = 1. What do you notice? of the form -1/P dr. TP da and integrals
Chapter3: Polynomial Functions
Section3.5: Mathematical Modeling And Variation
Problem 7ECP: The kinetic energy E of an object varies jointly with the object’s mass m and the square of the...
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Transcribed Image Text:·S=
Question 3. In this question you will examine relationships between integrals of the form
· 6² ==
(1) Compute these integrals for p = 2, p = 1/2, and p = 1. What do you notice?
of the form
x-¹/P dx.
TP dr and integrals
(2) Graph these areas on a single plot, and explain how they are related geometrically (i.e. in the picture).
![[Continued from previous page]
(3) Complete the text of this theorem to make it a true statement about the convergence/divergence of these
integrals:
1₁0 x
Theorem. Let p> 0 be a real number (the same p for both integrals). Then the integrals
1-¹/P dr... [have this convergence/divergence behaviour].
I
IP dx and
For example, do they both converge? both diverge? Can one converge but the other diverge? Do these
statements depend on p? Or is there no relationship at all?
(4) Prove your theorem algebraically by computing these integrals for a general p.
Hint. You may need to break it up into cases of p. If you are lost in the abstraction,
compute more cases of p, like in part 1, until you see what is happening. Don't include
these extra computations in your solution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fac5348b7-e00b-4a82-86d7-ca896458528d%2F29d358b2-2814-407c-b2e2-bf42d2db9c03%2Fwdn3cir_processed.png&w=3840&q=75)
Transcribed Image Text:[Continued from previous page]
(3) Complete the text of this theorem to make it a true statement about the convergence/divergence of these
integrals:
1₁0 x
Theorem. Let p> 0 be a real number (the same p for both integrals). Then the integrals
1-¹/P dr... [have this convergence/divergence behaviour].
I
IP dx and
For example, do they both converge? both diverge? Can one converge but the other diverge? Do these
statements depend on p? Or is there no relationship at all?
(4) Prove your theorem algebraically by computing these integrals for a general p.
Hint. You may need to break it up into cases of p. If you are lost in the abstraction,
compute more cases of p, like in part 1, until you see what is happening. Don't include
these extra computations in your solution.
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