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-2 (a) Consider the functions f(x) = 100 – zª and F(x) = [*¸ ƒ(t) dt. i. Write a simplified form of F(x) = f* f(t) dt. ii. Verify that F'(x) = f(x) and F(-2) = 0. = (b) Now consider the function g(x) = √100-24. Unlike the previous function, there is no closed- form "nice" way to write the anti-derivative G(x) where G'(x) = g(x). Instead, use the Fundamental Theorem of Calculus Part II to find an integral-defined function G(x) such that G'(x) = g(x) and G(-2) = 0. A little more about the difference between these two cases: We get lucky and F(x) can be written in terms of usual combinations of our usual functions. However, G(x) cannot be written any more simply than as "the definite integral from t = -2 to t = x of the function √100-4". It gets a little more interesting - it's not that nobody has figured out how yet, but in fact it's known that we'll never have a simplification for G(x) in the same way we do for F(x). However, we will have a completely different way of looking at these functions in Math 141 that puts them on a level playing field.

-2 (a) Consider the functions f(x) = 100 – zª and F(x) = [*¸ ƒ(t) dt. i. Write a simplified form of F(x) = f* f(t) dt. ii. Verify that F'(x) = f(x) and F(-2) = 0. = (b) Now consider the function g(x) = √100-24. Unlike the previous function, there is no closed- form "nice" way to write the anti-derivative G(x) where G'(x) = g(x). Instead, use the Fundamental Theorem of Calculus Part II to find an integral-defined function G(x) such that G'(x) = g(x) and G(-2) = 0. A little more about the difference between these two cases: We get lucky and F(x) can be written in terms of usual combinations of our usual functions. However, G(x) cannot be written any more simply than as "the definite integral from t = -2 to t = x of the function √100-4". It gets a little more interesting - it's not that nobody has figured out how yet, but in fact it's known that we'll never have a simplification for G(x) in the same way we do for F(x). However, we will have a completely different way of looking at these functions in Math 141 that puts them on a level playing field.

Advanced Engineering Mathematics
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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a and b and the paragraph below b 

Cuestion 1 (a) Consider the functions f(x) = 100 – 2¹ and F(x) = f*¸ ƒ(1) dt. i. Write a simplified form of F(x) = f*"* f(t) dt. -2 ii. Verify that F'(x) = f(x) and F(-2) = 0. (b) Now consider the function g(x) = 100 x4. Unlike the previous function, there is no closed- form "nice" way to write the anti-derivative G(x) where G'(x) = g(x). Instead, use the Fundamental Theorem of Calculus Part II to find an integral-defined function G(x) such that G'(x) = g(x) and G(-2) = 0. A little more about the difference between these two cases: We get lucky and F(x) can be written in terms of usual combinations of our usual functions. However, G(x) cannot be written any more simply than as "the definite integral from t = −2 to t = x of the function √100 – 4". It gets a little more interesting it's not that nobody has figured out how yet, but in fact it's known that we'll never have a simplification for G(x) in the same way we do for F(x). However, we will have a completely different way of looking at these functions in Math 141 that puts them on a level playing field. —
Transcribed Image Text:Cuestion 1 (a) Consider the functions f(x) = 100 – 2¹ and F(x) = f*¸ ƒ(1) dt. i. Write a simplified form of F(x) = f*"* f(t) dt. -2 ii. Verify that F'(x) = f(x) and F(-2) = 0. (b) Now consider the function g(x) = 100 x4. Unlike the previous function, there is no closed- form "nice" way to write the anti-derivative G(x) where G'(x) = g(x). Instead, use the Fundamental Theorem of Calculus Part II to find an integral-defined function G(x) such that G'(x) = g(x) and G(-2) = 0. A little more about the difference between these two cases: We get lucky and F(x) can be written in terms of usual combinations of our usual functions. However, G(x) cannot be written any more simply than as "the definite integral from t = −2 to t = x of the function √100 – 4". It gets a little more interesting it's not that nobody has figured out how yet, but in fact it's known that we'll never have a simplification for G(x) in the same way we do for F(x). However, we will have a completely different way of looking at these functions in Math 141 that puts them on a level playing field. —
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