Curve Sketching - 18.01 Calculus Jason Starr Fall 2005...

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18.01 Calculus Jason Starr Fall 2005 Lecture 9. September 29, 2005 Homework. Problem Set 2 all of Part I and Part II. Practice Problems. Course Reader: 2B-1, 2B-2, 2B-4, 2B-5. 1. Application of the Mean Value Theorem. A real-world application of the Mean Value Theorem is error analysis . A device accepts an input signal x and returns an output signal y . If the input signal is always in the range 1 / 2 x 1 / 2 and if the output signal is, 1 y = f ( x ) = 1 + x + x 2 + x 3 , what precision of the input signal x is required to get a precision of ± 10 3 for the output signal? If the ideal input signal is x = a , and if the precision is ± h , then the actual input signal is in the range a h x a + h . The precision of the output signal is f ( x ) f ( a ) . By the Mean Value | | Theorem, f ( x ) f ( a ) = f ( c ) , x a for some c between a and x . The derivative f ( x ) is, f ( x ) = (3 x 2 + 2 x + 1) . (1 + x + x 2 + x 3 ) 2 For 1 / 2 x 1 / 2, this is bounded by, 3(1 / 2) 2 + 2(1 / 2) + 1 | f ( x ) = 7 . 04 . | ≤ [1 + ( 1 / 2) + ( 1 / 2) 2 + ( 1 / 2) 3 ] 2 Thus the Mean Value Theorem gives, f ( x ) f ( a ) = f ( c 7 . 04 x a 7 . 04 h. | | | ) || x a | | | Therefore a precision for the input signal of, h = 10 3 / 7 . 04 10 4 guarantees a precision of 10 3 for the output signal. 2. First derivative test. A function f ( x ) is increasing , respectively decreasing , if f ( a ) is less than f ( b ), resp. greater than f ( b ), whenever a is less than b . In symbols, f is increasing, respectively decreasing, if f ( a ) < f ( b ) whenever a < b, resp. f ( a ) > f ( b ) whenever a < b.
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18.01 Calculus Jason Starr Fall 2005 If f ( a ) is less than or equal to f ( b ), resp. greater than or equal to f ( b ), whenever a is less than b , then f ( x ) is non-decreasing , resp. non-increasing . If f ( x ) is increasing, the
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Curve Sketching - 18.01 Calculus Jason Starr Fall 2005...

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