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HW1 - M i Show that ² b a f ≤ M b-a 2 MATH 186 –...

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MATH 186 – WINTER 2009 HOMEWORK SET 1 Due Friday 01/23 Be kind to your grader, please staple your work! What you need to know: - Partitions of an interval - Upper and lower sums - Definition of sup and inf - Definition of integrable functions - Vectors and matrices What you shouldn’t forget: - Definition and properties of continuous functions Ex 1. Let f : [ - 1 , 2] R be the function defined by f ( x ) = - 1 , - 1 x < 0 5 , x = 0; 3 , 0 < x 1 1 , 1 < x 2 . Compute 2 - 1 f using the definition (i.e., with ε ’s, partitions, upper sums, etc.). Ex 2. Let f : [ a, b ] R be an integrable function. In particular, f ( x ) M for every x

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Unformatted text preview: M . i) Show that ² b a f ≤ M ( b-a ) . 2 MATH 186 – WINTER 2009 HOMEWORK SET 1 ii) Show that if f is continuous on [ a, b ], then ± b a f ( x ) dx = ( b-a ) f ( ξ ) , for some ξ ∈ [ a, b ]. Ex 3. Let A be the 2 × 2 matrix 1 2-3 4 . i) Find a vector ±v 1 such that A±v 1 = [ 1 ]. ii) Find a vector ±v 2 such that A±v 2 = [ 1 ]. iii) Find a matrix B such that AB = 1 1 . iv) What is the relationship between AB and BA in this case? From Spivak: Chapter 13: Ex # 31 (a), 37. Please recycle!...
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HW1 - M i Show that ² b a f ≤ M b-a 2 MATH 186 –...

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