1007_Tut4_Sol - MATH 1007 A Tutorial#4 Solutions 1[4 marks...

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MATH 1007 A Tutorial #4 Nov 10, 2006 Solutions 1. [4 marks] Use substititution to find the required integrals: (a) (4 + x 2 ) 10 (2 x ) dx Hint: Set u = 4 + x 2 Solution: Then du dx = 2 x = du = 2 x dx (4 + x 2 ) 10 (2 x ) dx = u 10 du = 1 11 u 11 + C = 1 11 (4 + x 2 ) 11 + C (b) 2 x 4 - 1 (8 x 3 ) dx Hint: Set u = 2 x 4 - 1 Solution: Then du dx = 8 x 3 = du = 8 x 3 dx 2 x 4 - 1 (8 x 3 ) dx = u du = u 1 / 2 du = 1 3 / 2 u 3 / 2 + C = 2 3 (2 x 4 - 1) 3 / 2 + C 1
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2. [4 marks] Use the Fundamental Theorem of Calculus to solve the fol- lowing questions: (a) 4 1 (1 + 6 x ) dx Solution: 4 1 (1 + 6 x ) dx = x + 3 x 2 4 1 = (4 + 3 · 16) - (1 + 3 · 1) = (4 + 48) - 4 = 48 (b) If g ( x ) = e x 0 sin 3 t dt , find g ( x ). Hint: let u = e x so that dg dx = dg du · du dx Solution: dg dx = dg du · du dx = d du u 0 sin 3 t dt · d dx ( e x ) = (sin 3 u )( e x ) = e x sin 3 ( e x ) 2
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3. [2 marks] Consider the function f ( x ) = 1 x from x = 1 to x = 5. Use 4 approximating rectangles/intervals to give an expression for the area under f (a) using right end points of intervals (b) using left end points of intervals
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