MIT18_014F10_pset8sols

MIT18_014F10_pset8sols - 18.014 Problem Set 8 Solutions...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 18.014 Problem Set 8 Solutions Total: 24 points Problem 1: Compute 1 xf (2 x ) dx given that f is continuous for all x , and f (0) = 1, f (0) = 3, f (1) = 5, f (1) = 2, f (2) = 7, f (2) = 4. Solution (4 points) Applying integration by parts (theorem 5.5), we have 1 xf (2 x ) 1 1 1 1 1 1 1 1 xf (2 x ) dx = f (2 x ) dx = f (2) f (2) + f (0) = . 2 2 2 4 4 2 We can use this theorem because x is differentiable with constant derivative 1 that is continuous and never changes sign, and f (2 x ) is continuous by hypothesis. Problem 2: Use the definition a x = e x log a to derive the following properties of general exponentials: (b) ( ab ) x = a x b x . (c) a x a y = a x + y . (d) ( a x ) y = ( a y ) x = a xy (e) Suppose a > 0, a = 1. Then y = a x if and only if x = log a y . Solution (4 points) (b) By the definition of the exponential function, part (ii) of theorem 3 of course notes M, part (i) of theorem 2 of course notes M, and the definition of the exponential function, we have ( ab ) x = e x log( ab ) = e x log( a )+ x log( b ) = e x log( a ) e x log( b ) = a x b x . (c) By the definition of the exponential function, part (i) of theorem 2 of course notes M, and the definition of the exponential function, we have x y x log( a ) y log( a ) ( x + y ) log( a ) x + y a a = e e = e = a . (d) After twice using the definition of the exponential function, using that the ex- ponential function and the logarithmic function are inverses, and again using the definition of the exponential function, we obtain ( a x ) y = e y log( a x ) = e y log( e x log a ) = e yx log( a ) = a xy ....
View Full Document

This note was uploaded on 01/18/2012 for the course MATH 18.014 taught by Professor Christinebreiner during the Fall '10 term at MIT.

Page1 / 6

MIT18_014F10_pset8sols - 18.014 Problem Set 8 Solutions...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online