275_review_2

2 is it true that if f g is integrable on a b then

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2. Is it true that if f + g is integrable on [ a, b ] then both f and g must be integrable there? 3. Show that the following function is integrable on [0 , 2]: f ( x ) = ( x if x 6 = 1 4 if x = 1 Hint: this is almost continuous, except at x = 1. 4. More general version: show that if f is an integrable function on [ a, b ] and g is equal to f except for maybe finitely many points in [ a, b ] then g is also integrable. Hint: take the difference. 5. Compute the following integrals: (a) R 4 1 ( x 4 - 3 x 2 + 2 x - 1) dx (b) R 3 - 5 | x 2 - 4 | dx (c) R 7 2 ( x + 2 - x 1 / 3 ) dx (d) R 2 π/ 3 π/ 2 cos(2 x + π/ 6) dx (e) R 13 π/ 6 0 | sin x | dx 2

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6. Suppose that f is strictly increasing and continuous on [0 , ) with range [0 , ) (which also means f (0) = 0). Let g ( x ) be the inverse function of f and F ( x ) = R x 0 f ( y ) dy . Find an expression for R x 0 g ( y ) dy in terms of F, g . Hint: copy the proof of the integral of x 1 /p . 7. Find lim x 0 sin(2 x )+cos( x 3 ) x 2 x . 8. Find lim x →∞ x ( x + 2 - x ). 9. Show that if lim x a f ( x ) = 0 then lim x a 1 | f ( x ) | = . 10. Show that if lim x →∞ f ( x ) = 0 then lim x →∞ 1 f ( x ) might not exists (not even in the infinite sense). 11. Is there a function f which is continuous on R and f (1 /n ) = ( - 1) n for all positive integer n ? 12. Is there a function f which is continuous on R and f (1 /n ) = ( - 1) n /n for all positive integer n ? 13. We define the positive part of a number x as x + = ( x if x 0 0 if x < 0 (a) Show that the function x x + is continuous everywhere.
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