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f07HW7 - x and that F ′ t = x − t n − 1 n − 1 f n t...

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MAT 125A Homework 7 M.Fukuda Please submit your answers at the discussion session on December 6th Thursday. You can use theorems in the lecuture unless otherwise stated. When you do so write those statements clearly instead of quoting them by numbers. 1. Find the Taylor series for sinh x = 1 2 ( e x e x ) , cosh x = 1 2 ( e x + e x ) and show why they converge for all x R . 2. This is another proof for Thm 31.3. Let 0 < x < b and M be the same as in the proof of Them 31.3. Also, let F ( t ) = f ( t ) + n 1 summationdisplay k =1 ( x t ) k k ! f ( k ) ( t ) + M ( x t ) n n ! for t [0 ,x ]. i) Show that
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Unformatted text preview: , x ] and that F ′ ( t ) = ( x − t ) n − 1 ( n − 1)! ( f ( n ) ( t ) − M ) . ii) By showing F (0) = F ( x ) and applying Rolle’s theorem 29.2 to F prove that there exists y ∈ (0 , x ) such that f ( n ) ( y ) = M . 3. Let ( S 1 , d 1 ), ( S 2 , d 2 ) and ( S 3 , d 3 ) be metric spaces. Suppose two functions f : S 1 → S 2 and g : S 2 → S 3 are continuous. Show that g ◦ f is continuous by using the de²nition 21.1. 1...
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