Lecture Notes Chapter 0.pdf - CHAPTER 0 REVIEW 0.1 Summation E XAMPLE 0.5 Let Z x F(x = E XAMPLE 0.1 Find 7(a 5 ∑(2i(b i=3 ∑(3i2 i=1 E XAMPLE 0.2

Lecture Notes Chapter 0.pdf - CHAPTER 0 REVIEW 0.1...

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CHAPTER 0 REVIEW 0.1 Summation E XAMPLE 0.1. Find (a) 7 i = 3 ( 2 i ) (b) 5 i = 1 ( 3 i 2 ) E XAMPLE 0.2. Let f ( x , y ) be a discrete function with values f ( 0 , 1 ) = 0 . 1, f ( 0 , 2 ) = 0 . 2, f ( 1 , 1 ) = 0 . 3, f ( 1 , 2 ) = 0 . 4 and f ( x , y ) = 0 otherwise. Calculate (a) x f ( x , 2 ) (b) y f ( 0 , y ) (c) x y f ( x , y ) E XAMPLE 0.3. Let f ( x , y ) = x + y 30 x = 0 , 1 , 2 , 3; y = 0 , 1 , 2 0 otherwise Calculate (a) x f ( x , 2 ) (b) y f ( 0 , y ) (c) x x f ( x , 2 ) (d) y y f ( 0 , y ) (e) x y f ( x , y ) (f) x y xy f ( x , y ) 0.2 Derivatives E XAMPLE 0.4. Denote f ( · ) = F 0 ( · ) . Find f if (a) F ( x ) = x 3 + 1 9 (b) F ( y ) = 5 y 16 - 1 4 (c) F ( t ) = 1 - e - 8 t (d) F ( u ) = ( 1 when u 0 0 when u < 0 . E XAMPLE 0.5. Let F ( x ) = Z x - f ( t ) dt Find f ( x ) , if (a) F ( x ) = 0 , x 2 x 5 , 2 < x < 7 1 , x 7 (b) F ( x ) = 0 , x 0 x 4 , 0 < x < 1 1 , x 1 (c) F ( x ) = ( 0 , x 0 1 - e - x / 100 , x > 0 E XAMPLE 0.6. Find the first and second derivatives M 0 ( t ) and M 00 ( t ) for each of the following function M ( t ) . (a) M ( t ) = pe t 1 - ( 1 - p ) e t , for t < ln ( 1 - p ) (b) M ( t ) = e 3 t + 2 t 2 (c) M ( t ) = e 4 ( e t - 1 ) (d) M ( t ) = ( 1 - 2 t ) - ν / 2 0.3 Integrals Z x n dx = x n + 1 n + 1 + C , n 6 = - 1 E XAMPLE 0.7. Evaluate
ii Chapter 0. Review (a) Z 3 0 1 5 dy (b) Z 1 1

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