midterm2-sample.pdf - MIDTERM 2 – MATH 2065 – Fall 2013 – Version 1 Name Justify your answers to receive full credit No notes formula sheets

# midterm2-sample.pdf - MIDTERM 2 – MATH 2065 – Fall 2013...

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MIDTERM 2 – MATH 2065 – Fall 2013 – Version 1 Name: Justify your answers to receive full credit. No notes, formula sheets, phones allowed. Basic scientific calculator allowed. Good luck! Time: 50 minutes; Points: 100 (1) Fill in the blanks. (a) (6 points) If L = D 2 + 3 D + 2 and y = e t , then L ( y ) = . Is y = e t a solution to the differential equation y 00 + 3 y 0 + 2 y = 0? . (b) (8 points) If q ( s ) = ( s + 3) 3 ( s 2 - 2 s + 10) 2 then B q = . Laplace Transform Table f ( t ) ←→ F ( s ) = L { f ( t ) } ( s ) f ( t ) ←→ F ( s ) = L { f ( t ) } ( s ) 1. 1 ←→ 1 s 5. cos bt ←→ s s 2 + b 2 2. t n ←→ n ! s n +1 6. sin bt ←→ b s 2 + b 2 3. e at ←→ 1 s - a 7. e at cos bt ←→ s - a ( s - a ) 2 + b 2 4. t n e at ←→ n ! ( s - a ) n +1 8. e at sin bt ←→ b ( s - a ) 2 + b 2 Laplace Transform Principles Input Derivative Principles L { f 0 ( t ) } ( s ) = s L { f ( t ) } - f (0) L { f 00 ( t ) } ( s ) = s 2 L { f ( t ) } - sf (0) - f 0 (0) First Translation Principle L { e at f ( t ) } = F ( s - a ) Transform Derivative Principle L {- tf ( t ) } ( s ) = d ds F ( s ) Convolution Principle L { f * g )( t ) } = F ( s ) G ( s ) 1
(2) (14 points) Are the following sets linearly dependent or independent? Prove your answer using one of the methods from class. If linearly dependent, produce a linear combination that demonstrates this.

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