de-t3-a

# N 2 n 1 test3map2302 page 2 of 4 3 15 pts a if f t

This preview shows pages 1–4. Sign up to view the full content.

n 2)( n 1) .

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

View Full Document
TEST3/MAP2302 Page 2 of 4 ______________________________________________________________________ 3. (15 pts.) (a) If f ( t ) and g ( t ) are piecewise continuous functions defined for t 0, what is the definition of the convolution of f with g , ( f * g )( t )?? ( f g )( t ) t 0 f ( x ) g ( t x ) dx (b) Using only the definition of the convolution as a definite integral, not some fancy transform shenanigans, compute ( f * g )( t ) when f ( t ) = 3 t 2 and g ( t ) = 4 t . ( f g )( t ) t 0 f ( x ) g ( t x ) dx t 0 3 x 2 4( t x ) dx t 0 12 tx 2 12 x 3 dx 4 tx 3 3 x 4 t 0 t 4 . (c) Using the Laplace transform table, compute the Laplace transform of f * g when f ( t ) = t cos( t ) and g ( t ) = exp(2 t ). [Do not attempt to simplify the algebra after computing the transform.] {( f g )( t )}( s ) { f ( t )}( s ) { g ( t )}( s ) { t cos( t )}( s ) { e 2 t }( s ) s 2 1 ( s 2 1) 2 1 ( s 2) ______________________________________________________________________ 4. (10 pts.) (a) Suppose that f ( t ) is defined for t > 0. What is the definition of the Laplace transform of f , { f ( t )}, in terms of a definite integral?? { f ( t )}( s ) 0 f ( t ) e st dt lim R → ∞ R 0 f ( t ) e st dt for all s for which the integral converges. (b) Using only the definition, not the table, compute the Laplace transform of f ( t ) 0 , if 0 < t < 4 3 , if 4 < t . { f ( t )}( s ) 0 f ( t ) e st dt lim R → ∞ 4 0 0 e st dt R 4 3 e st dt lim R → ∞ 3 e 4 s s 3 e Rs s 3 e 4 s s provided s > 0. Note: You may, of course, check your "answer" using #15 in the table.
TEST3/MAP2302 Page 3 of 4 ______________________________________________________________________ 5. (10 pts.) The equation below has a regular singular point at

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 4

n 2 n 1 TEST3MAP2302 Page 2 of 4 3 15 pts a If f t and g t...

This preview shows document pages 1 - 4. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online