In problems 19 - 22, find L[Kt)] by first using an appropriate trigonometric identity. 19. f(t) = sin 2t cos 2t. 20. f(t) = cos?t. 21. f(t) = sin(4t +5). %3D %3D
In problems 19 - 22, find L[Kt)] by first using an appropriate trigonometric identity. 19. f(t) = sin 2t cos 2t. 20. f(t) = cos?t. 21. f(t) = sin(4t +5). %3D %3D
Chapter6: Exponential And Logarithmic Functions
Section6.1: Exponential Functions
Problem 60SE: The formula for the amount A in an investmentaccount with a nominal interest rate r at any timet is...
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Please Q19 to Q21
![In problems 10 – 18, find L[(t)J.
10. f(t) = 2t*.
11. f(t) = 4t – 10.
12. f(t) = t2 + 6t - 3.
13. f(t) = (t + 1)3.
14. f(t) = 1+ et.
15. f(t) = (1+ e2t)?.
16. f(t) = 4t2 - 5 sin 3t.
17. f(t) = k sinh kt.
18. f(t) = et sinh t.
In problems 19 – 22, find L[Kt)] by first using an appropriate trigonometric identity.
19. f(t) = sin 2t cos 2t.
20. f(t) = cos²t.
21. f(t) = sin(4t + 5).
22. f(t) = 10 cos (t -).
23. One definition of the gamma function r(a) is given by the improper integral
T(a) =
ta-le-t dt, a > 0.
Use this definition to show that r(a + 1) = ar(a).
24. Use problem 23 to show that
Г(а + 1)
L[t"] =
a > -1.
sa+1
This result is a generalisation of
L[t"] =
n!
n = 1,2,3, ..
sn+1'
In problems 25 – 28, use the results of 23 and 24 and the fact that r(-) = Vn to find the
transform of the given function.
25. f(t) = t.
26. f(t) = t7.
27. f(t) = t7.
28. f(t) = 6tz - 24tz.
29. Use L[eat =- to show that
s-a
s - a + ib
L[ea+ib)r] =
%3D
(s - a)2 + b2'
where a and b are real and i? = - 1. Show how Euler's fomula can be used to pro
the results
s - a
L[eat cos bt] =
(s - a)2 + b2
and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F0980cccd-2546-4880-814d-919b1581f406%2F697ba7f3-4f50-4b72-a4d8-7a4fc8493b6a%2Femzmhv_processed.jpeg&w=3840&q=75)
Transcribed Image Text:In problems 10 – 18, find L[(t)J.
10. f(t) = 2t*.
11. f(t) = 4t – 10.
12. f(t) = t2 + 6t - 3.
13. f(t) = (t + 1)3.
14. f(t) = 1+ et.
15. f(t) = (1+ e2t)?.
16. f(t) = 4t2 - 5 sin 3t.
17. f(t) = k sinh kt.
18. f(t) = et sinh t.
In problems 19 – 22, find L[Kt)] by first using an appropriate trigonometric identity.
19. f(t) = sin 2t cos 2t.
20. f(t) = cos²t.
21. f(t) = sin(4t + 5).
22. f(t) = 10 cos (t -).
23. One definition of the gamma function r(a) is given by the improper integral
T(a) =
ta-le-t dt, a > 0.
Use this definition to show that r(a + 1) = ar(a).
24. Use problem 23 to show that
Г(а + 1)
L[t"] =
a > -1.
sa+1
This result is a generalisation of
L[t"] =
n!
n = 1,2,3, ..
sn+1'
In problems 25 – 28, use the results of 23 and 24 and the fact that r(-) = Vn to find the
transform of the given function.
25. f(t) = t.
26. f(t) = t7.
27. f(t) = t7.
28. f(t) = 6tz - 24tz.
29. Use L[eat =- to show that
s-a
s - a + ib
L[ea+ib)r] =
%3D
(s - a)2 + b2'
where a and b are real and i? = - 1. Show how Euler's fomula can be used to pro
the results
s - a
L[eat cos bt] =
(s - a)2 + b2
and
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