Answer:
Step-by-step explanation:
By definition of Laplace transform we have
L{f(t)} = ![L{{f(t)}}=\int_{0}^{\infty }e^{-st}f(t)dt\\\\Given\\f(t)=7t^{3}\\\\\therefore L[7t^{3}]=\int_{0}^{\infty }e^{-st}7t^{3}dt\\\\](https://tex.z-dn.net/?f=L%7B%7Bf%28t%29%7D%7D%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7De%5E%7B-st%7Df%28t%29dt%5C%5C%5C%5CGiven%5C%5Cf%28t%29%3D7t%5E%7B3%7D%5C%5C%5C%5C%5Ctherefore%20L%5B7t%5E%7B3%7D%5D%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7De%5E%7B-st%7D7t%5E%7B3%7Ddt%5C%5C%5C%5C)
Now to solve the integral on the right hand side we shall use Integration by parts Taking 
as first function thus we have
![\int_{0}^{\infty }e^{-st}7t^{3}dt=7\int_{0}^{\infty }e^{-st}t^{3}dt\\\\= [t^3\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(3t^2)\int e^{-st}dt]dt\\\\=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\](https://tex.z-dn.net/?f=%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7De%5E%7B-st%7D7t%5E%7B3%7Ddt%3D7%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7De%5E%7B-st%7Dt%5E%7B3%7Ddt%5C%5C%5C%5C%3D%20%5Bt%5E3%5Cint%20e%5E%7B-st%7D%20%5D_%7B0%7D%5E%7B%5Cinfty%7D-%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5B%283t%5E2%29%5Cint%20e%5E%7B-st%7Ddt%5Ddt%5C%5C%5C%5C%3D0-%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B3t%5E%7B2%7D%7D%7B-s%7De%5E%7B-st%7Ddt%5C%5C%5C%5C%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B3t%5E%7B2%7D%7D%7Bs%7De%5E%7B-st%7Ddt%5C%5C%5C%5C)
Again repeating the same procedure we get
![=0-\int_{0}^{\infty }\frac{3t^{2}}{-s}e^{-st}dt\\\\=\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt\\\\\int_{0}^{\infty }\frac{3t^{2}}{s}e^{-st}dt= \frac{3}{s}[t^2\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t^2)\int e^{-st}dt]dt\\\\=\frac{3}{s}[0-\int_{0}^{\infty }\frac{2t^{1}}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{2}}[\int_{0}^{\infty }te^{-st}dt]\\\\](https://tex.z-dn.net/?f=%3D0-%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B3t%5E%7B2%7D%7D%7B-s%7De%5E%7B-st%7Ddt%5C%5C%5C%5C%3D%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B3t%5E%7B2%7D%7D%7Bs%7De%5E%7B-st%7Ddt%5C%5C%5C%5C%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B3t%5E%7B2%7D%7D%7Bs%7De%5E%7B-st%7Ddt%3D%20%5Cfrac%7B3%7D%7Bs%7D%5Bt%5E2%5Cint%20e%5E%7B-st%7D%20%5D_%7B0%7D%5E%7B%5Cinfty%7D-%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5B%28t%5E2%29%5Cint%20e%5E%7B-st%7Ddt%5Ddt%5C%5C%5C%5C%3D%5Cfrac%7B3%7D%7Bs%7D%5B0-%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B2t%5E%7B1%7D%7D%7B-s%7De%5E%7B-st%7Ddt%5D%5C%5C%5C%5C%3D%5Cfrac%7B3%5Ctimes%202%7D%7Bs%5E%7B2%7D%7D%5B%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7Dte%5E%7B-st%7Ddt%5D%5C%5C%5C%5C)
Again repeating the same procedure we get
![\frac{3\times 2}{s^2}[\int_{0}^{\infty }te^{-st}dt]= \frac{3\times 2}{s^{2}}[t\int e^{-st} ]_{0}^{\infty}-\int_{0}^{\infty }[(t)\int e^{-st}dt]dt\\\\=\frac{3\times 2}{s^2}[0-\int_{0}^{\infty }\frac{1}{-s}e^{-st}dt]\\\\=\frac{3\times 2}{s^{3}}[\int_{0}^{\infty }e^{-st}dt]\\\\](https://tex.z-dn.net/?f=%5Cfrac%7B3%5Ctimes%202%7D%7Bs%5E2%7D%5B%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7Dte%5E%7B-st%7Ddt%5D%3D%20%5Cfrac%7B3%5Ctimes%202%7D%7Bs%5E%7B2%7D%7D%5Bt%5Cint%20e%5E%7B-st%7D%20%5D_%7B0%7D%5E%7B%5Cinfty%7D-%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5B%28t%29%5Cint%20e%5E%7B-st%7Ddt%5Ddt%5C%5C%5C%5C%3D%5Cfrac%7B3%5Ctimes%202%7D%7Bs%5E2%7D%5B0-%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7D%5Cfrac%7B1%7D%7B-s%7De%5E%7B-st%7Ddt%5D%5C%5C%5C%5C%3D%5Cfrac%7B3%5Ctimes%202%7D%7Bs%5E%7B3%7D%7D%5B%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7De%5E%7B-st%7Ddt%5D%5C%5C%5C%5C)
Now solving this integral we have
![\int_{0}^{\infty }e^{-st}dt=\frac{1}{-s}[\frac{1}{e^\infty }-\frac{1}{1}]\\\\\int_{0}^{\infty }e^{-st}dt=\frac{1}{s}](https://tex.z-dn.net/?f=%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7De%5E%7B-st%7Ddt%3D%5Cfrac%7B1%7D%7B-s%7D%5B%5Cfrac%7B1%7D%7Be%5E%5Cinfty%20%7D-%5Cfrac%7B1%7D%7B1%7D%5D%5C%5C%5C%5C%5Cint_%7B0%7D%5E%7B%5Cinfty%20%7De%5E%7B-st%7Ddt%3D%5Cfrac%7B1%7D%7Bs%7D)
Thus we have
![L[7t^{3}]=\frac{7\times 3\times 2}{s^4}](https://tex.z-dn.net/?f=L%5B7t%5E%7B3%7D%5D%3D%5Cfrac%7B7%5Ctimes%203%5Ctimes%202%7D%7Bs%5E4%7D)
where s is any complex parameter
Answer and step-by-step explanation:
We learn that (a+b)^3 = a^3 + 3a^2b + 3ab^2 + b^3
So in this case, it would be:
= x^3 + (3*x^2*2) + (3*x*2^2) + 2^3
= x^3 + 6x^2 + 3x * 4 + 8
= x^3 + 6x^2 + 12x + 8
This is the standard form of the equation
Hope it help you :3
Assume that the budget is n.
Now we are given that 120% of the budget is equal to $60.
This means that:
60 = 1.2n
Solve the equation for n to know the budget:
60 = 1.2n
n = 60 / 1.2 = 50
Based on the above calculations, Jada's budget is 50$
Answer:
$2,980
Step-by-step explanation:
Mortgage points is a term used by a mortgage brokers and it is usually referred to the percentage of the loan amount.
From the given question:
Felicia decided to purchase 2 points in order to lower her interest rate on her $149,000 mortgage
the 2 points that Felicia purchased is equivalent to 2% i.e 0.02 points
Thus; the additional money she need to bring to closing is calculated by the formula:
cost = points × loan amount
= 0.02 × $149,000
= $2,980
