Answer:
Question 1) 72
Question 2) 40
Question 3) Both C and D.
Question 4) Both A and C.
Step-by-step explanation:
Question 1)
We are given that:
![\displaystyle \int_1^5f(x)\, dx=8\text{ and we want to find} \int_1^5xf'(x)\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_1%5E5f%28x%29%5C%2C%20dx%3D8%5Ctext%7B%20and%20we%20want%20to%20find%7D%20%5Cint_1%5E5xf%27%28x%29%5C%2C%20dx)
We will use integration by parts, given by:
![\displaystyle \int_a^b u\, dv=uv-\int_a^b v\, du](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_a%5Eb%20u%5C%2C%20dv%3Duv-%5Cint_a%5Eb%20v%5C%2C%20du)
We will let:
![u=x\Rightarrow du=dx\text{ and } dv=f'(x)\, dx \Rightarrow v=f(x)](https://tex.z-dn.net/?f=u%3Dx%5CRightarrow%20du%3Ddx%5Ctext%7B%20and%20%7D%20dv%3Df%27%28x%29%5C%2C%20dx%20%5CRightarrow%20v%3Df%28x%29)
Therefore:
![\displaystyle \int_1^5xf'(x)\, dx=xf(x)\Big|_1^5-\int_1^5f(x)\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_1%5E5xf%27%28x%29%5C%2C%20dx%3Dxf%28x%29%5CBig%7C_1%5E5-%5Cint_1%5E5f%28x%29%5C%2C%20dx)
Substitute:
![\displaystyle \int_1^5xf'(x)\, dx=(5(f(5))-(1(f(1))-(8)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_1%5E5xf%27%28x%29%5C%2C%20dx%3D%285%28f%285%29%29-%281%28f%281%29%29-%288%29)
Evaluate:
![\displaystyle \int_1^5xf'(x)\, dx=75-(-5)-8=72](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_1%5E5xf%27%28x%29%5C%2C%20dx%3D75-%28-5%29-8%3D72)
Question 2)
Similarly, we will let:
![\displaystyle u=x\Rightarrow du=dx\text{ and } dv=f'(x)\, dx \text{ so } v=f(x)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20u%3Dx%5CRightarrow%20du%3Ddx%5Ctext%7B%20and%20%7D%20dv%3Df%27%28x%29%5C%2C%20dx%20%5Ctext%7B%20so%20%7D%20v%3Df%28x%29)
Hence:
![\displaystyle \int_0^3 xf'(x)\, dx=xf(x)\Big|_0^3-\int_0^3f(x)\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E3%20xf%27%28x%29%5C%2C%20dx%3Dxf%28x%29%5CBig%7C_0%5E3-%5Cint_0%5E3f%28x%29%5C%2C%20dx)
Evaluate:
![\displaystyle \int_0^3 xf'(x)\, dx=(3f(3))-(0(f(0))-(2)](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E3%20xf%27%28x%29%5C%2C%20dx%3D%283f%283%29%29-%280%28f%280%29%29-%282%29)
Thus:
![\displaystyle \int_0^3 xf'(x)\, dx=3(14)-2=40](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cint_0%5E3%20xf%27%28x%29%5C%2C%20dx%3D3%2814%29-2%3D40)
Question 3)
We are given:
![\displaystyle g(x)=\int_4^xf(x)\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20g%28x%29%3D%5Cint_4%5Exf%28x%29%5C%2C%20dx)
By the Fundamental Theorem of Calculus:
![g'(x)=f(x)>0](https://tex.z-dn.net/?f=g%27%28x%29%3Df%28x%29%3E0)
The derivative of g is always positive. So, the values of g are always increasing.
The tables that reflect this are C and D.
And there are, as I understand it, no way to determine their exact values. Both C and D are correct.
Question 4)
Similarly, we are given:
![\displaystyle g(x)=\int_{-2}^xf(x)\, dx](https://tex.z-dn.net/?f=%5Cdisplaystyle%20g%28x%29%3D%5Cint_%7B-2%7D%5Exf%28x%29%5C%2C%20dx)
By the FTC:
![g'(x)=f(x)](https://tex.z-dn.net/?f=g%27%28x%29%3Df%28x%29%3C0)
So, g should be decreasing for all x.
The tables that reflect this are A and C.
So, both A and C are correct.
So what you will have to do is 45 alla you have to do is multiple
*(Respuesta)* =
* (Explicación) * = El motivo por el que necesita agregar
a
.
Espero que esto ayude
Persona que respondió: BangtanBoyScouts
Answer:
11.6cm
Step-by-step explanation:
a^2+b^2=c^2
11^2+b^2=16^2
121+b^2=256
b^2=135
b=11.6
Answer:
g=-3repeated
Step-by-step explanation: