Total-$31.23
explaination- you multiply 2.25 x4.8 and get 10.80 and then you multiply 2.25x7.3 and get16.43 then u add 4 to each of them and add them all up 14.80+16.43=31.23 and the 4 dollars are already included
No, for example 8-1. If you do the Commutative property, it would be 1-8; which would be a negative number. Associate property is basically the same thing; so no, you can't.
a. Given that y = f(x) and f(0) = -2, by the fundamental theorem of calculus we have
![\displaystyle \frac{dy}{dx} = 3x^2 + 4x + k \implies y = f(0) + \int_0^x (3t^2+4t+k) \, dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%20%5Cfrac%7Bdy%7D%7Bdx%7D%20%3D%203x%5E2%20%2B%204x%20%2B%20k%20%5Cimplies%20y%20%3D%20f%280%29%20%2B%20%5Cint_0%5Ex%20%283t%5E2%2B4t%2Bk%29%20%5C%2C%20dt)
Evaluate the integral to solve for y :
![\displaystyle y = -2 + \int_0^x (3t^2+4t+k) \, dt](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20-2%20%2B%20%5Cint_0%5Ex%20%283t%5E2%2B4t%2Bk%29%20%5C%2C%20dt)
![\displaystyle y = -2 + (t^3+2t^2+kt)\bigg|_0^x](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20-2%20%2B%20%28t%5E3%2B2t%5E2%2Bkt%29%5Cbigg%7C_0%5Ex)
![\displaystyle y = x^3+2x^2+kx - 2](https://tex.z-dn.net/?f=%5Cdisplaystyle%20y%20%3D%20x%5E3%2B2x%5E2%2Bkx%20-%202)
Use the other known value, f(2) = 18, to solve for k :
![18 = 2^3 + 2\times2^2+2k - 2 \implies \boxed{k = 2}](https://tex.z-dn.net/?f=18%20%3D%202%5E3%20%2B%202%5Ctimes2%5E2%2B2k%20-%202%20%5Cimplies%20%5Cboxed%7Bk%20%3D%202%7D)
Then the curve C has equation
![\boxed{y = x^3 + 2x^2 + 2x - 2}](https://tex.z-dn.net/?f=%5Cboxed%7By%20%3D%20x%5E3%20%2B%202x%5E2%20%2B%202x%20-%202%7D)
b. Any tangent to the curve C at a point (a, f(a)) has slope equal to the derivative of y at that point:
![\dfrac{dy}{dx}\bigg|_{x=a} = 3a^2 + 4a + 2](https://tex.z-dn.net/?f=%5Cdfrac%7Bdy%7D%7Bdx%7D%5Cbigg%7C_%7Bx%3Da%7D%20%3D%203a%5E2%20%2B%204a%20%2B%202)
The slope of the given tangent line
is 1. Solve for a :
![3a^2 + 4a + 2 = 1 \implies 3a^2 + 4a + 1 = (3a+1)(a+1)=0 \implies a = -\dfrac13 \text{ or }a = -1](https://tex.z-dn.net/?f=3a%5E2%20%2B%204a%20%2B%202%20%3D%201%20%5Cimplies%203a%5E2%20%2B%204a%20%2B%201%20%3D%20%283a%2B1%29%28a%2B1%29%3D0%20%5Cimplies%20a%20%3D%20-%5Cdfrac13%20%5Ctext%7B%20or%20%7Da%20%3D%20-1)
so we know there exists a tangent to C with slope 1. When x = -1/3, we have y = f(-1/3) = -67/27; when x = -1, we have y = f(-1) = -3. This means the tangent line must meet C at either (-1/3, -67/27) or (-1, -3).
Decide which of these points is correct:
![x - 2 = x^3 + 2x^2 + 2x - 2 \implies x^3 + 2x^2 + x = x(x+1)^2=0 \implies x=0 \text{ or } x = -1](https://tex.z-dn.net/?f=x%20-%202%20%3D%20x%5E3%20%2B%202x%5E2%20%2B%202x%20-%202%20%5Cimplies%20x%5E3%20%2B%202x%5E2%20%2B%20x%20%3D%20x%28x%2B1%29%5E2%3D0%20%5Cimplies%20x%3D0%20%5Ctext%7B%20or%20%7D%20x%20%3D%20-1)
So, the point of contact between the tangent line and C is (-1, -3).