Question

the function g(x) = 12x^2-sinx is the first derivative of f(x). If f(0)=-2 what is the value of f(2pi

Answer 1

`f(2 \pi)=32\pi^3-2`

Explanation:

Fundamental Theorem of Calculus

`\displaystyle \int \text{f}(x)\:\text{d}x=\text{F}(x)+\text{C} \iff \text{f}(x)=\frac{\text{d}}{\text{d}x}(\text{F}(x))`

If differentiating takes you from one function to another, then integrating the second function will take you back to the first with a constant of integration.

Given:

• 
  `g(x)=12x^2-\sin x`

• 
  `f(0)=-2`

If g(x) is the first derivative of f(x), we can find f(x) by integrating g(x) and using f(0) = -2 to find the constant of integration.

`\boxed{\begin{minipage}{4 cm}\underline{Integrating $x^n$}\\\\$\displaystyle \int x^n\:\text{d}x=(x^(n+1))/(n+1)+\text{C}$\end{minipage}}`
`\boxed{\begin{minipage}{4 cm}\underline{Integrating $\sin x$}\\\\$\displaystyle \int \sin x\:\text{d}x=-\cos x+\text{C}$\end{minipage}}`

`\begin{aligned} \displaystyle f(x)&=\int f'(x)\; \text{d}x\\\\&=\int g(x)\;\text{d}x\\\\&=\int (12x^2-\sin x)\;\text{d}x\\\\&=\int 12x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\int x^2\; \text{d}x-\int \sin x \; \text{d}x\\\\&=12\cdot (x^((2+1)))/(2+1)-(-\cos x)+\text{C}\\\\&=4x^(3)+\cos x+\text{C}\end{aligned}`

To find the constant of integration, substitute f(0) = -2 and solve for C:

`\begin{aligned}f(0)=4(0)^3+\cos (0) + \text{C}&=-2\\0+1+\text{C}&=-2\\\text{C}&=-3\end{aligned}`

Therefore, the equation of function f(x) is:

`\boxed{f(x)=4x^3+ \cos x - 3}`

To find the value of f(2π), substitute x = 2π into function f(x):

`\begin{aligned}f(2 \pi)&=4(2 \pi)^3+ \cos (2 \pi) - 3\\&=4\cdot 2^3 \cdot \pi^3+1 - 3\\&=32\pi^3-2\\\end{aligned}`

Therefore, the value of f(2π) is 32π³ - 2.

Answer 2

`f(2\pi) = 32\pi^3 - 2`

Explanation:

Main steps:

Step 1: Use integration to find a general equation for f

Step 2: Find the value of the constant of integration

Step 3: Find the value of f for the given input

Step 1: Use integration to find a general equation for f

If
`f'(x) = g(x)`, then
`f(x) = \int g(x) ~dx`

`f(x) = \int [12x^2 - sin(x)] ~dx`

Integration of a difference is the difference of the integrals

`f(x) = \int 12x^2 ~dx - \int sin(x) ~dx`

Scalar rule

`f(x) = 12\int x^2 ~dx - \int sin(x) ~dx`

Apply the Power rule & integral relationship between sine and cosine:

• Power Rule:
  `\int x^n ~dx=(1)/(n+1)x^(n+1) +C`
• sine-cosine integral relationship:
  `\int sin(x) ~dx=-cos(x)+C`

`f(x) = 12*((1)/(3)x^3+C_1) - (-cos(x) + C_2)`

Simplifying

`f(x) = 12*(1)/(3)x^3+12*C_1 +cos(x) + -C_2`

`f(x) = 4x^3+cos(x) +(12C_1 -C_2)`

Ultimately, all of the constant of integration terms at the end can combine into one single unknown constant of integration:

`f(x) = 4x^3 + cos(x) + C`

Step 2: Find the value of the constant of integration

Now, according to the problem,
`f(0) = -2`, so we can substitute those x,y values into the equation and solve for the value of the constant of integration:

`-2 = 4(0)^3 + cos(0) + C`

`-2 = 0 + 1 + C`

`-2 = 1 + C`

`-3 = C`

Knowing the constant of integration, we now know the full equation for the function f:

`f(x) = 4x^3 + cos(x) -3`

Step 3: Find the value of f for the given input

So, to find
`f(2\pi)`, use 2 pi as the input, and simplify:

`f(2\pi) = 4(2\pi)^3 + cos(2\pi) -3`

`f(2\pi) = 4*8\pi^3 + 1 -3`

`f(2\pi) = 32\pi^3 - 2`