7 Find $$ A $$ as a function of $$ x $$, given that $$ \frac{\mathrm{d} A}{\mathrm{~d} x}=3(x-1)^{2} $$ and that $$ A=10 $$ when $$ x=3 $$.

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Given that $$ \frac{\mathrm{d} A}{\mathrm{~d} x}=3(x-1)^{2} $$ and $$ A=10 $$ when $$ x=3 $$, we can find the function $$ A $$ as a function of $$ x $$ by integrating the given derivative.

The derivative of $$ A $$ with respect to $$ x $$ is $$ \frac{\mathrm{d} A}{\mathrm{~d} x}=3(x-1)^{2} $$.

To find the function $$ A $$, we need to integrate the derivative with respect to $$ x $$:

$$ A = \int 3(x-1)^{2} \, \mathrm{d}x $$

Now, we can integrate the expression to find the function $$ A $$ as a function of $$ x $$.
Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int 3\left(x-1\right)^{2} dx$$
- step1: Use properties of integrals:
  $$3\times \int \left(x-1\right)^{2} dx$$
- step2: Use the substitution $$dx=1 dt$$ to transform the integral$$:$$
  $$3\times \int \left(x-1\right)^{2}\times 1 dt$$
- step3: Simplify:
  $$3\times \int \left(x-1\right)^{2} dt$$
- step4: Use the substitution $$t=x-1$$ to transform the integral$$:$$
  $$3\times \int t^{2} dt$$
- step5: Evaluate the integral:
  $$3\times \frac{t^{2+1}}{2+1}$$
- step6: Simplify:
  $$3\times \frac{t^{3}}{3}$$
- step7: Reduce the fraction:
  $$1\times t^{3}$$
- step8: Multiply the terms:
  $$t^{3}$$
- step9: Substitute back:
  $$\left(x-1\right)^{3}$$
- step10: Add the constant of integral C:
  $$\left(x-1\right)^{3} + C, C \in \mathbb{R}$$
The integral of $$ 3(x-1)^{2} $$ with respect to $$ x $$ is $$ (x-1)^{3} + C $$, where $$ C $$ is an arbitrary constant.

Now, we are given that $$ A=10 $$ when $$ x=3 $$. We can use this information to find the value of the constant $$ C $$.

Substitute $$ A=10 $$ and $$ x=3 $$ into the function $$ A = (x-1)^{3} + C $$:

$$ 10 = (3-1)^{3} + C $$
$$ 10 = 8 + C $$
$$ C = 2 $$

Therefore, the function $$ A $$ as a function of $$ x $$ is $$ A = (x-1)^{3} + 2 $$.
$$ A = (x-1)^3 + 2 $$

7 Find as a function of , given that
and that when
.

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