[u]Calculus Round[/u]


[b]p1.[/b] Evaluate $$\int^2_{-2}(x^3 + 2x + 1)dx$$


[b]p2.[/b] Find $$\lim_{x \to 0} \frac{ln(1 + x + x^3) - x}{x^2}$$


[b]p3.[/b] Find the largest possible value for the slope of a tangent line to the curve $f(x) = \frac{1}{3+x^2}$ .


[b]p4.[/b] An empty, inverted circular cone has a radius of $5$ meters and a height of $20$ meters. At time $t = 0$ seconds, the cone is empty, and at time $t \ge 0$ we fill the cone with water at a rate of $4t^2$ cubic meters per second. Compute the rate of change of the height of water with respect to time, at the point when the water reaches a height of $10$ meters.


[b]p5.[/b] Compute $$\int^{\frac{\pi}{2}}_0 \sin (2016x) \cos (2015x) dx$$


[b]p6.[/b] Let $f(x)$ be a function defined for $x > 1$ such that $f''(x) = \frac{x}{\sqrt{x^2-1}}$ and $f'(2) =\sqrt3$. Compute the length of the graph of $f(x)$ on the domain $x \in (1, 2]$.


[b]p7.[/b] Let the function $f : [1, \infty) \to R$ be defuned as $f(x) = x^{2 ln(x)}$. Compute $$\int^{\sqrt{e}}_1 (f(x) + f^{-1}(x))dx$$


[b]p8.[/b] Calculate $f(3)$, given that $f(x) = x^3 + f'(-1)x^2 + f''(1)x + f'(-1)f(-1)$.


[b]p9.[/b] Compute $$\int^e_1 \frac{ln (x)}{(1 + ln (x))^2} dx$$


[b]p10.[/b] For $x \ge 0$, let $R$ be the region in the plane bounded by the graphs of the line $\ell$ : $y = 4x$ and $y = x^3$. Let $V$ be the volume of the solid formed by revolving $R$ about line $\ell$. Then $V$ can be expressed in the form $\frac{\pi \cdot 2^a}{b\sqrt{c}}$ , where $a$, $b$, and $c$ are positive integers, $b$ is odd, and $c$ is not divisible by the square of a prime. Compute $a + b + c$.


[u]Calculus Tiebreaker[/u]


[b]Tie 1.[/b] Let $f(x) = x + x(\log x)^2$. Find $x$ such that $xf'(x) = 2f(x)$.


[b]Tie 2.[/b] Compute $$\int^{\frac{\sqrt2}{2}}_{-1} \sqrt{1 - x^2} dx$$


[b]Tie 3.[/b] An axis-aligned rectangle has vertices at $(0,0)$ and $(2, 2016)$. Let $f(x, y)$ be the maximum  possible area of a circle with center at $(x, y)$ contained entirely within the rectangle. Compute the expected value of $f$ over the rectangle.


PS. You should use hide for answers.
### Tie 1
We are given the function \( f(x) = x + x(\log x)^2 \) and need to find \( x \) such that \( xf'(x) = 2f(x) \).

1. **Compute the derivative \( f'(x) \):**
   \[
   f(x) = x + x(\log x)^2
   \]
   Using the product rule for differentiation:
   \[
   f'(x) = \frac{d}{dx}[x] + \frac{d}{dx}[x(\log x)^2]
   \]
   \[
   f'(x) = 1 + \left( (\log x)^2 \cdot \frac{d}{dx}[x] + x \cdot \frac{d}{dx}[(\log x)^2] \right)
   \]
   \[
   f'(x) = 1 + (\log x)^2 + x \cdot 2(\log x) \cdot \frac{1}{x}
   \]
   \[
   f'(x) = 1 + (\log x)^2 + 2\log x
   \]

2. **Set up the equation \( xf'(x) = 2f(x) \):**
   \[
   x(1 + (\log x)^2 + 2\log x) = 2(x + x(\log x)^2)
   \]
   Simplify the right-hand side:
   \[
   x + x(\log x)^2 + 2x\log x = 2x + 2x(\log x)^2
   \]

3. **Simplify and solve for \( x \):**
   \[
   x + x(\log x)^2 + 2x\log x = 2x + 2x(\log x)^2
   \]
   Subtract \( x + x(\log x)^2 \) from both sides:
   \[
   2x\log x = x + x(\log x)^2
   \]
   Divide both sides by \( x \):
   \[
   2\log x = 1 + (\log x)^2
   \]
   Let \( y = \log x \):
   \[
   2y = 1 + y^2
   \]
   Rearrange to form a quadratic equation:
   \[
   y^2 - 2y + 1 = 0
   \]
   Factorize the quadratic equation:
   \[
   (y - 1)^2 = 0
   \]
   Thus, \( y = 1 \).

4. **Solve for \( x \):**
   \[
   \log x = 1 \implies x = e
   \]

The final answer is \( \boxed{e} \).