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11. Widget output difference on two days ●●○○○ 20 ptsOn Monday, David produces $w$ widgets per hour and works for $t$ hours. On Tuesday, he works for $2$ fewer hours than on Monday, but produces $4$ additional widgets per hour. If $w = 2t$, how many more widgets did David produce on Monday than on Tuesday?
12. Cube root plus square root equation ●●○○○ 20 ptsSolve the equation $\sqrt[3]{x} + \sqrt{x+12} = 0$ for $x$.
13. Equation with square and cube roots ●●○○○ 20 ptsSolve the equation $\frac{\sqrt{x}+\sqrt[3]{x}}{\sqrt{x}-\sqrt[3]{x}}=3$ for real $x$.
14. Value of f(3) from symmetry conditions ●●○○○ 20 ptsLet $f(x)$ be a function with domain $\mathbb{R}$ such that $f(x+1) = f(-x-1)$ and $f(2-x) = -f(x)$ for all real $x$. Find the value of $f(3)$.
15. Real roots of an exponential-cosine equation ●●○○○ 20 ptsFind the number of real roots of the equation $e^{\cos x} - e^{-\cos x} - 4 = 0$.
16. Sum of coefficients in a shifted cubic expansion ●●○○○ 20 ptsIf the equation $1+x+x^{2}+x^{3}=a_{0}+a_{1}(1-x)+a_{2}(1-x)^{2}+a_{3}(1-x)^{3}$ holds for all real $x$, where $a_{0}$, $a_{1}$, $a_{2}$, $a_{3}$ are real constants, find the value of $a_{1}+a_{2}+a_{3}$.
17. Quadratic expression in shifted variables ●●○○○ 20 ptsGiven $a=2009x+2007$, $b=2009x+2008$, $c=2009x+2009$, find the value of $a^{2}+b^{2}+c^{2}-ab-bc-ca$.
18. Points where f equals its average value ●●○○○ 20 ptsLet $f(x) = (x-2)^2$. Find all values of $c$ such that $f(c)$ equals the average value of $f$ over the closed interval $[-1, 5]$. Round each answer to two decimal places.
19. Fractional-power equation with substitution ●●○○○ 20 ptsSolve the equation $(x - 2)^{2/3} + (x - 2)^{1/3} - 2 = 0$, using real cube roots.
20. Evaluate a function from two values ●●○○○ 20 ptsIf $f(1)=5$, $f(2)=8$ and $f(x)=ax+bx+2$, what is the value of $f(3)$?

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