Tutor profile: Kevin B.

Write a pseudocode function, fib(n), to find the $$n^{th}$$ number in the fibonacci sequence, $$F_{n} = F_{n-1} + F_{n-2} $$, starting with $$F_{0} = 0, F_{1} = 1$$ and using recursion.

fib(n): if (n == 0): return 0 else if (n == 1): return 1 else: return fib(n-1) + fib(n-2)

Find the area bounded by the curves $$y=x^{2}$$ and $$y=x$$.

First, find the points at which the two curves intersect. $$x^{2}=x \Rightarrow x=0,1$$ Next, take the integral of the upper bound minus the lower bound on the interval. $$Area = \int_{0}^{1}(x-x^{2})dx = \left[ \frac{x^{2}}{2} - \frac{x^3}{3} \right]_{0}^{1} = \left( \frac{1}{2} - \frac{1}{3} \right) - 0 = \frac {1}{6} $$

A ball is thrown horizontally at 10 m/s off a roof that is 120 m tall. What horizontal distance does the ball travel before hitting the ground? (Use g = 10 m/s^2 and neglect air resistance)

First, find the time it will take for the ball to hit the ground. $$V_{0y} = 0 m/s$$ $$y_{0} = 120 m$$ $$y_{f} = 0 m$$ $$ \left( y_{f} - y_{0}\right) = v_{0y} * t - \frac{1}{2} * g * t^{2}$$ $$\Rightarrow \left( 0 - 120 \right) = 0 - \frac{1}{2} * 10 * t^{2}$$ $$\Rightarrow -120 = -5 * t^{2}$$ $$\Rightarrow t^{2} = 24$$ $$\Rightarrow t=\sqrt{24} = 2\sqrt{6} s$$ Then, find the horizontal distance travelled in this time. $$V_{x} = 10m/s$$ $$x_0 = 0 m/s$$ $$\left( x_{f} - x_{0} \right) = v_{x} * t$$ $$\Rightarrow \left( x_{f} - 0 \right) = 10 * 2\sqrt{6}$$ $$\Rightarrow x_{f} = 20\sqrt{6} \approx 48.99m$$ Therefore, the ball will travel approximately 48.99 m before hitting the ground.