Question

Find the average rate of change of the function over the given interval

Answer 1

`\bf slope = {{ m}}= \cfrac{rise}{run} \implies \cfrac{{{ f(x_2)}}-{{ f(x_1)}}}{{{ x_2}}-{{ x_1}}}\impliedby \begin{array}{llll} average\ rate\\ of\ change \end{array}\\\\ -------------------------------\\\\`


`\bf h(t)=cot(t)\implies h(t)=\cfrac{cos(t)}{sin(t)}\quad \begin{cases} t_1=(\pi )/(4)\\ t_2=(3\pi )/(4) \end{cases}\implies \cfrac{h\left( (3\pi )/(4) \right)-h\left( (\pi )/(4) \right)}{(3\pi )/(4)-(\pi )/(4)} \\\\\\`


`\bf \cfrac{(cos\left( (3\pi )/(4) \right))/(sin\left( (3\pi )/(4) \right))-(cos\left( (\pi )/(4) \right))/(sin\left( (\pi )/(4) \right))}{(\pi )/(2)}\implies \cfrac{-1-1}{(\pi )/(2)}\implies \cfrac{-2}{(\pi )/(2)}\implies -\cfrac{4}{\pi }\\\\\\ -------------------------------\\\\`


`\bf h(t)=cot(t)\implies h(t)=\cfrac{cos(t)}{sin(t)}\quad \begin{cases} t_1=(\pi )/(3)\\ t_2=(3\pi )/(2) \end{cases}\implies \cfrac{h\left( (3\pi )/(2) \right)-h\left( (\pi )/(3) \right)}{(3\pi )/(2)-(\pi )/(3)} \\\\\\`


`\bf \cfrac{(cos\left( (3\pi )/(2) \right))/(sin\left( (3\pi )/(2) \right))-(cos\left( (\pi )/(3) \right))/(sin\left( (\pi )/(3) \right))}{(9\pi -2\pi )/(6)}\implies \cfrac{(0)/(-1)-((1)/(2))/((√(3))/(2))}{(7\pi )/(6)}\implies\cfrac{-(1)/(√(3))}{(7\pi )/(6)}\implies -\cfrac{√(3)}{3}\cdot \cfrac{6}{7\pi } \\\\\\ -\cfrac{2√(3)}{7\pi }`