Determine \( \frac{d y}{d x} \) if \( f(x)=\cos x \) using first principles. Show all intermediate steps HINT: \( \lim _{x \rightarrow 0} \frac{\cos h-1}{h}=0 \) and \( \lim _{x \rightarrow 0} \frac{\sin h}{h}=1 \)

\( \lim _ { t \rightarrow 0 ^ { + } } \frac { \log ( t ) } { t } \)

A waste management company is designing a rectangular construction dumpster that will be twice as long as it is wide and must hold \( 15 \mathrm{yd}^{3} \) of debris. Find the dimensions of the dumpster that will minimize its surface area.

6) Calcule las deruadas suce sivas o de orden superior de las funciones \( \begin{array}{ll}\text { a) } y=3 x^{4}-2 x^{3}+5 x^{2}-6 x+2 & \text { b) } y=x^{4}+2 x\end{array} \)

Tính tổng Riemann cho hàm \( f(x) = 2x + 1 \) trên khoảng \([1,3]\) với số phân chia là 4 phần.

\( 1 \leftarrow \) Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 33 in. by 18 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume. The dimensions of box of maximum volume are (Round to the nearest hundredth as needed. Use a comma to separate answers as needed.)