1. Які вирази не є первісною для заданої функції \( f(x)=\sin 2 x \) ? \( F(x)=4-\frac{1}{2} \cos 2 x \quad F(x)=-\frac{1}{2} \cos 2 x \)

A particle \( P \) moves in a straight line so that, \( t \) seconds after passing a fixed point \( O \), its displacement, \( s \) metres, is given by \( s=4 t^{3}-15 t^{2}+18 t \). Find (i) the velocity of \( P \) when \( t=3 \), (ii) the acceleration of \( P \) when \( t=3 \), (iii) the values of \( t \) when \( P \) is at instantaneous rest, (iv) the distance travelled by \( P \) in the first 2 seconds of its motion, (v) the distance of \( P \) from \( O \) when \( t=2 \).

If \( \lim _{x \rightarrow a} \frac{x^{8}-a^{8}}{x^{6}-a^{6}}=48 \), then \( a \)

c. The value of output which equates average product and marginal product Use the product rule to find the first-order partials for each of the following functions a. \( z=3 x^{2}(5 x+7 y) \) b. \( z=(9 x-4 y)(12 x+2 y) \)

Given \( \mathrm{TP}=90 \mathrm{~K}^{2}-\mathrm{K}^{3} \) find a. Marginal product b. Average product c. The value of output which equates average product and marginal product

4) \( \lim _{x \rightarrow 0} \frac{2 x+5 x^{-2}}{3 x+x^{-2}}= \)