10. A company manufactures and sales \( X \) units of DSTV per month. If the monthly cost and demand equations are: \[ P(X)=10,000+4 x \] \( P(X)=20-\frac{X}{2,000} P \) is price \( [0,4000] \) For each month find:- 1) The production level that leads to maximum revenue and the maximum revenue. 2) The production levels that leads to maximum profit and the maximum profit. 3) The production level that leads to minimum cost and the minimum cost.

4. Let \( g \) be a function such that \( g(1)=3 \) and whose derivative is \( g^{\prime}(x)=6 \cos (2 \pi x)+3 \). Solve analytically a. Use a linear approximation to estimate the value of \( g(1.2) \). b. Find the solution to the differential equation \( d g=(6 \cos (2 \pi x)+3) d x \)

5. Given \( h(x)=\int_{0}^{4 x} \sqrt{t-2} d t \), evaluate the following analytically. \( \begin{array}{ll}\text { a. Determine when } h(x)=\frac{16}{3} & \text { b. Determine when } h^{\prime}(x)=8\end{array} \)

23-24 ¿Cómo podría "remover la discontinuidad" en cada una de las siguientes funciones? En otras palabras, ¿cómo redefiniría \( f(2) \) a fin de que sean continuas en \( x=2 \) ? \( \begin{array}{ll}\text { 23. } f(x)=\frac{x^{2}-x-2}{x-2} & \text { 24. } f(x)=\frac{x^{3}-8}{x^{2}-4}\end{array} \)

Find equations of the tangent line and normal line to \( x^{2}+4 x y+y^{2}=13,(2,1) \)

\( f ( x ) = 4 x ^ { 3 } - 3 x ^ { 2 } - x - 1 , x = - 1 \)