Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit. Assume that revenue, \( R(x) \), and cost, \( C(x) \), are in thousands of dollars, and \( x \) is in thousands of units. \( R(x)=8 x-4 x^{2}, C(x)=x^{3}-5 x^{2}+3 x+1 \)

\( \lim _ { n \rightarrow \infty } [ \sum _ { i = 1 } ^ { n } \frac { 3 - ( - 1 ) } { n } \cdot f ( ( - 1 ) + i \cdot \frac { 3 - ( - 1 ) } { n } ) ] \)

El volumen se calcula mediante una integral doble sobre la base del plano (zy) y la alurara \( z(z, y) \). Ya sabemos que lo equación del plano es: \[ z=11-\frac{11 e}{0}-\frac{11 y}{10} . \]

\( 1 \leftarrow \) The brain mass of a fetus can be estimated using the total mass of the fetus by the function \( \mathrm{b}=0.22 \mathrm{~m}^{0.87} \), where m is the mass of the fetus (in grams) and b is the brain mass (in grams). Suppose the brain mass of a 27 - g fetus is changing at a rate of 0.21 g per day. Use this to estimate the rate of change of the total mass of the fetus, \( \frac{\mathrm{dm}}{\mathrm{dt}} \). \( \frac{\mathrm{dm}}{\mathrm{dt}}=\square \mathrm{g} \) per day (Round to five decimal places as needed.)

\( \lim _ { n \rightarrow \infty } [ \sum _ { i = 1 } ^ { n } \frac { 3 - ( - 1 ) } { n } \cdot f ( - 1 ) + i \cdot \frac { - ( - 1 ) } { n } ) ] \)

The brain mass of a fetus can be estimated using the total mass of the fetus by the function \( b=0.22 \mathrm{~m}^{0.87} \), where m is the mass of the fetus (in grams) and b is the brain ma: grams). Suppose the brain mass of a \( 21-\mathrm{g} \) fetus is changing at a rate of 0.26 g per day. Use this to estimate the rate of change of the total mass of the fetus, \( \frac{\mathrm{dm}}{\mathrm{dt}} \). \( \frac{\mathrm{dm}}{\mathrm{dt}}=\square \mathrm{g} \) per day (Round to five decimal places as needed.)