Cálculo preguntas y respuestas

A rectangular box with a volume of \( 540 \mathrm{ft}^{3} \) is to be constructed with a square base and top. The cost per square foot for the bottom is \( 20 \notin \), for the top is \( 10 \notin \), and for the sides is \( 2.5 \notin \). What dimensions will minimize the cost?

Analiza el siguiente procedimiento de derivación y selecciona correcto ó Incorrecto \( \begin{array}{l}\text { Formula: } \\ y=\log _{a} u \leftrightarrow y^{\prime}=\left(\frac{1}{\ln a}\right)\left(\frac{u^{\prime}}{u}\right) \\ \begin{array}{l}\text { Función: } \\ f(x)=\log _{3}\left(5 x^{2}+3\right)^{2} \\ a=3 \\ u=\left(5 x^{2}+3\right)^{2} \\ u^{\prime}=2\left(5 x^{2}+3\right)(10 x)=100 x^{3}+60 x \\ \text { Procedimiento: } \\ f^{\prime}(x)=\left(\frac{1}{\ln 3}\right)\left(\frac{100 x^{3}+60 x}{\left(5 x^{2}+3\right)^{2}}\right) \\ f^{\prime}(x)=\frac{100 x^{3}+60 x}{\ln 3\left(25 x^{4}+15 x^{2}+9\right)}\end{array}\end{array} \).

10 The diagram shows a right circular cone with radius 10 cm and height 30 cm . The cone is imitially completely fllled with water. Water leaks out of the cone through a small hole at the vertex at a rate of \( 4 \mathrm{~cm}^{-1} \mathrm{~s}^{-1} \). a Show that the volume of water in the cone. \( V \mathrm{~cm}^{3} \), when the height of the water is \( h \mathrm{~cm} \) is given by the formula \( V=\frac{\pi h}{27} \). b Find the rate of change of \( h \) when \( h=20 \).

\( \int _ { 0 } ^ { 1 } ( 1 - x ^ { 2 } ) ^ { \frac { 7 } { 2 } } d x \)

Analiza el siguiente procedimiento en el \( * \) que se deriva una función logarítmica, selecciona Correcto o Incorrecto según sea el caso \[ y=\log _{a} u \leftrightarrow y^{\prime}=\left(\frac{1}{\ln a}\right)\left(\frac{u^{\prime}}{u}\right) \] \( \begin{array}{l}f(x)=\log _{6}\left(4 x^{2}-2\right)^{5} \\ a=6 \\ u=\left(4 x^{2}-2\right)^{5} \\ u^{\prime}=(5)\left(4 x^{2}-2\right)^{4}(8 x) \\ f^{\prime}(x)=\left(\frac{1}{\ln 6}\right)\left(\frac{(5)\left(4 x^{2}-2\right)^{4}}{\left(4 x^{2}-2\right)^{5}}\right)\left(\frac{8 x}{1}\right) \\ f^{\prime}(x)=\left(\frac{40 x}{\ln 6}\right)\left(\frac{\left(4 x^{2}-2\right)^{4}}{\left(4 x^{2}-2\right)^{5}}\right)=\left(\frac{40 x}{\ln 6\left(4 x^{2}-2\right)^{4}}\right)\end{array} \)

What is the general solution of the equation: \[ \frac{d^{2} y}{d x^{2}}-4 y=5 \cos 3 x \]

soit \( f \) la fonction définie sur \( ]-\infty ; 2[U] 2 ;+\infty[ \) par \( f(x)=\frac{x+1}{2-x} \). Démontrer que \( f \) est dérivable en 3 et donner la valeur de \( f^{\prime}(3) \).

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. \( R(x)=8 x-4 x^{2}, C(x)=x^{3}-5 x^{2}+3 x+1 \)

\( y^{\prime} \) si \( x^{4}+y^{4}=16 \).

Obtener la derivada de la siguiente Función Exponencial \( y=e^{x^{2}+5 x+1} \)