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Advanced Math Archive: Questions from October 22, 2022

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  2. Use the product rule to differentiate (a) \( y=x(3 x+4)^{2} \) (b) \( y=x^{2}(x-2)^{3} \) (c) \( y=x \sqrt{(x+2)} \) (d) \( y=(x-1)(x+6)^{3} \) (e) \( y=(2 x+1)(x+5)^{3} \) (f) \( y=x^{3}(2 x-5)^{4
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  1. Use the chain rule to differentiate (a) \( y=(5 x+1)^{3} \) (b) \( y=(2 x-7)^{n} \) (c) \( y=(x+9)^{5} \) (d) \( y=\left(4 x^{2}-7\right)^{3} \) (e) \( y=\left(x^{2}+4 x-3\right)^{4} \) (f) \( y=\s
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• Let ℓ1 be the line passing through the points A=(9,8,3) and B=(11,7,4), and let ℓ2 be the line with symmetric equations ..... where c is a constant. (a) (2pts) Write the vector equation of ℓ1. (
  Sea \( \ell_{1} \) la recta que pasa por los puntos \( A=(9,8,3) \) y \( B=(11,7,4) \) y sea \( \ell_{2} \) la recta con ecuaciones simétricas \( \frac{x-1}{3}=\frac{y-10}{-1}=\frac{z-c}{-7} \). dond
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• number 27 and 31 please
  In Exercises 23-34, find \( f_{x}, f_{y} \), and \( f_{z} \). 23. \( f(x, y, z)=1+x y^{2}-2 z^{2} \) 24. \( f(x, y, z)=x y+y z+x z \) 25. \( f(x, y, z)=x-\sqrt{y^{2}+z^{2}} \) 26. \( f(x, y, z)=\left(
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• The temperature at a point (x, y) is T (x, y), measured in degrees Celsius. An insect crawls in such a way that its position after t seconds is given by x = √1 + t, y = 2 + (t, where x and y are me
  [7 pts.] La temperatura en un punto \( (x, y) \) es \( T(x, y) \), medida en grados Celsius. Un insecto se arrastra de tal manera que su posición después de \( t \) segundos es dada por \( x= \) \(
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  Assuming that \( \alpha \neq 0 \) and \( \beta \neq 0 \), then the solution of the system \[ \begin{array}{l} 3 \alpha x+\beta y=2 \\ 2 \beta x-\alpha y=1 \end{array} \] is ... \[ \ldots x=\frac{3 \al
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• Study whether or not the following families of vectors are linearly independent.
  (a) \( \left\{\left(\begin{array}{rr}1 & -1 \\ 0 & 1\end{array}\right),\left(\begin{array}{rr}1 & 0 \\ -1 & 1\end{array}\right)\right\} \subset \mathcal{M}_{2}(\mathbb{R}) \) (b) \( \left\{x-1, x+1, x
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  Given magnitudes \( w, x, y, z \), Euclid defines \( w: x
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  Graph each of the following: a) \( \mathrm{y}=\sqrt{\mathrm{x}-3} \) b) \( y=3 \sqrt{x}-5 \) c) \( y=\sqrt{2 x-8} \) d) \( y=\sqrt{\frac{1}{2} x+3} \) e) \( y=-\sqrt{-x+1} \) f) \( x=\sqrt{y}+4 \)
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• Differentiate the function
  b. \( y=\frac{2 t^{2}+t^{3}}{5 e^{t} \cos t} \)
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  sec 3.1: Problem 2 (1 point) Solve \( \begin{array}{ccr} & x^{\prime}=y-x+t & x(0)=8 \\ & y^{\prime}=y & y(0)=2 \\ x(t)= & & \text { help (formulas) } \\ & & \\ y(t)= & \text { help (formulas) }\end{a
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