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rusak2 [61]
3 years ago
15

(( PLEASE I NEED HELP ASAP THIS IS REALLY IMPORTANT ))

Mathematics
1 answer:
docker41 [41]3 years ago
8 0
1. Complementary angles are any 2 angles whose sum is 90°

    Complementary angles are any 2 angles whose sum is 180°

a) BXC and CXD add up to 90°, so they are 2 complementary angles

b) AXC and CXD add up to 180°, so they are 2 supplementary angles

c) 2x+40°=90° (as BXC and CXD are complementary)

     2x=90°- 40°
      2x=50°
        x=50°/2=25°
     
      5y=90°
        \frac{5y}{5}= \frac{90}{5}
          y=18°
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Compute the matrix of partial derivatives of the following functions.
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For a vector-valued function

\mathbf f(\mathbf x)=\mathbf f(x_1,x_2,\ldots,x_n)=(f_1(x_1,x_2,\ldots,x_n),\ldots,f_m(x_1,x_2,\ldots,x_n))

the matrix of partial derivatives (a.k.a. the Jacobian) is the m\times n matrix in which the (i,j)-th entry is the derivative of f_i with respect to x_j:

D\mathbf f(\mathbf x)=\begin{bmatrix}\dfrac{\partial f_1}{\partial x_1}&\dfrac{\partial f_1}{\partial x_2}&\cdots&\dfrac{\partial f_1}{\partial x_n}\\\dfrac{\partial f_2}{\partial x_1}&\dfrac{\partial f_2}{\partial x_2}&\cdots&\dfrac{\partial f_2}{\partial x_n}\\\vdots&\vdots&\ddots&\vdots\\\dfrac{\partial f_m}{\partial x_1}&\dfrac{\partial f_m}{\partial x_2}&\cdots&\dfrac{\partial f_n}{\partial x_n}\end{bmatrix}

So we have

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D f(x,y)=\begin{bmatrix}\dfrac{\partial(e^x)}{\partial x}&\dfrac{\partial(e^x)}{\partial y}\\\dfrac{\partial(\sin(xy))}{\partial x}&\dfrac{\partial(\sin(xy))}{\partial y}\end{bmatrix}=\begin{bmatrix}e^x&0\\y\cos(xy)&x\cos(xy)\end{bmatrix}

(b)

D f(x,y,z)=\begin{bmatrix}\dfrac{\partial(x-y)}{\partial x}&\dfrac{\partial(x-y)}{\partial y}&\dfrac{\partial(x-y)}{\partial z}\\\dfrac{\partial(y+z)}{\partial x}&\dfrac{\partial(y+z)}{\partial y}&\dfrac{\partial(y+z)}{\partial z}\end{bmatrix}=\begin{bmatrix}1&-1&0\\0&1&1\end{bmatrix}

(c)

Df(x,y)=\begin{bmatrix}y&x\\1&-1\\y&x\end{bmatrix}

(d)

Df(x,y,z)=\begin{bmatrix}1&0&1\\0&1&0\\1&-1&0\end{bmatrix}

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Step-by-step explanation:

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