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Transformations of quadratic functions

Cameron wants to prove the Alternate Interior Angles Theorem. In the diagram, PQundefined∥RSundefined \overleftrightarrow{P Q} \| \overleftrightarrow{R S} \newlineComplete Cameron's proof that alternate interior angles ∠PVW \angle P V W and ∠VWS \angle V W S are congruent.\newlineConstruct the midpoint M M of WV‾ \overline{W V} , then rotate PQundefined,RSundefined \overleftrightarrow{P Q}, \overleftrightarrow{R S} , and TUundefined180∘ \overleftrightarrow{T U} 180^{\circ} about M M to get P′Q′undefined,R′S′undefined \overleftrightarrow{P^{\prime} Q^{\prime}}, \overleftrightarrow{R^{\prime} S^{\prime}} , and T′U′undefined \overleftrightarrow{T^{\prime} U^{\prime}} . Since M M lies on ∠PVW \angle P V W 11 and the rotation is ∠PVW \angle P V W 22 coincides with ∠PVW \angle P V W 33 are equidistant from M M on ∠PVW \angle P V W 11, so ∠PVW \angle P V W 66 coincides with ∠PVW \angle P V W 77 and ∠PVW \angle P V W 88 coincides with ∠PVW \angle P V W 33\newlineA ∠VWS \angle V W S 00 rotation of a line is a parallel line, and the only line parallel to ∠VWS \angle V W S 11 that passes through ∠PVW \angle P V W 77 is ∠PVW \angle P V W 33 Therefore, ∠PVW \angle P V W 33 coincide, and ∠VWS \angle V W S 55 and ∠VWS \angle V W S 66 coincide. So, the rotation maps ∠PVW \angle P V W 33\newlineSince rotation ∠PVW \angle P V W 33 , ∠VWS \angle V W S 99.
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22. Dengan metode Lagrange cari maksimum dan m m imum f(x1,x2)=x12+x22 f\left(x_{1}, x_{2}\right)=x_{1}^{2}+x_{2}^{2} titik-titik pada ellip den a persamaan x12+2x22=1 x_{1}^{2}+2 x_{2}^{2}=1 .\newline33. Diberikan masalah konsumen dengan fungsi utility u(x1,x2)=x1x2 u\left(x_{1}, x_{2}\right)=x_{1} x_{2} dengan x1 x_{1} dan x2 x_{2} masing-masing jumlah barang pertarna dan kedua dalam kg \mathrm{kg} . Diketahui harga barang pertama 22 juta rupiah per kg \mathrm{kg} dan harga barang kedua 11 juta rupiah per kg \mathrm{kg} . Jika maksimum pembelian adalah 66 juta rupiah, modelkan masalah konsumen di atas dengan masalah optimisasi pemrograman cekung! Selesaikan masalah optimisasi pemrograman cekung dengan metode Kuhn Tucker (Petunjuk :Tuliskan fungsi Langrange ca.m rala Kuhn Tucker). Interpretasikan jawaban anda untuk masalah konsumen, tas.
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包不武非\newlineFacultatea de Calculatoare, Informatică și Microelectronică\newlineTestarea nr. 11 la Analiza Matematică 22\newlineVarianta 1616\newline11. (66p) Aflați masa arcului de curbă AB A B cu densitatea μ(x,y,z)=3x−5y+4z \mu(x, y, z)=3 x-5 y+4 z , ce unește punctele A(−1,2,5) A(-1,2,5) si B(1,−2,4) B(1,-2,4) .\newline22. (55p) Calculați integrala curbilinie\newline∫AB(2x+5y)dx+8xdy \int_{A B}(2 x+5 y) d x+8 x d y \newlinede-a lungul curbei y=x3 y=x^{3} ce unește punctele A(−2,−8) A(-2,-8) și B(1,1) B(1,1) .\newline33. (10p) (10 p) Să se determine aria suprafeței paraboloidului de rotaţie y=x2+z2 y=x^{2}+z^{2} , decupată (tăiată) de planul y=2 y=2 .\newline44. (1414p) Să se calculeze integrala de suprafață\newlineI=∬S3xdydz−ydzdx−2zdxdy, I=\iint_{S} 3 x d y d z-y d z d x-2 z d x d y, \newlineande μ(x,y,z)=3x−5y+4z \mu(x, y, z)=3 x-5 y+4 z 00 este faţa de sus a părţii planului μ(x,y,z)=3x−5y+4z \mu(x, y, z)=3 x-5 y+4 z 11, decupată (tăiată) de planurile le coordonate μ(x,y,z)=3x−5y+4z \mu(x, y, z)=3 x-5 y+4 z 22 şi situată în al patrulea octant.\newline(55p) Calculați derivata funcției μ(x,y,z)=3x−5y+4z \mu(x, y, z)=3 x-5 y+4 z 33 în punctul μ(x,y,z)=3x−5y+4z \mu(x, y, z)=3 x-5 y+4 z 44 direcția vectorului μ(x,y,z)=3x−5y+4z \mu(x, y, z)=3 x-5 y+4 z 55.
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Let G=C4×C2=⟨σ,τ∣σ4=τ2=e,στ=τσ⟩ G=C_{4}\times C_{2}=\langle \sigma,\tau \mid \sigma^{4}=\tau^{2}=e, \sigma\tau=\tau\sigma \rangle . Consider the matrices\newlineS=(10 0i)andT=(−10 01) S=\begin{pmatrix} 1 & 0 \ 0 & i \end{pmatrix} \quad \text{and} \quad T=\begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} \newlineVerify that sending σ↦S \sigma \mapsto S and τ↦T \tau \mapsto T defines a representation of G G . Now let\newlineQ=(i0 11)andR=(−10 i+11) Q=\begin{pmatrix} i & 0 \ 1 & 1 \end{pmatrix} \quad \text{and} \quad R=\begin{pmatrix} -1 & 0 \ i+1 & 1 \end{pmatrix} \newlineVerify that sending σ↦Q \sigma \mapsto Q and τ↦R \tau \mapsto R also defines a representation of G G . Show that S S is conjugate to S=(10 0i)andT=(−10 01) S=\begin{pmatrix} 1 & 0 \ 0 & i \end{pmatrix} \quad \text{and} \quad T=\begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} 00. Show that S=(10 0i)andT=(−10 01) S=\begin{pmatrix} 1 & 0 \ 0 & i \end{pmatrix} \quad \text{and} \quad T=\begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} 11 is conjugate to S=(10 0i)andT=(−10 01) S=\begin{pmatrix} 1 & 0 \ 0 & i \end{pmatrix} \quad \text{and} \quad T=\begin{pmatrix} -1 & 0 \ 0 & 1 \end{pmatrix} 22. Are these two representations equivalent?
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