ICE agents are part of the federal government and therefore…

Questions

ICE аgents аre pаrt оf the federal gоvernment and therefоre do not need to abide by the 4th amendment.

Instructiоns: On а sepаrаte sheet оf paper, answer each оf the exam problems shown below. Write your answers clearly. Unless otherwise stated, you will need to justify your answers to get the full credit. Problem 1. (10 pts) Compute the linear, l(x1,x2){"version":"1.1","math":"(l(x_1,x_2))"}, and quadratic, q(x1,x2){"version":"1.1","math":"( q(x_1,x_2))"}, approximations of the function f ( x 1 , x 2 ) = 5 x 1 2 x 2 − x 1 4 x 2 − 3 x 1 + 1 , {"version":"1.1","math":"f(x_1,x_2)=5x_1^2x_2 - x_1^4x_2 - 3x_1+1, "} at the point x ( 0 ) = [ 0 1 ] ⊤ {"version":"1.1","math":"x^{(0)}=begin{bmatrix} 0 & 1end{bmatrix}^top"} Problem 2. (10 pts) Given a point x ( 0 ) = 0 {"version":"1.1","math":"x^{(0)}=0"} and the equation f = f ( x ) = − x 3 + 5 x 2 − 2 x − 3. {"version":"1.1","math":"f=f(x)=-x^3+5x^2-2x-3."}Find x(1){"version":"1.1","math":"( x^{(1)})"}using Newton's method of tangents for finding a root of f = 0. {"version":"1.1","math":"(f=0.)"} Problem 3. (10 pts) Consider a rectangle with the shorter side a = 1 {"version":"1.1","math":"( a=1)"} and the longer side b . {"version":"1.1","math":"(b.)"} Find b {"version":"1.1","math":"(b)"} for which the sides of the rectangle satisfy the golden section. In your manipulations you may find useful that 5 = 2.236 {"version":"1.1","math":"( sqrt{5}=2.236 )"} Problem 4. (10 pts) What is the largest a {"version":"1.1","math":"(a )"} for which the quadratic form, f = f ( x 1 , x 2 , x 3 ) = x 1 2 − 2 x 1 x 2 − 2 x 1 x 3 + a x 2 2 + 2 x 3 2 , {"version":"1.1","math":"f=f(x_1,x_2,x_3)=x_1^2-2x_1x_2-2x_1x_3+ax_2^2+2x_3^2,"}is positive semi-definite and not positive definite? Problem 5. (20 pts) (10 pts) Does the function f ( x 1 , x 2 ) = 1 2 x 1 2 − x 1 x 2 + 3 2 x 2 2 + x 2 + 3 {"version":"1.1","math":"f(x_1,x_2)=frac{1}{2}x_1^2-x_1x_2+frac{3}{2}x_2^2 + x_2 +3 "} have a minimizer or a maximizer? If it does, then find it; otherwise explain why it does not. (10 pts) Does the function f ( x 1 , x 2 ) = − x 1 2 + x 1 x 2 − x 2 2 − x 2 + 1 {"version":"1.1","math":"f(x_1,x_2)=-x_1^2+x_1x_2-x_2^2 - x_2 +1 "}{"version":"1.1","math":"f(x_1,x_2)=2x_1^2+4x_1x_2+2x_2^2 - x_1 +3 "}have a minimizer or a maximizer? If it does, then find it; otherwise explain why it does not. Problem 6. (15 pts) Bracket the minimizer of f = 2 x 1 2 + x 2 2 {"version":"1.1","math":"f=2x_1^2+x_2^2 "} on the line passing through the point x ( 0 ) = [ − 5 0 ] ⊤ {"version":"1.1","math":"x^{(0)}=left[begin{array}{cc} -5 & 0 end{array}right]^{top}"} in the direction d = [ 10 10 ] ⊤ {"version":"1.1","math":"( d =left[begin{array}{cc} 10 & 10 end{array}right]^{top})"}. Use ε = 0.1 {"version":"1.1","math":"(varepsilon=0.1)"}. Problem 7. (15 pts) It is well known that Newton's method (also known as the Newton-Raphson method) seeks a point that satisfies the FONC for an extremizer. Apply the Newton's algorithm to the function, f ( x 1 , x 2 ) = − 1 2 x 1 2 + 1 2 x 2 2 − 12 x 1 + x 2 + 5. {"version":"1.1","math":" f(x_1,x_2)= -frac{1}{2}x_1^2 + frac{1}{2}x_2^2 -12x_1+ x_2 + 5. "} The starting point is x ( 0 ) = [ 0 − 1 ] ⊤ {"version":"1.1","math":"(x^{(0)}=begin{bmatrix} 0 & -1end{bmatrix}^top )"}. Is the point that you obtain a maximizer, a minimizer, or neither? Justify your answer. Problem 8. (10 pts) Given Q = [ 1 1 1 a ] , {"version":"1.1","math":"Q=begin{bmatrix} 1 & 1\1 & a end{bmatrix},"}where a ∈ R {"version":"1.1","math":"( ain mathbb{R} )"}, and vectors d ( 0 ) = [ − 3 1 ] and d ( 1 ) = [ − 1 2 ] . {"version":"1.1","math":"d^{(0)}=begin{bmatrix} -3\1end{bmatrix}quad mbox{and}quad d^{(1)}=begin{bmatrix} -1\2end{bmatrix}."}Find a {"version":"1.1","math":"(a)"} for which vectors d ( 0 ) {"version":"1.1","math":"( d^{(0)})"} and d ( 1 ) {"version":"1.1","math":"( d^{(1)})"} are Q − {"version":"1.1","math":"( Q-)"}orthogonal. *** Congratulations, you are almost done with Midterm Exam 1. DO NOT end the Honorlock session until you have submitted your work to Gradescope. When you have answered all questions: Use your smartphone to scan your answer sheet and save the scan as a PDF. Make sure your scan is clear and legible. Submit your PDF to Gradescope as follows: Email your PDF to yourself or save it to the cloud (Google Drive, etc.). Click this link to go to Gradescope: Midterm Exam 1 Click the button below to agree to the honor statement. Click Submit Quiz to end the exam. End the Honorlock session.