Instructions:  On a separate sheet of paper, answer each of 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 & 1\end{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”:”\(x^{(0)}=\begin{bmatrix} 0 & -1\end{bmatrix}^\top \)”}. 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”:”\( a\in \mathbb{R} \)”}, and vectors  d ( 0 ) = [ − 3 1 ] and d ( 1 ) = [ − 1 2 ] . {“version”:”1.1″,”math”:”d^{(0)}=\begin{bmatrix} -3\\1\end{bmatrix}\quad \mbox{and}\quad d^{(1)}=\begin{bmatrix} -1\\2\end{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. 

When elements combine to form compounds,

Determine the answer to the following equation with correct number of significant figures: 106 ÷ 9.02 x 1.9 = ________

A nurse is collecting data from a 12-year-old child who reports experiencing mild, aching pain daily for the last 9 months. In which of the following ways should the nurse classify the pain?

A nurse is collecting data on a 4-month-old infant’s dietary intake. Which of the following findings demonstrates healthy eating habits?

A parent of a nonverbal 2-year-old suspected of having autism spectrum disorder (ASD) presents to the pediatric clinic to meet with the genetic nurse. The parent states “My spouse had all the prenatal tests possible while pregnant. Why didn’t anything show up then?” Which of the following responses should the nurse provide to the parent?

A nurse is observing a preschool classroom. A teacher asks the nurse what they would see in a child that is emotionally abused. Which of the following responses should the nurse make?

A nurse is providing care for a toddler who was recently diagnosed with autism spectrum disorder (ASD). The toddler’s caregiver states “I am happy we received an early diagnosis so we can begin treatment immediately! The earlier we treat it, the sooner they can be cured.” Which statement should the nurse use to respond to the caregiver?

A nurse is reinforcing teaching for the parents of a child who has Duchenne muscular dystrophy (DMD) about the testing indicated with this disorder. Which of the following diagnostic tests should the nurse include in the teaching?

A nurse in an urgent care unit receives a call from a concerned caregiver stating that their toddler ingested cleaning supplies and is asking what they should do. Which of the following statements should the nurse respond with?