Damage to which of the following cranial nerves may result i…
Questions
Dаmаge tо which оf the fоllowing crаnial nerves may result in weakness in the sternocleidomastoid muscle?
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. (15 pts) A discrete-time system with the input u[k]{"version":"1.1","math":"( u[k] )"} and output y[k]{"version":"1.1","math":"( y[k] )"} is given above. The two blocks marked with ''T{"version":"1.1","math":"( T )"}'' are time-delay units that delay their input signals by one time step (e.g., input f[k]{"version":"1.1","math":"( f[k] )"} results in the output f[k−1]{"version":"1.1","math":"( f[k-1] )"} ). Find the transfer function relating {"version":"1.1","math":""}Y(z){"version":"1.1","math":"Y(z)"} and U(z){"version":"1.1","math":"U(z)"}, where Y(z){"version":"1.1","math":"Y(z)"} and U(z){"version":"1.1","math":"U(z)"} are the Z{"version":"1.1","math":"Z"}-transforms of the output y(k){"version":"1.1","math":"y(k)"} and input u(k){"version":"1.1","math":"u(k)"}, respectively. Problem 2. (15 pts) Find the transfer function matrix relating Y(s){"version":"1.1","math":"( Y(s) )"} and U(s){"version":"1.1","math":"( U(s) )"} of the following system: [x˙1(t)x˙2(t)]=[−20−91][x1(t)x2(t)]+[1011]u(t),y(t)=[01][x1(t)x2(t)].{"version":"1.1","math":"begin{eqnarray*} left[begin{array}{c} dot{x}_1(t)\ dot{x}_2(t) end{array}right]&=&left[begin{array}{cc} -2 & 0\ -9 & 1 end{array}right]left[begin{array}{c} {x}_1(t)\ {x}_2(t) end{array}right] + left[begin{array}{cc} 1 & 0\ 1& 1 end{array}right]u(t),\ y(t) &=& left[begin{array}{cc} 0 & 1 end{array}right]left[begin{array}{c} {x}_1(t)\ {x}_2(t) end{array}right]. end{eqnarray*}"} Here, Y(s){"version":"1.1","math":"( Y(s) )"} and U(s){"version":"1.1","math":"( U(s) )"} are the Laplace transforms of the output y(t){"version":"1.1","math":"( y(t) )"} and input u(t){"version":"1.1","math":"( u(t) )"}, respectively. Problem 3. (20 pts) Suppose the characteristic polynomial of an unknown matrix A∈R3×3{"version":"1.1","math":"( A in {mathbb R}^{3times 3} ) "} is χA(λ)=det(λI3−A)=λ3−4λ2+5λ−2.{"version":"1.1","math":"chi_A(lambda)=det(lambda I_3-A)=lambda^3 - 4lambda^2 +5lambda -2. "} (5 pts) One of the zeros of{"version":"1.1","math":""}the characteristic polynomial is 2. What are the eigenvalues of A{"version":"1.1","math":"( A )"}? Is A{"version":"1.1","math":"( A )"} one-to-one? onto? (10 pts) For given t≥0{"version":"1.1","math":"( tge 0 )"}, express eAt{"version":"1.1","math":"( e^{At} )"} as a linear combination of the identity matrix I{"version":"1.1","math":"( I )"}, A{"version":"1.1","math":"( A )"}, and A2{"version":"1.1","math":"( A^2 )"}. (5 pts) From the given information, can you tell if the continuous-time LTI system x˙=Ax{"version":"1.1","math":"( dot x = Ax )"} is stable, marginally stable, or unstable? If so, what is the conclusion? If not, explain why. What about the discrete-time LTI system x[k+1]=Ax[k]{"version":"1.1","math":"( x[k+1]=A x[k] )"}? Problem 4. (25 pts) Consider the following two problems whose solutions are related. (15 pts) Given a matrix A=[−310−1]{"version":"1.1","math":"( A=begin{bmatrix} -3& 1\ 0&-1end{bmatrix} )"}, find eAt{"version":"1.1","math":"( e^{At} )"} for t≥0{"version":"1.1","math":"( tge 0 )"}. (10 pts) Find the state transition matrix Φ(t,s){"version":"1.1","math":"( Phi(t,s) )"}, s,t≥0{"version":"1.1","math":"( s,tge0 )"}{"version":"1.1","math":"( s,tge0 )"}, for the following system: x˙(t)=[−3e−te−t0−e−t]x(t).{"version":"1.1","math":"begin{align*} dot x(t) = begin{bmatrix} -3e^{-t} & e^{-t} \ 0 & -e^{-t} end{bmatrix} x(t). end{align*}"} Problem 5. (25 pts) Consider a linear system x˙=Ax{"version":"1.1","math":"( dot x=Ax )"} with A∈R4×4{"version":"1.1","math":"( Ainmathbb R^{4times 4} )"} given by A=[v1v2v3v4]⏟T[−1000001000010000]⏟J[w1Tw2Tw3Tw4T]⏟T−1{"version":"1.1","math":"begin{align*} A=underbrace{begin{bmatrix} v_1&v_2&v_3&v_4 end{bmatrix}}_T underbrace{begin{bmatrix} -1&0&0&0\0&0&1&0\0&0&0&1\0&0&0&0end{bmatrix}}_J underbrace{begin{bmatrix} w_1^T\w_2^T\w_3^T\w_4^Tend{bmatrix}}_{T^{-1}} end{align*}"} for some vi,wi∈R4{"version":"1.1","math":"( v_i, w_i in mathbb R^4 )"}, i=1,2,3,4{"version":"1.1","math":"( i=1,2,3,4)"}. (6 pts) Find all the modes of the solutions to the system x˙=Ax{"version":"1.1","math":"( dot x=Ax )"}. (3 pts) Given the initial condition x(0)=v1−2v3+3v4∈R4{"version":"1.1","math":"( x(0)=v_1-2v_3+3v_4inmathbb R^4 )"}, find the corresponding solution x(t){"version":"1.1","math":"( x(t) )"} and write it as a proper linear combination of the modes obtained in the previous sub-problem. (6 pts) Find three nonzero initial conditions x(0){"version":"1.1","math":"( x(0) )"} from which the solution x(t){"version":"1.1","math":"( x(t) )"} as t→∞{"version":"1.1","math":"( ttoinfty )"} will: (i) converge to 0; (ii) remain constant; (iii) diverge to infinity, respectively. In the next two subproblems, consider the discrete-time system x[k+1]=Ax[k]{"version":"1.1","math":"( x[k+1]=Ax[k] )"} with A{"version":"1.1","math":"( A )"} given above. (6 pts) Find all the modes of the solutions to the system x[k+1]=Ax[k]{"version":"1.1","math":"( x[k+1]=Ax[k] )"}. (4 pts) Is the discrete-time system x[k+1]=Ax[k]{"version":"1.1","math":"( x[k+1]=Ax[k] )"} stable, marginally stable, or unstable? *** Congratulations, you are almost done with Midterm Exam 1. DO NOT end the Examity 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. 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An 86-yeаr-оld client presents tо the emergency depаrtment (ED) with the chief cоmplаint of headache. The client describes the headache as sudden in onset 4 hours ago, located in the front of the head, and with pain that is constant, stabbing, and unrelieved by acetaminophen extra-strength 1000 mg. The client is accompanied by an adult family member, who reports that the client sustained a fall 2 days ago; the family member thinks it is possible that the client hit their head on a piece of furniture. The family member has noticed periods of confusion in the client since the fall and states that they have been uncomfortable leaving the client alone since that time. The family member reports that the confusion seems to be worsening and that they are concerned about an injury from the fall and concerned for the client’s safety. The ED physician examines the client and notes the presence of ipsilateral dilation (widening of the pupil on the same side of the body) of the right pupil and prescribes a computed tomography (CT) scan of the brain to rule out a subdural hematoma. While waiting for the CT scan, the licensed practical nurse (LPN) assisting in caring for the client is directed to manage care based on the head injury. Complete the following sentence by selecting from the list of Nursing Actions. Answer may only be used once. In the immediate care of the client, the nurse would first _______________, second ________________, and third ______________. FIRST SECOND THIRD A. Stabilize the cervical spine A. Stabilize the cervical spine A. Stabilize the cervical spine B. Apply O2 via nonrebreather mask B. Apply O2 via nonrebreather mask B. Apply O2 via nonrebreather mask C. Assist to establish intravenous access with two large-bore catheters C. Assist to establish intravenous access with two large-bore catheters C. Assist to establish intravenous access with two large-bore catheters D. Assess airway, breathing, and circulation D. Assess airway, breathing, and circulation D. Assess airway, breathing, and circulation