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Compute the gradient of the following function \(f(x,y,z)=xyz\).

Sketch the level curves and gradient vector field of the function \(f(x,y)=x^2+3y^2\)

What is the max of the following function on the unit disk \(f(x,y)=x^2+y^2+x.\)

Select the correct graph and the correct contour plot of level curves for the function  \(f(x,y) = x^2-y^2 \)  

Two very long wires carry a current i = 10 A due to the emf from the battery. The wire has two semi-circular kinks about point P, one with a radius of curvature of R = 3 cm and the other 2R. What is the magnitude and direction of the magnetic field at point P?

If the Clearview Mono test is positive, #1. What information is most diagnostic? #2. Based on all the data provided, what condition is most likely?    

Detection of immunoglobulins by electrophoresis and testing for specific antibodies by other lab methods are very helpful in the diagnosis of many types of diseases. Immunoglobulins are divided into five major classes (IgG, IgM, IgA, IgD, and IgE). Ig is the abbreviation for immunoglobulin.  Case study: History and Physical Examination Phong is a 5-year-old boy who has suffered from repeated upper respiratory infections since he was 16 months old. At 4 years of age, he was diagnosed with allergic asthma. The following winter, at age 5, he was admitted to the hospital twice for acute asthma with a secondary bacterial pulmonary infection. At the second admission, a complete blood count (CBC), sputum culture, and immune status workup were ordered. Laboratory Data Assay Patients results Reference range HemoglobinHematocritTotal WBC countAbsolute lymphocyte countT lymphocytesB lymphocytes 13.540%16.2 × 109/L4.3 × 109/L2.4 × 109/L1.0 × 109/L    2.7–5.4 × 109/L2.7–5.3 × 109/L0.6–1.4 × 109/L Antibodies to normal childhood immunization—presentSerum immunoglobulin profileIgM, IgG, and IgA bands: normalIgE and IgD: no bands observed Serum total protein IgMIgGIgA 2.4 g/L18.9 g/L0.020 g/L 0.5–2.0 g/L4.9–16.1 g/L0.4–2.0 g/L Questions What abnormal laboratory data does the patient demonstrate? Decreased T lymphocyte count Decreased IgA level Decreased total protein Both A and B What is the cause of this patient’s recurring infections? Neutropenia Lymphocytosis Impaired phagocytosis IgA deficiency Which clinical conditions are associated with this patient’s immune problem? Recurrent infections Allergies Increased incidence of inflammatory conditions All of the above. Bonus Question (2 points): What is the presumptive diagnosis?

Use the chain rule to compute \(\frac{d f(x(t),y(t),z(t))}{dt}\bigg|_{t=0}\) where \(f(x,y,z)=x^2+xy-z\) and \((x(t),y(t),z(t))=(t,t,1)\).

The following dilutions were set up to titer an antibody. The following results were obtained: 1:4+, 1:8+, 1:16+. 1:32+, 1:64+. 1:128-. How should the titer be reported out?

Heather, a 37-year-old woman, is a first-time donor at your blood center. She is a healthy donor, has an unremarkable medical history, and is not taking any medications. Initial ABO phenotyping (RBC’s antigens) results are as follows: ABO Testing Results Donor Red Cells Antigen testing (Front Type)   Antiserum-A reagent Antiserum-B reagent Antiserum-D reagent 0 reaction or negative 4+ reaction 4+ reaction     1. With the antiserum-B reagent and antiserum-D reagent reacting with Heather’s antigens on her RBC’s,  what ABO blood type does Heather presumably possess? 2. What does the antiserum-A and the zero or negative reaction indicate? 3. Name 3 functions of an antigen.