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

What is \(\frac{\partial \cos(xyz)}{\partial x}\)?

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

A 19-year-old man complains of severe fatigue, sore throat, and swollen lymph nodes. Manual differential counts: WBC: 11.3                                      Band neutrophils: 8 RBC: 4.99                                       Segmented neutrophils: 12 Hgb: 16.1                                        Lymphocytes: 78 Hct: 49.0                                         Monocytes: 2 MCV: 98.2                                      Eosinophils: MCH: 32.3                                      Basophils: MCHC: 33.3                                   Metamyelocytes: RDW: 11.9                                     Myelocytes: PLT: 138.9                                      Promyelocytes: The pic below demonstrates many of the lymphocytes seen on the peripheral smear.      What additional test should be performed?    

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?

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 \((x(t),y(t),z(t))=(t,t,1)\).