What is the approximate direction of the gradient vector at each of the following points?

Which of the following solids has a volume equal to the integral ∫0π∫01∫r1-r2 r dzdrdθ{“version”:”1.1″,”math”:”∫0π∫01∫r1-r2 r dzdrdθ”}

The function g(x, y) has tangent lines at the point (1, 2) with directions 2, 0, 8{“version”:”1.1″,”math”:”2, 0, 8″}  and 0, -13, 2{“version”:”1.1″,”math”:”0, -13, 2″}. Find  ∇g1,2{“version”:”1.1″,”math”:”∇g1,2″} . Write you answer using the greater and less than symbols for brackets, commas, numbers and no spaces. _______

Which point has the gradient vector with the smallest magnitude?

Briefly explain why antibiotics have no effect against viral infections (e.g. influenza).  To get full credit, you must show that: (1) you’re familiar with at least one of the ways in which antibiotics work, and (2) you can relate this to the processes involved in viral replication.

Which of the following solids has a volume equal to the integral ∫01∫-1-x21-x2∫x2+y21  1  dzdydx{“version”:”1.1″,”math”:”∫01∫-1-x21-x2∫x2+y21  1  dzdydx”}

Suppose that someone accidentally breathes in some tree pollen.  After a sufficient period of time passes, this person shows signs of being allergic to tree pollen.  In the space below, describe the entire sequence of events leading to the development of an allergic reaction to tree pollen. This sequence should start with being exposed to tree pollen and end with showing the symptoms/signs of tree pollen allergy.  Your answer should be comprehensive. Include as many details as you can about the cells, molecules, receptors and specific events that are involved in this process. Hint: It might be helpful for you to draw out this process first on your piece of scratch paper (similar to a concept map); you can then use that to help you answer this question. You won’t be asked to turn in the drawing on your scratch paper.

Which of the following solids has a volume equal to the integral ∫π/23π/2∫01∫r1 r dzdrdθ{“version”:”1.1″,”math”:”∫π/23π/2∫01∫r1 r dzdrdθ”}

There are two options for the next question and you can choose which one you will answer. Your options are shown below. Read these questions, then take mental note of which one you prefer to answer: Option A: Suppose that a new disease has swept across Washington state. Those who are infected with this disease develop lesions on the skin. Many scientists suspect that the new disease is caused by a pathogen called Bobo Virus. Scientists have found that this virus is relatively easy to grow in the laboratory using cell culture. There is also a murine (mouse) model for studying the pathogenic effects of this virus. Through their work, scientists have found that Bobo Virus causes respiratory illness (instead of skin lesions) in 3% of the mice infected. (a) In correct order, list all of Koch’s postulates. (b) Based on the information provided above, do you see any problems with identifying Bobo Virus as the cause of the new Washington State disease? Why or why not? Option B: Describe one way that regulatory T-cells become activated by normal flora.  Include all the cells/molecules/receptors that are involved. Go into as much detail as you can to comprehensively describe this process.

Which of the following solids has a volume equal to the integral ∫-11∫01-x2 y dydx{“version”:”1.1″,”math”:”∫-11∫01-x2 y dydx”}