Answer the following questions. Upload your solutions as a single pdf file to LMS. Include your last name and HW1 in the name of the file, eg, mitchellHW1.pdf. No late homework accepted.

1. Section 14.1, pages 747–750:
   1. What are the domain and range of the function g(r,s) = cos ^-1(rs)?
   2. Sketch the contour map of f(x,y) = x² + y² with level curves c = 0, 4, 8, 12, 16.
2. Section 14.3, pages 764–767:
   1. Compute the first order partial derivatives of z = sin(r²s).
   2. Compute the first order partial derivatives of z = e .
   3. Let W = 13.1267 + 0.6215T - 13.947v^0.16 + 0.486Tv^0.16. Calculate ∂W∕∂v at (T,v) = (-10, 15), and use this value to estimate ΔW if Δv = 2.
   4. Compute of g(x,y) = .
   5. Let

      with t > 0 and x ≥ 0. Show that for any fixed x, the function u(x,t) is maximized by t = x².

3. Section 14.4, pages 772–774:
   1. Find an equation of the tangent plane of g(x,y) = e^x∕y at (x,y) = (2, 1).
   2. Find the linearization to f(x,y,z) = xy∕z at (2,1,2). Use it to estimate f(2.05, 0.9, 2.01). How close is your estimate to the true value (found using a calculator)?
4. Compute the first and second partial derivatives of f(r,θ) = r^θ.

Homework submission guidelines

Mandatory: you must only submit one pdf file for each HW or exam. The single pdf file submitted will be a multipage document, and the pages must be in the correct order.

Producing a pdf file: the homework assignments will involve extensive mathematical formulas and expressions, and will often require sketches (of curves, domains, etc).

• Suggestion 1 (free):

  a) write out the solutions neatly, and clearly, on regular paper

  b) use a pdf scanner on your phone, and produce one multipage pdf file. Recommended scanning apps include Adobe Scan (highly recommended), ABBYY FineScanner, CamScanner

• Suggestion 2 (not free):

  use an iPad, or something similar, along with a pdf producing note taking app (like Notability)

• Suggestion 3 (free):

  use LaTeX. This produces easy to read text but doing sketches is more complicated. The reason is that you will need to make pdf images of your sketches, and then include them in the LaTeX output. This is not difficult to do, and if you want to try this just ask and we will provide more information on what to do. Again, there is to be only one pdf file submitted per assignment.