MATH 2010 Multivariable Calculus and Matrix Algebra, Homework 1.

Due: Thursday, May 28, 2020, 11.59pm.

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  -----
√ y2+z2 .
    3. Let W = 13.1267 + 0.6215T - 13.947v0.16 + 0.486Tv0.16. Calculate ∂W∕∂v at (T,v) = (-10, 15), and use this value to estimate ΔW if Δv = 2.
    4. Compute ∂2g-
∂x∂y of g(x,y) = -xy-
x- y.
    5. Let
u (x,t) = -√----e-x2∕4t
          2  πt

      with t > 0 and x 0. Show that for any fixed x, the function u(x,t) is maximized by t = 1

  3. Section 14.4, pages 772–774:
    1. Find an equation of the tangent plane of g(x,y) = ex∕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) = rθ.

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).

    John Mitchell
    Amos Eaton 325
    mitchj at rpi dot edu
    Office hours: Mon, Tues, Thurs, Fri: 2.30–3.30pm, WebEx.
    TA: Rachel Wesley.
    Office hours: available on Slack an WebEx.