Math 312 Bard College

Homework 3

Due Date: Friday, September 22

Instructions: Feel free to work together with other students in the class, though you must turn in your own copy of the solutions, and you must acknowledge anyone that you worked with.

  1. The following figure shows the image of a square grid under a nonlinear function \(\mathbf{f}\colon\mathbb{R}^2\to\mathbb{R}^2\). Each square of the original grid had side length \(0.2\).
    1. The point \(\mathbf{f}(1.6,1.4)= (0.8,0.7)\) is shown in green. Estimate the matrix for \([D\mathbf{f}(1.6,1.4)]\) as accurately as you can.
    2. Use linear approximations to estimate \(\mathbf{f}(1.62,1.4)\), \(\mathbf{f}(1.6,1.42)\), and \(\mathbf{f}(1.63,1.39)\).
    3. The point \(\mathbf{f}(1.8,0.2)= (0.5,0.2)\) is shown in yellow. Estimate the matrix for \([D\mathbf{f}(1.8,0.2)]\) as accurately as you can.
    4. Use linear approximations to estimate \(\mathbf{f}(1.82,0.2)\), \(\mathbf{f}(1.8,0.22)\), and \(\mathbf{f}(1.83,0.21)\).
    5. Use a linear approximation to estimate the point \((x_0,y_0)\) for which \(\mathbf{f}(x_0,y_0) = (0.5,0.22)\).
  2. A function \(\mathbf{f}\colon \mathbb{R}^2\to\mathbb{R}^2\) satisfies \(\mathbf{f}(2,4) = (5,7)\), \(\mathbf{f}(2.2,4.1) = (5.5,7.4)\), and \(\mathbf{f}(2.1,4.2) = (5.1,7.5)\). Use this information to estimate the matrix for \([D\mathbf{f}(2,4)]\).
  3. Recall that a \(2\times 2\) matrix \[ \begin{bmatrix}x_1 & x_2 \\ x_3 & x_4\end{bmatrix} \] can be viewed as a point \((x_1,x_2,x_3,x_4)\) in \(\mathbb{R}^4\). Let \(\mathbf{f}\colon \mathbb{R}^4\to\mathbb{R}^4\) be the function that squares a \(2\times 2\) matrix, i.e. \(\mathbf{f}(A) = A^2\) for any \(2\times 2\) matrix \(A\).
    1. Write an explicit formula for \(\mathbf{f}(x_1,x_2,x_3,x_4)\), and compute the \(4\times 4\) Jacobian matrix \([D\mathbf{f}(x_1,x_2,x_3,x_4)]\).
    2. Compute \([D\mathbf{f}(1,2,3,4)]\), and use a linear approximation to estimate \(\mathbf{f}(1.01,2.02,3.01,4.03)\). How does this compare with the actual value of \(\begin{bmatrix}1.01 & 2.02 \\ 3.01 & 4.03\end{bmatrix}^2\)?
    3. Use a linear approximation to find a \(2\times 2\) matrix \(A\) for which \[ A^2 \approx \begin{bmatrix}\phantom{1}7.2 & 10.4 \\ 15.2 & 22.6\end{bmatrix}. \] (Feel free to use a calculator or computer for the row reduction.) How close is the square of your answer to the desired value?