Worksheet 1-3

Worksheet 1-3#

Download: CMSE382-WS1-3.pdf

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Worksheet 1-3: Q1#

Circle all the properties of each given set.

  1. \(S_1 = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 < 1\}\)

    Open   Closed   Bounded   Compact

  2. \(S_2 = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 \leq 1\}\)

    Open   Closed   Bounded   Compact

  3. \(S_3 = \{(x, y) \in \mathbb{R}^2 : x^2 + y^2 = 1\}\)

    Open   Closed   Bounded   Compact

  4. \(S_4 = [0, 1) \subset \mathbb{R}\)

    Open   Closed   Bounded   Compact

  5. \(S_5 = (0, 1) \cup (2, 3) \subset \mathbb{R}\)

    Open   Closed   Bounded   Compact

  6. \(S_6 = [0, 1] \cup [2, 3] \subset \mathbb{R}\)

    Open   Closed   Bounded   Compact

  7. \(S_7 = \{(x,y) \in \mathbb{R}^2 \colon x \geq 0, y\geq 1 \}\)

    Open   Closed   Bounded   Compact

Worksheet 1-3: Q2#

  1. Find the gradient for the scalar-valued function \(f(x,y,z)=x^4 + 3yz\) at \((1,2,3)\).

  2. Find the directional derivative of the function at \((1,2,3)\) in the direction of a unit vector parallel to \(\mathbf{u}=(1,-2,2)\).

  3. Find the Hessian for \(f\) at \(\mathbf{x}=(1,2,3)\).

Worksheet 1-3: Q3#

Consider the function \(f(x_1, x_2) = x_1^2 + 2x_1 x_2 + x_2^2\) and the point \(\mathbf{x} = (1, 1)\).

  1. Compute \(\nabla f(\mathbf{x})\) at \(\mathbf{x} = (1, 1)\).

  2. Write the linear approximation \(L(\mathbf{y})\) to \(f\) at \(\mathbf{x} = (1, 1)\):

    \[L(\mathbf{y}) = f(\mathbf{x}) + \nabla f(\mathbf{x})^T (\mathbf{y} - \mathbf{x})\]

  3. Use the linear approximation \(L(\mathbf{y})\) to estimate \(f(1.1, 0.9)\).

  4. Compute the actual value \(f(1.1, 0.9)\).

  5. What is the error in the linear approximation? (i.e., \(|f(1.1, 0.9) - L(1.1, 0.9)|\))

  6. Compute the Hessian matrix \(\nabla^2 f(\mathbf{x})\) at \(\mathbf{x} = (1, 1)\).

  7. For \(\mathbf{y} = (1.1, 0.9)\), compute the (approximate) quadratic error term:

    \[E_2(\mathbf{y}) \approx \frac{1}{2}(\mathbf{y} - \mathbf{x})^T \nabla^2 f(\mathbf{x}) (\mathbf{y} - \mathbf{x})\]

  8. For \(\mathbf{y} = (1.1, 1.1)\), compute the (approximate) quadratic error term \(E_2(\mathbf{y})\).

  9. What does an error \(E_2(\mathbf{y}) = 0\) tell us? What does a non-zero error term tell us? What does the error depend on?

    (Hint: desmos.com/3d/esp2pdudke)

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