Category: 11 HL Homework

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11 HL Working with Derivatives

Following on from our discussion from class, complete the following question.

Let \(f(x)=\frac{1}{3}x^3-2x^2\). Find the coordinates of the local maximum and the local minimum of \(f\).

Are you interested in seeing something else that’s sort of neat? Read on.

We’ll eventually be discussing something called the second derivative. Once you’ve found the derivative of a given function, you can then go on to find the derivative of that derivative. For example, if \(f(x)=2x^5\), then \(f'(x)=10x^4\). The second derivative is represented as \(f^{\prime \prime}(x)\); in this case, we have \(f^{\prime \prime}(x)=40x^3\).

Find the second derivative of the function \(f(x)=\frac{1}{3}x^3-2x^2\), then solve the equation \(f^{\prime \prime}(x)=0\). On the graph of \(f\), plot the point on \(f\) whose \(x\)-coordinate is the solution you found to \(f^{\prime \prime}(x)=0\). What do you notice about the location of this point?

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11 HL Limits

Complete the following before our next class.

Exercise 17A questions 1, 2c, 3, 5ai
Exercise 17B.1 question 2
Exercise 17B.2 question 1

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11 HL Planes and Matrices

  1. Use row reduction to express the matrix below in row echelon form (or reduced row echelon form).
    \[\left[\begin{array}{@{}ccc|c@{}}
    1 & 3 & -2 & 4 \\
    -2 & -6 & 4 & 9 \\
    2 & 1 & 1 & -2
    \end{array}\right]\]
  2. Use row reduction to express the matrix below in row echelon form (or reduced row echelon form).
    \[\left[\begin{array}{@{}ccc|c@{}}
    1 & 3 & -2 & 4 \\
    -2 & -6 & 4 & 8 \\
    2 & 1 & 1 & -2
    \end{array}\right]\]
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11 HL Vectors Assignment [updated]

Complete the questions here and submit your work to me as an electronic file before the end of the day on Monday, February 5th.

You can use any software you like to create your file, but your submission should be sent to me as a PDF document.

Update: Are you trying to use Google Docs for this? Surprisingly, the Google Docs equation editor doesn’t support vectors (or matrices)! If you don’t have \(\LaTeX\), Word, Or Pages available, you could also use LibreOffice (which is free and has an equation editor). If you can’t find an alternative, handwritten work (hard copies) will be accepted.

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11 HL Lines in 3D

Consider the three lines defined below.

\[L_1: \vec{r}=\begin{bmatrix}1\\2\\3\end{bmatrix}+\lambda \begin{bmatrix}1\\-3\\-4\end{bmatrix}\]

\[L_2: \vec{r}=\begin{bmatrix}-2\\-3\\0\end{bmatrix}+\lambda \begin{bmatrix}4\\4\\0\end{bmatrix}\]

\[L_3: \vec{r}=\begin{bmatrix}2\\-5\\-3\end{bmatrix}+\lambda \begin{bmatrix}0\\2\\1\end{bmatrix}\]

Show that \(L_1\) and \(L_2\) are skew lines, then find the point of intersection of \(L_1\) and \(L_3\).

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11 HL Planes

Here are a couple of short questions to look at before our next lesson.

  1. Verify that the points \(A(1,2,3)\), \(B(-2,0,0)\), and \(C(3,-2,-1)\) are not collinear.
  2. Find the vector equation of the plane that contains all three points from question 1.
  3. Find the Cartesian equation of the plane you determined in question 2.
  4. Verify your answers using GeoGebra.