Category: Mathematics

Complete the following question for tomorrow’s lesson.

Use the derivative of the function \[f(x)=x^3-x^2+2x-1\] to find the coordinates of the (local) extrema of \(f\).

Update: Oops! The function above has no extrema! (How can you tell from the first derivative?) However, it does have what’s called a point of inflexion, which is a point at which the function changes concavity (from concave up to concave down, or vice versa). Can you find the coordinates of the point of inflexion?

At any rate, the function \(g\) below does have extrema, so you should find the coordinates of the extrema of \(g\). Can you also find the coordinates of its point of inflexion?

\[g(x)=x^3-x^2-2x-1\]

This was going to be part of a test, but instead let’s use the questions below for an assignment. Have this completed and ready to hand in by Wednesday this week. Electronic submissions are accepted, but not required.

Lines and Planes Assignment

Below is the slide of questions we began today. Complete these for our lesson tomorrow.

Questions

Here are the questions we considered in today’s class. See if you can answer these (with algebraic solutions) for tomorrow’s lesson. GeoGebra will be useful to check you answers, and to give you some insight into the question if you get stuck.

1. Find a vector normal to both
   \[\vec{a}=\begin{bmatrix}1\\2\\-2\end{bmatrix} \text{ and }\vec{b}=\begin{bmatrix}3\\4\\1\end{bmatrix}.\]
2. a) Find a vector normal to the plane \[\vec{r}=\begin{bmatrix}1\\4\\-2\end{bmatrix} +\lambda \begin{bmatrix}1\\1\\-1\end{bmatrix} + \mu \begin{bmatrix}-3\\1\\2\end{bmatrix}\] b) Using your answer to part a), can you find the distance of the point \(A(1, 1, 1)\) to the given plane?

Try to complete the following questions for our lesson on Monday.

1. Find a vector equation of the plane passing through \(A(1,2,3)\), \(B(3,1,-2)\), and \(C(4,4,4)\).
2. Find a vector equation of the plane with Cartesian equation \(3x+2y-z=1\).
3. Find a vector equation of the plane containing the line \[\vec{r}=\begin{bmatrix}-1\\-1\\4\end{bmatrix}+\lambda \begin{bmatrix}-2\\1\\1\end{bmatrix}\] and passing through the point \(A(6,-3,2)\).

The attached Vectors Assignment contains a long answer question that you can work on during Tuesday’s lesson. Complete this question for Wednesday’s lesson. Your answers will be collected and marked. You can use GeoGebra to confirm you answers, but full algebraic solutions must be included.

Try to complete the following questions for Monday’s lesson.

1. Find the distance of the point  \(A(-1,1,2)\) to the line \[\displaystyle{\vec{r}=\begin{bmatrix}1\\-2\\3\end{bmatrix}+\lambda \begin{bmatrix}-2\\1\\1\end{bmatrix}}\]
2. The lines below intersect at a point \(A\). Use an algebraic method to find the coordinates of  \(A\), then verify your answer using GeoGebra.
   \[L_1: \vec{r}=\begin{bmatrix}-1\\-1\\4\end{bmatrix}+\lambda \begin{bmatrix}-2\\1\\1\end{bmatrix}\] \[L_2: \vec{r}=\begin{bmatrix}-2\\0\\7\end{bmatrix}+\lambda \begin{bmatrix}-6\\4\\8\end{bmatrix}\]
3. In 2D, any two non-parallel lines will have a point of intersection. In 3D, two lines can be non-parallel and have no point of intersection; such lines are called skew lines.
   Show that \(L_1\) from question 2 and \(L_3\) below are skew lines.
   \[L_3: \vec{r}=\begin{bmatrix}-2\\5\\12\end{bmatrix}+\lambda \begin{bmatrix}-6\\4\\8\end{bmatrix}\]