Category: 11 Mathematics HL

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Review: The Derivative Part 2

Hope you’re all enjoying the break!

Attached below is the second part of your summer review notes on the derivative (note that the document attached below contains the previous review notes as well). Again, some of this material will be review from the last few lessons, but there are a few new elements that you should get acquainted with now, and which we’ll be spending more time with at the start of next term.

Also as before, there are some questions at the end of the new section that you should complete (but which will not be collected at the start of the term). If you have any trouble with these, please do post your questions in the comments section below.

There will be one final set of additional notes (again, focused on review from the last few lessons), which will be posted, along with your short summer assignment, sometime on or before the 7th of August.

Derivatives Review Notes (updated)

 

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Review: The Derivative

Over the summer I’ll be posting a few review notes, and (later in July) a short assignment for you to complete for the end of the summer. The first topic for review is the definition of the derivative.

I suggest that you read through the attached notes and complete the questions in each section. (Your answers to the questions included in the note will not be collected, but you really must complete them in order to best understand this material.) Note that there are no diagrams included in these notes, and I suggest that you create your own supplemental diagrams as you’re reading through the material.

As always, post any questions you have in the comments section below, or send them to me in an email if you’d prefer.

Derivatives Review Notes

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Derivatives and Curves [Updated]

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?

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

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New HL Resource

I’ve added a new resource to the HL Resources page that will be useful when studying. Note that (as with the resources you’ve already got), you will be able to find past paper questions that will be especially useful when studying for our end-of-year exam.

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Planes and the Cross Product

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?
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Planes

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 \(A(6,-3,2)\).
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Vectors Homework Assignment

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.