Category: 12 Mathematics SL

After today’s review of techniques for differentiation, try the questions listed below before our next class.

We’ve already seen how to find the equation of the tangent line to a curve that passes through a given point, and finding a normal through a given point on a curve is essentially the same. A normal is a line that is perpendicular to a tangent, and you will recall that if one line has slope \(m\), any line perpendicular to it will have slope \(-\frac{1}{m}\). For example, the equation of the line tangent to the function \(f(x)=x^2\) that passes through the point \((1,1)\) has slope \(2\) (found by taking the derivative), and so has equation \(y=2x-1\). Similarly, the normal to the function \(f(x)=x^2\) that passes through the point \((1,1)\) has equation \(y=-\frac{1}{2}x-1\).

We’ve now covered the derivatives of logarithmic and trigonometric functions, and the questions below involve applications of those derivative results.

For logarithmic functions, you may find it easier to simplify some expressions using the properties of logarithms before you try to differentiate. See the list of properties of logarithms at the bottom of page 376, and you can see an example of how these can simplify your calculations in Example 12 on page 377.

Exercise 15F 1ghk, 2adeh, 3abegi, 5
Exercise 15G 1adgh (see page 379 for more about the derivative of \(\tan x\), 2adgk, 3bek, 4b

Complete the following questions before our next lesson.

Exercise 15D (the quotient rule) questions 1abf, 2ad, and 4.
Exercise 15E (the derivative of the exponential function) questions 1ijno, 2acg, 3a, and 5.

Complete the following questions for our next class.

Exercise 15B.2 questions 1ad, 2abcdfi, 3acef, 4, 5

Have a look at the questions in the section below prior to tomorrow’s class, and we’ll be working on these during part of tomorrow’s class.

Exercise 15A questions 1aejno, 2ab, 3bf, 4abc, 5, 7.

In today’s class we saw that, given the function \(f(x)=x^2\), we could use a limit to show that the slope of the tangent at any point on that function could be calculated using \(2x\). This new function is called the derivative of \(f\), and \(f'(x)\) is typically used to represent that new function.

So, if \(f(x)=x^2\), we have shown that \(f'(x)=2x\).

Read Section E on pages 355–357, paying particular attention to the examples (which essentially follow the method we used in class).

Complete questions 1, 2, 5cd and 6 on page 357.

Before completing these questions you should review your notes on rational functions from last year (in particular, focus on horizontal asymptotes). Also read the section on asymptotes and the two examples shown on pages 348–349.

Complete questions 1–3 on page 349.

Remember, if you get stuck or have any other questions you can post a comment below!