Math 006 - Introduction to Calculus

Lecture 11

New Stuff

In this class we will do more work with exponential and logrithmic functions.

We will go over the formular for the derivative of the logrithmic function, and more examples of exponential and logrithmic function.

We will also cover a special case of differential equations, where the solution is the exponential function.

What we are doing this class is section 8.5 of the text book.

The basics of differentiation of logrithmic and exponential functions are the following two rules:

Note that e is just a number when it's on it's own, so the derivative of something like ex² is just 2ex, just like the derivative of 3x² is 6x.

Find the derivatives of the following functions:

• 3^x
• e^{(cos3(x) + x sin(x))}
• ln(3x² + 4) e^{(4x2 - 7x +1)}

Note, some of these are a little difficult, but if you can do them you should have no trouble in the exam. The main thing is keeping track of all the bits and pieces.

3^x

The derivative of this function is

for any function of the form a^x, the derivative is ln(a) a^x, provided a is a constant.

ln(3x² + 4) e^{(4x2 - 7x +1)}

The derivative is found using the chain rule, product rule, etc, and you get:

The steps of getting to the answer are illustrated in the following picture:

e^{(cos3(x) + x sin(x))}

The derivative is:

The steps of getting to the answer are illustrated in the following picture:

There will be no more assignements. The last one is due next week.

Sample Quiz
There will be a quiz next week, which will be similar to the following question:

The population of a certain city grows at 10% per year. The population in 1990 was 200 000. What will the popultion be in the year 2000? (Assume the rate of growth remains constant.)

You can find the solution on the quiz page.

What will be on the exam (1) Find the derivative of list of functions, including functions involving sine, cosine, exponential and logrithmic fuunctions, in combination with polynomials. (2) Find the equations for tangents of graphs, and tangents with given slope. (3) Long detailed question to sketch a curve, including finding convave up, concave down, like assignment 8 examples. (4) given a sketch of a graph, sketch the derivative (like homework: page 75, 2.1, Ex 2, page 77 Ex 13) (5) Related rates problem. (6) Problem about finding absolute max and min on an interval.

Next week will be a review class - please let me know what you want me to concentrate on!

See previous weeks handout for tutorial times etc. I will update the marks page over the weekend . Please check your marks at http://www.mast.queensu.ca/~helena/math006/marks.html

handout

Please send corrections to spelling mistakes, and any other comments, to me.