Math 006 - Introduction to Calculus

Lecture 10

New Stuff

We will cover the following topics in this class:

• Exponential functions Definition and explaination of exponential and logarithmic functions, and their derivatives

First we will spend some time reviewing the idea of exponents and exponential functions, and then cover sections 8.1 to 8.4.

In this course, we started with polynomial functions, and looked at the calculus of these functions - finding the derivatives, equations for tangents, how to sketch the curves.

We can do the same kind of things for lots of other functions.

Last lecture we looked at sine and cosine.

Before we begin looking at the exponential function we will have another look at some of the properties of sin and cosine.

Most of this lecture will be about the exponential function.

The exponential function is important in measuring growth; it's inverse is the logrithmic function, which is also an important function.

This lecture we'll just look at the exponential function. Next lecture we'll look at the logrithmic function.

In the class we will go over properties of exponential functions, which are from the review section of chapter 8, and we'll look at the graph of exponential functions.

An exponential function is of the form

for example,

There is a special number, e, such that the derivative of e^x is also e^x. We use this to show how to find the derivative of other exponential functions.

handout

Note, assignment from week 9 is due in on Wednesday April 1, not March 25 as stated on handout for week 9.

Homework (not to be handed in)	Assignment Due date: Wednesday April 8.
Please read pages 350 - 152 of text book Ex 2 Page 352, Ex 1.(b) Ex 2.(b) Ex 3.(b) Ex 4.(b) Ex 3 Page 354, Ex 1.(b) Ex 2.(b) Ex 3.(b) Ex 8.1 Page 361, Ex 2. Ex 8.2 Page 366, A. 1(a)-(e), Ex 4.(a)-(d)	(1) For f(x) = x + 3sin(x), what are the absolute maximum maximum and minimum on [0,pi]? (2) Find x in each of the following: (i) x = log3(1/3) (ii) log2(x) = 5 (iii) 3 = 10x (iv) logx125 = 3 (3) Using graph paper, sketch the graphs of (a) y = 6x (b) y = (1/4)x on the interval [-1,2]. Make a table with at least 5 values of each function, and clearly and neatly plot and label these points.	(4) Differentiate the following function: x(ex + 1) e(x2 + 3x +1) e(x - 1/x) (xe(x+1) - 1)e2x

Sample Quiz
There will be a quiz next week, which will be similar to the following question:
Find the critical values of the function

What you are expected to know about trigonometric functions (1) d/dx(sin(x)) = cos(x) (2) d/dx(cos(x)) = -sin(x) (3) derivatives of functions involving sine and cosine, eg, x2cos(3x+1) (4) Equation of tangent at a point for a function involving sine and cosine. (5) If sin-1(a) = b is one value of sin-1(a), what are the others in [0,pi]? (6) Using (5) to find the max and min of simple functions involving sine and cosine (you won't need to learn double angle formulars).

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