Math 006 - Introduction to Calculus

Lecture 7

New Stuff

We will cover the following topics in this class:

• Curve Sketching Sketching polynomial functions; asymtopes, concave up and down functions, second derivative test; putting this together to sketch a graph of a function.

We will continue with some ideas of chapter 4, going over how we use the ideas there to sketch curves, and also covering more examples of maximum/minimum problems, from 4.4. Then we will start chapter 5, and see more ideas used for sketching curves. We will cover 5.1, 5.2, 5.3, and next week we will show how to use these ideas for a proceedure for sketching curves.

Max and Min:

Curve sketching

Case where nothing is infinite

Once you know where the maximum and minimum of a function over some range, then you've got a pretty good idea what the graph looks like. Eg, you've used the method described before to find max and min values of a function f(x), over some inverval, eg, [0,5]; say local maximum are at 0, 2, and local minimum at 1,4. Suppose you found values at these critical points, and at the boundary, as shown in the table below, and plot the points:

Now you can fill in between some how... But how do you know which of the following might be the way the curve fills in?

different possible ways to "join the dots" drawn in different colours

The idea is the difference between concave up and cocave down:

You can test for concave up or down by finding the second derivative:

Case where something is infinite

Suppose you have something infinite, eg, you want to know what happens when x gets very big, even when x gets infinite... For example, suppose f(x) represents the amount of some chemical produced in some reaction. At the beginning, the amount of the chemical grows pretty quickly, but then the reaction slows down, and even if you leave it going forever, you'll never get more than say 7 grams of the chemical:

Another way you might get infinity is if the function has an infinite value, eg, f(x)=infinity for some x, eg, if f(x)=1/x, then f(0) = infinity:

This situation is described by asymptotes; these are the vertical or horizontal lines that tell you what the function is doing as either x or f(x) gets infinite:

Next week we'll put together the above ideas to find how to sketch the graph of a function.

Tutorial and Examples

The following questions from the text book should be done for practice, but are not to be handed in for marking.

• Ex 4.3 Page 182, B Ex 2.a,b, C Ex 5
• Ex 4.4 Page 189, Ex 2, Ex 3, 15, 20
• Ex 5.1 Page 213, Ex 3
• Ex 5.2 Page 223, Ex 4
• Ex 5.3 Page 229, A Ex 1.a,b, B Ex 2a,b,c,j,k

The following questions from the text book should be handed in Wednesday 17 March.

• Ex 4.3 Page 182, B Ex 2.c
• Ex 4.4 Page 189, Ex 4
• Ex 5.2 Page 223, Ex 1.a.b
• Ex 5.3 Page 229, B Ex 2h

Sample Quiz

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

(a) What is the horizontal asymptote of the function (2x+1)/(x+1) ?

(b) What is the vertical asymptote of the function (2x+1)/(x+1) ?

(c) State whether this function is concave up or down on the interval [2,infinity].

You can find the solution here.

handout

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