Secant & Tangent Lines

Math 31 Lesson #1: Limits

Objective: By the end of this lesson, you will be able to:

“Calculus was the first achievement of modern mathematics and it is difficult to overestimate is importance.” - John Von Neumann What is calculus?  Calculus is a branch of mathematics which looks at things that change over time. It analyzes the change using functions at the exact moment at which the change is taking place.  Allows precise calculation of rates of change and of the total amount of change in a system.  Two primary tools: ______and ______.  The derivative and the integral grew out of the idea of a limit.

Calculus is based on two main concepts: ______and ______.

What is calculus used for?  physics, astronomy, biology, engineering, economics, medicine, sociology, computer science, statistics, engineering, business, and in other fields wherever a problem can be mathematically modeled and an optimal solution is desired.

Recall: In order to calculate the slope of a line we need two points. Slope Formula:

Finding the slope of a tangent line

Definitions:  Secant line:

 Tangent Line: Math 31 Lesson #1: Limits e.g. 1) Determine the equation of the line tangent to y= x2 , at the point (2, 4).

 To find the equation of a line we need the ______and ______.  To find the slope we need ______on the line. *Problem:

To try to get around this, let’s look at the points close to the one we are given. Hence, consider slopes of secant lines close to the point of tangency. The secant line goes through point P(2, 4) and point Q(x, y). Since y= x2 , the point Q can also be written as P ______.

Try where x = 3:

When x = 2.5:

When x = 2.1:

When x = 2.01:

The slope gets closer to ____ as we pick coordinates closer and closer to the one we were given. Math 31 Lesson #1: Limits

What if we pick coordinates on the other side and repeat the process?

When x = 1.5:

When x = 1.9:

When x = 1.99:

If the slope of the line approaches the ______from the left and right side, then that value is the ______of the tangent line. Put another way, the slope of the tangent line is the ______of the slopes of the secant lines as x approaches 2. Therefore, the slope of the tangent line to y= x2 at the point (2, 4) is ______. Now use the slope and the point (2, 4) to find the equation of the tangent line. Write the final answer in slope-intercept form.

e.g. 2) Determine the slope and equation of the tangent line to y= x where x = 9. Write the answer in general form.

Left side: Right side: x y Slope 9.1 9.001 9.00001 Math 31 Lesson #1: Limits

Assignment: p. 9-10 #7, 9, 10