1. CALCULUS
AND
ANALYTIC GEOMETRY
CS 001
LECTURE 03

2. Two goals of calculus What is the slope of a curve ?
What is the slope of a curve at point p ?

3. Two goals of calculus
What is the area of a region bounded by a curve ?

4. Average Rates of Change and Secant Lines
Given an arbitrary function y = ƒ(x)
we calculate the average rate of change of y with respect to x over the interval [x1, x2] by dividing the change in the value of y, y = ƒ(x2) – ƒ(x1), by the length x = x2 - x1 = h of the interval over which the change occurs.

5. Average Rates of Change and Secant Lines

6. Defining the Slope of a Curve
what is meant by the slope of a curve at a point P on the curve?
If there is a tangent line to the curve at P—a line that just touches the curve like the tangent to a circle—it would be reasonable to identify the slope of the tangent as the slope of the curve at P.

7. The tangent to a curve
you could pick a point Q and construct the line through P and Q.
This line is called a “secant” and it is of course not the tangent that you're looking for. But if you choose Q to be very close to P then the secant will be close to the tangent.

8. EXAMPLE: Tangent to a parabola
Find the slope of the parabola f(x) = x2, at the point P(2, 4). Write an equation for the tangent to the parabola at this point. Solution : We begin with a secant line through P(2, 4) and Q(2 + h, 2 + h2) nearby. We then write an expression for the slope of the secant PQ

9. Solution : If h> 0, then Q lies above and to the right of P, as in Figure. If h< 0, then Q lies to the left of P (not shown). In either case, as Q approaches P along the curve, h approaches zero and the secant slope h + 4 approaches 4. We take 4 to be the parabola’s slope at P.
h  0 , slope  4

10. Solution : The tangent to the parabola at P is the line through P with slope 4:

11. Limits
Read "the limit of f(x) as x approaches c is L.“It means that if you choose values of x which are close but not equal to a, then f(x) will be close to the value L.

12. Estimating a Limit Numerically
Use a table to estimate the limit numerically.
Solution
Let f(x) = 3x - 2.Then construct a table that shows values of f(x) for two sets of x values —one set that approaches 2 from the left and one that approaches 2 from the right.
From the table, it appears that the closer gets to 2, the closer f(x) gets to 4. So, you can estimate the limit to be 4.

13. Estimating a Limit Numerically
Use a table to estimate the limit numerically.
Solution
Let
We first try to substitute x = 2, but this leads to
which does not exist. Next we try to substitute values of x close but not equal to 2.
So that,
f(x) approaches 0:5.

14. One-Sided Limits

15. One-Sided Limits

16. Properties of Limits

17. Examples Using Limit Rule

18. Examples Using Limit Rule
Find limit of function : Solution

19. Algebraic Limits (Substitution Method)
The limit can be found directly by substituting the value of x.

20. Algebraic Limits (Factorization Method)
When we substitute the value of x in the rational expression it takes the form

21. (Rationalization Method)
Algebraic Limits
(Rationalization Method)
When we substitute the value of x in the rational expression it
takes the form

22. More Examples

23. Limit as x   of rational functions
A rational function is the quotient of two polynomials, so
We have seen that
To find limit divide numerator and denominator by xm

24. Example
let's compute Remember the trick and divide top and bottom by x2, and you get

25. Try this
A :let's compute
B: let's compute C: