Calculus And Analytical Geometry lecture week 2.pptx

Calculus Lecture for 2nd week in Engineering programs
This lecture can be used for computer science as well

1. Welcome!!

2. Calculus and Analytical Geometry

3. Learning Objectives Understanding of Limits Continuity and Discontinuity in Functions Learning Outcomes Students will be able to define limits Students will be able to discuss types of continuity and discontinuity and their plots

4. Two Basic Problems of Calculus The concept of a “limit” is the fundamental building block on which all calculus concepts are based. Many of the ideas of calculus originated with the following two geometric problems; The solution to both of these problems requires the use of limits Traditionally, that portion of calculus arising from the tangent line problem is called differential calculus and that arising from the area problem is called integral calculus

5. Limits In Mathematics, concept of limits is used to describe the “behaviour of a function” as it’s input approaches or get close to a particular value. 1.1.1 (pg# 52)

6. Limits 1.1.2 (pg# 54) ‏

7. Limits 1.1.2 (pg# 55) ‏

8. ‏ Calculating Limits – Basic Cases

9. Calculating Limits – Algebra Rules

10. CONTINUITY Intuitively, the graph of a function can be described as a “continuous curve” if it has not breaks or holes. The graph of a function has a break or hole if any of the following conditions occur: The function f is undefined at c The limit of f(x) does not exist as x approaches c The value of the function and the value of the limit at c are different.

11. Continuous Function

12. Example: Determine whether the following functions are continuous at x=-3. Solution: Observe that f(x) is not continuous at x=-3 since it’s undefined at x=-3, g(x) is not continuous at x=-3 since h(x) is continuous

13. Some properties of continuous functions

14. Types of Discontinuities There are 4 types of discontinuities Jump Point Essential Removable The first three are considered non removable

15. Jump Discontinuity Occurs when the curve breaks at a particular point and starts somewhere else Right hand limit does not equal left hand limit

16. Point Discontinuity Occurs when the curve has a “hole” because the function has a value that is off the curve at that point. Limit of f as x approaches x does not equal f(x)

17. Essential Discontinuity Occurs when curve has a vertical asymptote Limit dne due to asymptote

18. Removable Discontinuity Occurs when you have a rational expression with common factors in the numerator and denominator. Because these factors can be cancelled, the discontinuity is removable.

19. Places to test for continuity Rational Expression Values that make denominator = 0 Piecewise Functions Changes in interval Absolute Value Functions Use piecewise definition and test changes in interval Step Functions Test jumps from 1 step to next.

20. Continuous Functions in their domains Polynomials Rational f(x)/g(x) if g(x) ≠0 Radical trig functions

21. Thank you