![•Adding or subtracting complex number, [Combine
like terms].
Example: Simplify (8 – 3i) + (2+25i) - (12
– 3i)
•Multiplying complex number
Example: Solve: – 4i . 7i
Example: Simplify (3 –2i) (4+5i)
•Division in Complex Numbers
Complex Conjugate (Examples)
Example: Write the complex number
in Standard form: 8i / (6-5i)](https://image.slidesharecdn.com/lecture1-160116084955/85/Calculus-Lecture-1-12-320.jpg)
Calculus Lecture 1
- 1. Calculus Instructor: Faiqa Ali Student :Badar Shahzad
- 2. Contents •Equations •Complex Numbers •Quadratic Expressions •Inequalities •Absolute Value Equations & Inequalities •Applications
- 3. Equations Key Operation: To Solve an Equation?
- 4. Linear equations – Linear Equation in One Variable – Eq. does not have product of Two or more variables Examples •x2 + 5x -3 = 0 •5 = 2x •5 = 2/x •3 – s = ¼ •3 – t2 = ¼ •50 = ¼ r2
- 5. Examples •Linear equation in one variable Example 1: 3x + 17 = 2x – 2. Find x. •Linear equation in two or more variable Example 2 : The eq. x = (y – b)/m has 4 variables. Make y as a subject of Equation (Express Eq. in terms of y)
- 6. Word Problem (Very Important: Tagging quantities with variables) Example: At a meeting of the local computer user group, each member brought two nonmembers. If a total of 27 people attended, how many were members and how many were nonmembers? Solution: •Let x = no. of members, 2x = no. of nonmembers
- 7. Solve problems involving consecutive integers. Consecutive integers: Two integers that differ by 1. e.g.. 3 and 4. In General: x= an integer, x+1= next greater consecutive integer. Consecutive even integers: such as 8 and 10, differ by 2. Consecutive odd integers: such as 9 and 11, also differ by 2.
- 8. Example: Two pages that face each other have 569 as the sum of their page numbers. What are the page numbers? Solution: •Let x = the lesser page no. •Then x + 1= the greater page no. •The lesser page number is 284, and the greater page number is 285.
- 9. Do by yourself : 1. Two consecutive even integers such that six times the lesser added to the greater gives a sum of 86. Find integers. 2. The length of each side of a square is increased by 3 cm, the perimeter of the new square is 40 cm more than twice the length of each side of the original square. Find dimensions of the original square. 3. If 5 is added to the product of 9 and a number, the result is 19 less than the number. Find the number.
- 10. Complex numbers General form of a Complex Number: a+bi, •a and b are reals •i is an imaginary number. What is an imaginary number? A number for when squared gives – 1
- 11. Algebraic Operations on Complex Numbers
- 12. •Adding or subtracting complex number, [Combine like terms]. Example: Simplify (8 – 3i) + (2+25i) - (12 – 3i) •Multiplying complex number Example: Solve: – 4i . 7i Example: Simplify (3 –2i) (4+5i) •Division in Complex Numbers Complex Conjugate (Examples) Example: Write the complex number in Standard form: 8i / (6-5i)
- 13. Quadratic equation Second degree polynomial equation in ONE Variable General form: Example: Using the zero factor property Solve:
- 14. Square Root Property Very Important: Example: Solve x2 = 49 Method of Completing Square Example: Solve y2 - 2y = 3 Solution: y2 - 2y + 1 = 3 + 1 (y + 1)2 = 4 y = 3 or 1
- 15. Quadratic Formula •Do you know how do we get to Quadratic Formula: Example: Solve the Eq. by using Quadratic Formula: 3y2 + 9y = 2 Benefit of QUADRATIC FORMULA? Discriminant: D =b2 – 4ac It tells about Nature of the Roots: Method: Check if D = 0, >0 or <0 Example: Discuss the nature of the roots of 2x2 +7x – 11 = 0 Solution:
- 16. Equations Reducible to Quadratic Equations 1.1. Equation with Rational Expression 1.2. Equations with Radical Signs 1.3. Equations with Fractional Powers 1.4. Equations with integer powers
- 17. Inequalities •Inequality Signs •Rules of Algebraic Operations •Linear Inequalities •Quadratic Inequalities •Absolute Value Equations •Absolute Value Inequalities •Compound Inequalities
- 18. Examples Linear Inequality: 4q+3 < 2(q+3) • Quadratic Inequalities: • Absolute Value Equations: Absolute Value Inequalities: Compound Inequalities: Rational Inequality:
- 19. •Example: River Cruise A 3 hour river cruise goes 15 km upstream and then back again. The river has a current of 2 km an hour. What is the boat's speed and how long was the upstream journey? Hints: •Let x = the boat's speed in the water (km/h) •Let v = the speed relative to the land (km/h) going upstream, v = x-2 going downstream, v = x+2 Answer: x = -0.39 or 10.39 Time = distance / speed total time = time upstream + time downstream
- 20. •Boat's Speed = 10.39 km/h •upstream journey = 15 / (10.39-2) = 1.79 hours = 1 hour 47min •downstream journey = 15 / (10.39+2) = 1.21 hours = 1 hour 13min Question: The current in a river moves at 2mph. a boat travels 18 mph upstream and 7mph down stream in a total 7 hours. what the speed of the boat in still water? Ans: x = 3.8 mph in still
- 21. Example: Two Resistors In Parallel Total resistance is 2 Ohms, and one of the resistors is known to be 3 ohms more than the other. Find: Values of the two resistors? Solution: Formula: •R1 cannot be negative, so R1 = 3 Ohms is the answer. •The two resistors are 3 ohms and 6 ohms.
- 22. Others •Quadratic Equations are useful in many other areas: • For a parabolic mirror, a reflecting telescope or a satellite dish, the shape is defined by a quadratic equation. •Quadratic equations are also needed when studying lenses and curved mirrors. •And many questions involving time, distance and speed need quadratic equations.
Editor's Notes
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