Calculus

Instructor: Faiqa AliStudent :Badar Shahzad

Contents

•Equations

•Complex Numbers

•Quadratic Expressions

•Inequalities

•Absolute Value Equations & Inequalities

•Applications

Equations

Key Operation: To Solve an Equation?

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

•xy + x = 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)

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

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.

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.

Do by yourself :1. Two consecutive even integers such that sixtimes 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.

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

Real and Complex part of a Complex Number: a+bi

Algebraic Operations on Complex Numbers

•Adding or subtracting complex number, [Combine like terms].

Example: Simplify (8 – 3i) + (2+25i) - (12 – 3i)

•Multiplying complex numberExample: 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)

Quadratic equation

Second degree polynomial equation in ONE Variable

General form:

Example: Using the zero factor property Solve:

Square Root Property Very Important:

Example: Solve x2 = 49Method of Completing Square

Example: Solve y2 - 2y = 3Solution:

y2 - 2y + 1 = 3 + 1 (y + 1)2 = 4

y = 3 or 1

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 = 0Solution:

Equations Reducible to Quadratic Equations

1.1. Equation with Rational Expression

2.2. Equations with Radical Signs

3.3. Equations with Fractional Powers

4.4. Equations with integer powers

Inequalities

•Inequality Signs•Rules of Algebraic Operations•Linear Inequalities•Quadratic Inequalities•Absolute Value Equations•Absolute Value Inequalities•Compound Inequalities

ExamplesLinear Inequality: 4q+3 < 2(q+3)• Quadratic Inequalities:• Absolute Value Equations:Absolute Value Inequalities: Compound Inequalities:Rational Inequality:•

•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-2going downstream, v = x+2

Answer: x = -0.39 or 10.39

Time = distance / speed

total time = time upstream + time downstream

•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 water

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.

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.