## Monday, April 28, 2008

### Conic Sections - Parabola

Conic section

Definition: the locus of a moving point P such that its distance from a fixed point called focus bears a constant ratio 'e' to its distance from a fixed line called directinx is a called a conic section.

The ratio e is called eccentricity of the conic.

e = Distance of a point (P) from focus/distance of the point (P) from directrix

this conic section is called

parabola is e = 1.
ellipse if e<1. and
hyperbola if e>1.

Parabola

Equation y² = 4ax

Some important results connected with y² = 4ax

1. Vertex A(0,0)
2. Focus S (a,0)
3. Foot of the directrix Z (-a,0)
4. Equation of directrix x = -a
5. Equation of latus rectum (L.R.) x = a
6. Equation of axis y = 0
7. Equation of tangent at vertex x = 0
8. Length of latus rectum = 4a
9. Extremities of L.R = (a,2a) and (a, -2a)
10. Focal distance of any point (x,y) is a+x

For parabola equation x² = 4ay

1. Vertex A(0,0)
2. Focus S (0,a)
3. Foot of the directrix Z
4. Equation of directrix y = -a
5. Equation of latus rectum (L.R.)
6. Equation of axis x = 0
7. Equation of tangent at vertex y = 0
8. Length of latus rectum = 4a
9. Extremities of L.R = (2a,a) and (2a, -a)
10. Focal distance of any point (x,y) is

Other forms: y² = -4ax
x² = -4ay

The general form of a parabola with axis parallel to x axis is of the form

ay² + by + c = x

It reduces to the form

(y-k) ² = 4A(x-h)
Points on a parabola y² = 4ax
If (x1, y1) are points on a parabola

y1² = 4ax1
or x1 = y1²/4a

Hence points are on a parabola can be expressed as (y1²/4a,y1)

Equation of a chord joining two points y1 and y2.

The points will be (y1²/4a,y1), (y2²/4a,y2)

y(y1+y2)-4ax-y1y2 = 0

If the chord is a focal chord, it passes through focus which is (a,0)

Hence the condition is y1y2 = -4a²

Equation of a tangent at the point (x1,y1)

yy1 = 2a(x+x1)

Equation of a tangent at the point (x1,y1) in terms of slope

M = 2a/y1

y = mx+a/m

Condition for the line y = mx+c to be a tangent to the parabola y² = 4ax

c = a/m is the condition

Hence the line y = mx+a/m is always a tangent to the parabola y² = 4ax and its point of contact is (a/m²,2a/m)

Normal to the parabola y² = 4ax

y-y1 = -y1/2a(x-x1)

Normal in terms of slope
m = -y1/2a

Hence equation is
y = mx-2am-am³

Points of intersection of two tangents to y² = 4ax
Tangents are
yy1 = 2a(x+x1)
yy2 = 2a(x+x2)

[y1y2/4a,(y1+y2)/2]

Perpendicular tangents to a parabola intersect on the directrix.

From any point (h,k) three normals can be drawn to a parabola

### Conic Sections - Ellipse

Ellipse
Equation x²/a² + y²/b² = 1

Some important results connected with x²/a² + y²/b² = 1

1. Vertex A(-a,0), A’(a,0)
2. Focus S
3. Foot of the directrix Z
4. Equation of directrix
5. Equation of latus rectum (L.R.)
6. Equation of axis
7. Equation of tangent at vertex
8. Length of latus rectum = 2b²/a
9. Extremities of L.R = (-ae,b²/a) and (-ae,-b²/a)
10. Focal distance of any point

the sum of th focal distances of a point on the ellipse is constant and is equal to the length of the major axis.

Equation of a tangent at the point (x1,y1)

xx1/a² + yy1/b² = 1

Normal

x-x1/(x1/a²) = y-y1/(y1/b²)

Conormal points are those points, the normals at which pass through the same point. eg-P,Q,R are conormal points if normals at P,Q,R pass through the same point S.

Conidition for tangent

If y = mx+c is to be a tangent

C = ±SQRT(a²m²+b²)

Hence equation of a tangent

Y = mx±SQRT(a²m²+b²)

The points of contact

(-a²m/ SQRT(a²m²+b²), b²m/ SQRT(a²m²+b²)) or

(-a²m/c, b²/c)

Director circle

The locus of the point of intersection of two perpendicular tangents to an ellipse is called director circle.

The equation of director circle is

x² + y² = a²+b²

chord of contact points of the tangents drawn from the point (x1,y1)

xx1/a² + yy1/b² = 1

Polar of a point (x1,y1) w.r.t. ellipse is

xx1/a² + yy1/b² = 1

Polar is the locus of point, that is an intersection of tangents drawn from the extreme points of the chords which are drawn from a given point (x1,y1). The given point is called the pole of the polar.

Chord with a given middle point (h,k)

hx/a² + ky/b² = h²/a²+k²/b²

Diameter: Any line passing through the centre (0,0) of an ellipse is called a diameter hence its equation is of the form
y = mx.

### Conic Sections - Hyperbola

Hyperbola
Equation x²/a² - y²/b² = 1

Some important results connected with x²/a² - y²/b² = 1

1. Vertex A(a,0) and A’(-a,0), AA’ is called the transverse axis whose length is 2a.
2. Focus S
3. Foot of the directrix Z
4. Equation of directrix
5. Equation of latus rectum (L.R.) x = ae
6. Equation of axis
7. Equation of tangent at vertex
8. Length of latus rectum = 2b²/a
9. Extremities of L.R = (ae,b²/a) and (ae, -b²/a)
10. Focal distance of any point (x,y)

Most of the equations for various elements like tangents, normals etc. of the ellipse x²/a² + y²/b² = 1 are true for the hyperbola x²/a² - y²/b² = 1 if we replace b² by -b².

1. Tangent at (x1,y1) is

xx1/a²-yy1/b² = 1

2. Normal at (x1,y1) is

x-x1/(x1/a²) = -(y-y1)/(y1/b²)

3. The line y = mx+c is a tangent to the hyperbola if c² = a²m²-b² .
4. Hence, the line y = mx+SQRT(a²m²-b²) is always a tangent.
5. The points of contact are (-a²m/c,-b²/c)
6. The lx + my = n will be a tangent to the hyperbola is

a²l² = b²m² = n²
7. From any pont (h,k) outside the hyperbola two tangents (real) can be drawn . The slopes are given by the equation
m²(h²-a²) – 2mhk + (k²+b²) = 0
8. Polar of any point (h,k) in relation to the hyperbola is

hx/a²-ky/b² = 1

### Conic Sections - Problems

If x = 9 is the chord of contact of the hyperbola x²-y² = 9, then the equation of the corresponding pair of tangents is

a. 9x²-8y²+18x-9 = 0
b.9x²-8y²-18x+9 = 0
c. 9x²-8y²-18x-9 = 0
d. 9x²-8y²+18x=9 = 0

Comparing chord of contact of point (h,k) i.e.. hx-ky = 9 with x = 9, we get h =1, and k = 0.

Therefore pair of tangents is SS1 = T²

### Limits and Continuity - Part 1

Good online material

http://tutorial.math.lamar.edu/Classes/CalcI/LimitsIntro.aspx

## Sunday, April 27, 2008

### Applications of Derivatives - Part 1

Good online material

http://tutorial.math.lamar.edu/Classes/CalcI/DerivAppsIntro.aspx

List of the topics

Rates of Change The point of this section is to remind us of the application/interpretation of derivatives that we were dealing with in the previous chapter. Namely, rates of change.

Critical Points In this section we will define critical points. Critical points will show up in many of the sections in this chapter so it will be important to understand them.

Minimum and Maximum Values In this section we will take a look at some of the basic definitions and facts involving minimum and maximum values of functions.

Finding Absolute Extrema Here is the first application of derivatives that we’ll look at in this chapter. We will be determining the largest and smallest value of a function on an interval.

The Shape of a Graph, Part I We will start looking at the information that the first derivatives can tell us about the graph of a function. We will be looking at increasing/decreasing functions as well as the First Derivative Test.

The Shape of a Graph, Part II In this section we will look at the information about the graph of a function that the second derivatives can tell us. We will look at inflection points, concavity, and the Second Derivative Test.

The Mean Value Theorem Here we will take a look that the Mean Value Theorem.

Optimization Problems This is the second major application of derivatives in this chapter. In this section we will look at optimizing a function, possible subject to some constraint.

More Optimization Problems Here are even more optimization problems.

L’Hospital’s Rule and Indeterminate Forms This isn’t the first time that we’ve looked at indeterminate forms. In this section we will take a look at L’Hospital’s Rule. This rule will allow us to compute some limits that we couldn’t do until this section.

Linear Approximations Here we will use derivatives to compute a linear approximation to a function. As we will see however, we’ve actually already done this.

Differentials We will look at differentials in this section as well as an application for them.

Newton’s Method With this application of derivatives we’ll see how to approximate solutions to an equation.

Business Applications Here we will take a quick look at some applications of derivatives to the business field.

### Indefinite Iintegrals - Part 1

Basic integrals

Add C (constant) to given ∫f(x)dx

S.no. f(x) ∫f(x)dx

1. 0 ... C (constant)

2. xn (n not equal to -1) ... xn+1/(n+1)

3. 1/x ... ln|x|

4. ex ... ex

5. ax ... ax/ln a

Trigonometric functions

6. sin x ... -cos x

7. cos x ... sin x

8. Cosec²x ...-cot x

9. sec²x ... tan x

Good online material

http://tutorial.math.lamar.edu/Classes/CalcI/IntegralsIntro.aspx

### Definite Integrals - Part 1

If F(x) is the antiderivative of a function f(x) continuous on (a,b)which means F'(x) = f(x) (a
∫f(x)dx {from a to b} = F(x) from a to b = F(b) - F(a)

Definite integral - Geometrical interpretation

the definite integral represents the algebraic sum of the areas of the figures bounded by

the graph of the function y = f(x)
the x axis
the straight line x =a and x = b.

if the curve goes above and below the x axis in the interval a to b, the areas of above the x axis enter this sum with a plus sign, while those below the x axis enter it with a minus sign.

Good online material

http://tutorial.math.lamar.edu/Classes/CalcI/DefnofDefiniteIntegral.aspx

### Differential Equations - Part 1

Differential equation is an equation involving derivatives of a dependent variable with respect to one or more independent variables.

Example: d²y/dx²+ y = x²

Order and degree are two attributes of a differential equation.

The order of a differential equation is the order of the highest differential coefficient involved. If second order derivative is present in the differential equation, the order of the equation is two or it is of second order. The equation give above as an example is a second order equation as d²y/dx² a second order derivative of y is present in the equation.

In the equation, the power to which the higher differential coefficient or derivative is raised is known as the degree of the equation.

Examples (d³y/dx³)4 + (d²y/dx²)² +y² = 0 is a 3rd order and 4th degree equation. 3rd order because d³y/dx³ is present in the equation and it is the highest order derivative in the equation. 4th degree because d³y/dx³ is raised to the fourth power.

Formation of a differential equation

If a polynomial f(x,y,c1, c2...cn) is the solution of a differential equation, where x and y are variables and c1,c2...cn are constants, differentiate the equation n times successively and eliminate the constants.

Solution of a differential equation
It is f(x,y, c1, c2,..cn) which does not involve derivatives and the derivatives of f(x,y,c1,c2...cn) satisify the differential equation.

Solution of first order and first degree differential equations

Equations in which the variables are separable

If the given differential equation can be written in the form

f(x)dx + g(y)dy = 0

we can see that variables x and y are separated in the left hand side expression.

We can write the equation as

f(x)dx = -g(y)dy

integrating both sides we get the solution.

Soluton of linear differential equation

In a linear differential equation, the dependent variable and its derivatives occur in the first power only and are not multiplied together.

Good online material
http://tutorial.math.lamar.edu/Classes/DE/DE.aspx