Differential Calculus 1

Point

No length, no width, no thickness ( 0 dimension)

Line

With length, no width, no thickness (1-dimension)

Plane

With length and width but no thickness (2-dimension)

Solids

With length, width and thickness (3D)

Counterclockwise

Rotation in the coordinate system

Straight line

Shortest distance between two points

Pythagorean Theorem

Most proved theorem in math

Fermat’s Last Theorem

Least proved theorem in math

Formula for distance between 2 points

Locus

Set of points, path of a point

Division of Line Segment method 1

Division of Line Segment method 2

+slope

+rise, +run or -rise, -run

-slope

-rise, +run or +rise, -rrun

angle value of the slope

formula of inclination (equivalent to slope)

most important formula in lines

general form

with slope m=-A/B

standard form

Point slope form

Given a point and slope

two-point form

Given 2-points

slope intercept form

given slope and y-intercept

x-intercept

y=0; (x=value, y=0)

y-intercept

x=0; (x=0, y=value)

Intercept form

given x and y-intercept

Angles between 2 lines formula 1

Angles between 2 lines formula 2

perpendicular lines relationship with slope

negative reciprocal

equation of perpendicular lines

“bisector”

midpoint

Distance between a Point and a Line

Distance between 2 parallel lines

Conics

The locus of a point which moves so that its distance from a fixt point (focus) is in constant ratio to its distance from a fixed straight line (Directrix)

Conic section

• parabola, ellipse, hyperbola

• They can be obtained geometrically by cutting a cone with a plane

Degenerate conic

Happens when plane intersects the bicone at the apex (middle point of 2 cones)

Circle by cutting plane of bicone

• Special case of ellipse

• Parallel to the base

• angle 0 degrees

Parabola by cutting plane of bicone

• Parallel to the slant edge

• angle = alpha

Hyperbola by cutting plane of bicone

• Parallel to the axis (not necessary exactly @ the axis)

• angle greater than alpha

Ellipse by cutting plane of bicone

• Any angle provided not parallel to base, element, axis

• angle less than alpha

Principal Axis

Divides the conic symmetrically

Eccentricity

Measure of uncircleness

Directrix

Line perpendicular to the axis of symmetry and does not pass through the connic section

eccentricity of circle

0

eccentricity of ellipse

< 1

eccentricity of parabola

1

eccentricity of hyperbola

>1

eccentricity of line

infinity ∞

with xy term

circle condition in equation

parabola condition in equation

ellipse condition in equation

hyperbola condition in equation

if there is a Bxy term then use formula for discriminant

parabola discriminant

ellipse discriminant

hyperbola discriminant

xy = 0 (not necessarily a line)

Use discriminant equation

Parabola

Set of points in a plane that are equidistant from a fixed point and a fixed line

length of latus rectum of parabola

focal length in parabola

distance from vertex to the focus

for a parabola opening up/down

for `1 value of y, there are 2 values of x
(h,k) = vertex

for a parabola opening right/left

application of parabola

used as satellite dish because it collects the signals for transmission

Circle

A plane figure obtained as a locus of a point whose distance from a fixed point (center) is constant

equation of circle

simplified equation in circle

center equation possible for ellipse and hyperbola (not parabola)

Ellipse

A set of points in a plane when the SUM of whose distances from 2 fixed points is constant

measurements in ellipse

equation for an ellipse with horizontal principal axis

equation for an ellipse with vertical principal axis

latus rectum of ellipse

Hyperbola

A set of all pints in a plane when the DIFFERENCE of distances from two fixed points(foci) is constant

measurements in hyperbola

equation of hyperbola with horizontal principle axis

equation of hyperbola with vertical principle axis

latus rectum of hyperbola

transverse and conjugate axis formula

asymptote line equation