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Prerequisites to Calculus - Coggle Diagram
Prerequisites to Calculus
LINEAR FUNCTIONS
Increments:
If a particle moves from a point (x1,y1) to the point (x2,y2). Increments for change in x = x2 - x1. Increments for change in y = y2 - y1
EXAMPLE:
Find the coordinates increments From A (5,3) to B (8,5).
X: 8 - 5 = 3 and Y: 5 - 3 = 2
Slope:
Let P1(x1,y1) and P2(x2,y2) be points on a non-vertical line. m=rise/run = (y2 - y1)/(x2 - x1)
Example:
Find the slope for the points C(1,4) and D(-2,-5).
Slope: (-5-4)/(-2-1) = 3
Parallel Lines
- Have the same slope (m1-m2).
Perpendicular Lines
- Have Reciprocal slopes (m1)(m2) = -1
EXAMPLE:
Parallel Lines y=4x+3, y=4x-5. m1=m2 ---- 4x=4x
EXAMPLE:
Perpendicular. y=3x-6, y= (-1/3) + 4
(m1)(m2) = -1. ---(3)(-1/3) = -1
Vertical Lines
: x=a, all points have coordinates (x,b)
Horizontal Lines:
y=b all points have coordinates (x,b)
Horizontal Line: y= -4
Vertical Line: x = 8
Point-slope equation
: If a line has a slope m and passes through the point P(x1,y1), then the equation of this line is: (y-y1)=m(x-x1)
Example
: P(2,3) and m= (-3/2) y-3=(-3/2)(x-2)
Slope-intercept equation:
If a line has a slope m and has y-intercept b (0,b), then the equation of the this line is y=mx+b
EXAMPLE
: m=(-3/2) and b = -2 y=(-3/2)x -2
General Linear Equation:
The equation Ax+By=C (A and B not both zero).
EXAMPLE
: m=-1 and b=5 x+y=5
Functions & Graphs
Ways to Represent Functions
Closed Intervals:
[ ] for equal and greater or less than. For Open Intervals: ( ) for - or + infinity or greater or less than, not equal.
Even Functions:
f(-x) = f(x). Function is symmetrical about the y axis
EXAMPLE
: y=x^2 -2 f(-x)=(-x)^2 -2
f(-x) = f(x)
Odd Functions:
if f(-x)=-f(x). The function is symmetrical about the orgin
EXAMPLE:
f(x) =x^5+x
f(-x)=-x^5-x ----> f(-x)=-f(x)
Piecewise Functions:
different segments on the same graph
Composite Functions
: two functions can be composed when a portion fo the range of the first lies in the domain of the second
Example:
If f(x)=x^2+5x and g(x) =2x+3. 1) f(g(2)) ----> g(2) =2(2)+3 =7.
--------------> f(g(2)) = f(7) ----> (7)^2+5(7) = 84
Exponential Functions:
. -----------------------Translations can be made. 2^(x-3) + 4 ---> 3 is the horizontal translation. 4 is the vertical translation
Compound Interest:
A=A0(1+i/n)^nt
Example:
$5000 is invested at 7.2% compounded annually for 4 years. What is the amount of money at the end of the 4 years. ----------------------------------A=5000(1+0.072)^4 ------- A =$6603.12
Compound Interest
Continuously: A=A0(e)^it
Example:
The population is a certain city increase by 2.3% per year. Assuming the population in 2000 was 7.6 million, what is the population expected to be in 2010. A=7.6mill(e)^(10)(0.023) --------- A = 9.565 million
Exponential growth and decay:
A=A0(B)^(t/p)
Example:
A bacterial culture doubles every 2 hours. If the culture started with 24000 bacteria, how many bacteria will be present in 5 hours? A = 24000(2)^(5/2) -------- A = 135764 bacterias
Parametric Equations
Relations:
Set of ordered pairs (x, y) of real numbers. The graph of a relation is a set of points in a plane that correspond to the ordered pairs of the relation. If x and y are functions of a third variable t, called a parameter. Then parametric mode is used.
Ellipses:
Parametrization's of ellipses are similer to parametrizations of circles.
Example
x=3cost, y = 4sint, 0<t<2pi ((3cost)^2)/3^2 + ((4sint)^2)/4^2 ------ cos^2(t) + sin^2 (t) = 1. ------
Cartesian equation
(x^2)/9 + (y^2)/16 = 1
Parametrizing a Line Segment
Find a Parametrization for the line segment with endpoints (2,1) and (3,5). -------- T=0 (2,1) ---- T=1 (3.5) ------------x = 2+a(0) =2 ------------------- y =1+b(0)=1 -------------- 3 = 2 + a(1) = 1 ----------- 5 = 1 + b(1) = 4 -------- x(t) = 2+t ---- y(t) = 1+ 4t
Functions and Logarithms
One to one Function
: If a function has no two ordered pairs with different first coordinates and the same second coordinate.
Example
: Y=x^3
Example
f(x)=2x+3 2x+3=2x+3 ---- x1=x2
NOT ONE TO ONE
Inverses
-- each input of a one to one functions comes back with just one output, so it can be reversed
Example
y=2x+3 f(x)=2x+3 ----- x=2y+3 --- x-3 = 2y --- (x-3)/2 = f^-1(x)
Logarithmic Function:
y=logx/loga is the inverse of the exponential function y=a^x, where a>0, a doesn't equal 1
LOG RULES
Example
Solve for x --- logx/log2 = 3t +5 2^(3t + 5) = x
Trigonometric Funcitons
Arc Length
S = r x θ
Example
Radius = 10, Angle = 80 degrees. S=10(80pi/180) ---- S = 13.96 Radians
Trigonometric Ratios
Sinθ, Cosθ, Tanθ. Cscθ, Secθ, Cotθ.
Example
Point A(-3/5,-4/5), Determine all trigonometric rations. ------ Cosθ = -3/5, Sinθ = -4/5, Tanθ= 4/3, Secθ = -5/3,
Cscθ = -5/4, Cotθ = 3/4
Periodicity
When an angle of measure θ and an angle of measure θ +2pi are in standard position, their terminal rays coincide.
Transformations of trigonometric Graphs
Inverse Trigonometric Functions
: Only if one to one functions the graph can be inversed
Example
Solve for x -- sinx = 0.7 between 0 and 2pi ----- arcsin(sinx) = arcsin(0.7) --------x = 44 degrees = 0.76 radians ----- x1=0.76 radians. x2 = 180-44 = 136 degrees ----- x2 = 2.37 radians
