## MAT211 Review

##### Created by:

Duganium  on January 31, 2012

##### Description:

Calculus 1

MWCC Prof. Y. Elinevsky

Briggs, W. & Cochran, C. (2010). Calculus. Single variable ed. Addison-Wesley. ISBN-13: 9780321664075.

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# MAT211 Review

 SetsShown as it would appear on a graph, in interval notation, and it set notation Closed, bounded ⋅--⋅ or [a, b] or {x: a≤x≤b} Bounded ⊶ or (a, b] or {x: aa} ⋅→ or [a, ∞) or {x: x≥a} ←∘or (-∞, a) or {x: x
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#### Definitions

Sets Shown as it would appear on a graph, in interval notation, and it set notation
Closed, bounded
⋅--⋅ or [a, b] or {x: a≤x≤b}

Bounded
⊶ or (a, b] or {x: a<x≤b}
⊷ or [a, b) or {x: a≤x<b}

Unbounded
∘→ or (a, ∞) or {x: x>a}
⋅→ or [a, ∞) or {x: x≥a}
←∘or (-∞, a) or {x: x<a}
←⋅or (-∞, a] or {x: x≤a}
↔ or (-∞, ∞)
Interval Notation [ or ] indicates that x could be equal to the side bounded by the square bracket
( or ) indicates that x cannot be equal to the side bounded by the parenthesis
[a, b] or {x: a≤x≤b}
(a, b] or {x: a<x≤b}
[a, b) or {x: a≤x<b}
Set notation Use braces
{x: a≤x≤b}
Function test A function is valid only if any vertical line intersects the function only once
Transformations A graph of a function can be shifted various ways:
f(x)∓c -down or +up c units
f(x∓c) -right or +left c units
y=cf(x) 0<c (vertical shrink)<1<c(vertical stretch)
y=f(cx) 0<c(horizontal stretch)<1<c(horizontal shrink)
y=-f(x) reflect across x axis
y=f(-x) reflect across y axis
Transformation example
(image not to scale) Label a to c, peaks left to right
a is a vertical stretch of b
c is a vertical and horizontal stretch of b
Piecewise function a function that is not contiguous
Set notation uses a single vertically stretched brace, with the results for x shown, stacked, according to ascending values of x
f(x)=
{x if x<2
{3 if x=2
{-1/2x+t if x > 2
polynomial a mathematical expression involving a sum of powers in one or more variables multiplied by coefficients
x²-8x+24
rational number A rational number is a number that can be expressed as a fraction p/q where p and q are integers and q≠0.
Irrational number a real number that cannot be written as a simple fraction
pi/2
Real Line a line with a fixed scale so that every real number corresponds to a unique point on the line
consists of rational numbers ∪ irrational numbers
The set of rationals is so small that it measures 0 along this continuum
Trigonometric Functions Ratios of the sides of a right triangle
hypotenuse: the side opposite the right angle
The other two sides are either:
opposite: with respect to Θ
Θ: the angle in question

Sin = o/h; Cosecant: Csc = h/o
Cos = a/h; Secant: Sec = h/a
Tan = o/a; Cotangent: Cot = a/o
Radians Working with a circle of radius r, the radian measure of angle Θ is the length of the arc associated with Θ, denoted by s, divided by the radius of the circle r
Unit Circlecircle whose radius = 1
centered on a cartesian graph
(x, y) points on the arc are (cos Θ, sin Θ)
tan Θ = y / x Usually, the denominators will cancel leaving numerator of y / numerator of x
csc = 1 / y
sec = 1 / x
cot = x / y

The origin from which angles are measured is along the x axis from 0 to 1.
Positive: counterclockwise
Negative: clockwise
> 2π = greater than a full revolution
Unit Circle Quadrants Numbered 1 to 4 in Roman numerals moving counterclockwise from upper right.
I
+: x, y, all trig functions
-: nothing
II
+: y, sin, csc
-: x, cos, sec, tan, cot
III
+: tan, cot
-: x, y, sin, csc, cos, sec
IV
+: x, cos, sec
-: y, sin, csc, tan, cot
0; 0
30; π/6
45; π/4
60; π/3
90; π/2
120; 2π/3
135; 3π/4
150; 5π/6
180; π
Reciprocal Trigonometric Identities tan Θ = sin Θ / cos Θ
cot Θ = 1 / tanΘ = cos Θ / sin Θ
csc Θ = 1 / sin Θ
sec Θ = 1 / cos Θ
Pythagorean Trigonometric Identities sin² Θ + cos² Θ = 1
1 + cot² Θ = csc² Θ
tan² Θ + 1 = sec² Θ
Double and Half-Angle formulas sin 2Θ = 2 sin Θ cos Θ
cos 2Θ = cos² Θ - sin² Θ
cos² Θ = (1 + cos 2Θ) / 2
sin² Θ = (1 - cos 2Θ)/2
Power Function f(x)=x^y
A polynomial that graphs as a parabola that is flatter in the middle and climbs steeply on the sides. Rate of climb and value of y are directly related.
y=even: parabola with both legs pointing in the same direction, a u shape
y=odd: parabola with one leg pointing in the opposite direction
Root functionf(x)=x^1/2 or f(x)=√x
f(x)=x^1/3 or f(x)=∛x
Denominator (or root) must be greater than 1. These have a small steep rise near 0, then flatten as the root increases. The root value and flattening of climb are directly related.
Even root: begins at zero, climbs steeply, flattens quickly. Domain and range consist only of nonnegative numbers.
Odd root: Same as positive, but reflects across x/y, so it decreases sharply then flattens quickly.
Roots and Fractions √(a/b) = √a / √b
E.g.
√(1/2) = √1 / √2 = 1 / √2 = √2 / 2

∛(xy) = ∛x*∛y (where root is any real number)
∛(x²) = (∛x)²
x^n/2*x^m/2 = x^n+m/2

x√x = √(x³) = x^2/2 * x^1/2

x^n/m such that n/m > 1
factor out an x for each time m divides completely into n
multiply that result by the remainder
e.g.
x^5/3 = x^3/3x^2/3 or xx^2/3 or x(∛(x²))
Fractions a/b / c/d = ad / bc
sine wave
f(x)=sin(x)
Domain: all real numbers
Range: (-1, 1)
On a cartesian axis, the wave is evenly bisected by the x axis, with (0, 0) occurring halfway up a direct slope
The first peaks on either side of 0 occur at ∓π/2
The wave intercepts x again at ∓π
The first troughs on either side of 0 occur at ∓3π/2
The wave intercepts x again at ∓2π
Period: 2π
cosine waveIdentical to sine wave but shifted π/2 units left
cos x = sin (x + π/2)
On a cartesian axis, the wave is evenly bisected by the x axis, with (0, 0) occurring at a peak
The wave intercepts x at ∓π/2
The first troughs on either side of 0 occur at ∓π
The wave intercepts x again at ∓3π/2
The wave peaks again at ∓2π
Period: 2π
tangent curvef(x)=tan x = sin x / cos x
On a cartesian graph, this appears as a series of lines with a positive slope where either end stretches steeply to ∞ with a slight flattening between y (-3, 3).
Working rightward from x=0, the first portion continues from (0, 0) to grow steeper toward ∞.
It reaches ∞ (undefined) just left ∓π/2
Just right of ∓π/2 it stretches upward from -∞
It intercepts x again at ∓π.
Period: π
Amplitude & Period in Trig Function f(x)= v + a sin p(hx-π)
v, vertical translation: downward < 0 < upward
a, amplitude: |a|
p, period: 2π/p
h, horizontal stretch: compressed < 2π < stretched
phase shift: ∓π (-right, +left)
Works also for cosine
Sector of a circle A=Θ2r²
C=Θr
Dividing with Fractions (a / b) / (c / d) = ad / bc = (a / b)*(d / c)
a / b / c = a / (b*c) is a form of this with the d being an implicit 1 as c's denominator
a / c / d = (a*d) / c is a form of this with the b being an implicit 1 in a's denominator
Exponents (xy)² = x² * y²
(x/y)² = x² / y²

x² * x = x³ (add powers for like bases)
x³ / x = x² (cancel like bases, subtract powers)
Quadratic formula x = -b ± √(b² - 4ac)/2a
Law of Cosines relates the length of the sides of a planar triangle to the cosine of one of the angles

If the sides are a-b-c, the angles are C-A-B. The angle's letter belongs to the leg that doesn't form the angle.

a²=b²+c² - 2bc cos A
b²=a²+c² - 2ac cos B
c²=a²+b² - 2ab cos C
Law of Sines relates the lengths of the sides of a planar triangle to the sines of its angles

If the sides are a-b-c, the angles are C-A-B. The angle's letter belongs to the leg that doesn't form the angle.

a / sin A = b / sin B = c / sin C
Trigonometric Relationships sin(90∘ - Θ) = cos Θ
sin(α + β) = sin α cos β + cos α sin β
sin(45∘) = cos(45∘) = √2 / 2

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