MAT211 Review
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35 terms
Terms | 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: adjacent: with respect to Θ 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 Circle | circle 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 |
| Convert between degrees and radians | 1 degree = π/180 radians 1 radian = 180/π degrees |
| Unit Circle Degrees v Radians |  Degrees; Radians 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 function | f(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 wave | Identical 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 curve | f(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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