Mr. Beck
Review Sheet
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Test
Lines
Vertical Lines have no slope. (0 in denominator is undefined)
Horizontal Lines have 0 slope.
𝑥 +𝑥 𝑦 +𝑦
Midpoint formula: ( 1 2 2 , 1 2 2 )
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Slope formula: 𝑚 = 𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑋
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PreCalc
𝐶ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑌
Parallel lines have the same slope.
Perpendicular lines have slopes of negative reciprocal.
Distance formula: 𝑑 = √(𝑥2 − 𝑥1 )2 + (𝑦2 − 𝑦1 )2
Point Slope Form: 𝑦 − 𝑦1 = 𝑚(𝑥 − 𝑥1 )
Form of line: 𝑦 = 𝑚𝑥 + 𝑏
System of Equations can have:
A. Infinite solutions- (Same Equation)
𝑦 = 3𝑥 − 7
2𝑦 = 6𝑥 − 14
B. Parallel- (Never meet, therefore no solutions)
𝑦 = 3𝑥 + 2
𝑦 = 3𝑥 + 5
C. 1 Solution𝑦 = 3𝑥 − 1
𝑦 = −𝑥 + 5
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What is a function?
A set of ordered pairs where for each element in the domain there is only 1
element in the range.
Vertical line test can be used to identify function
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Standard form of a Parabola
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
(ℎ, 𝑘) = 𝑡𝑢𝑟𝑛𝑖𝑛𝑔 𝑝𝑜𝑖𝑛𝑡
−𝑏
Axis of Symmetry 𝑥 = 2𝑎
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Form of parabola: 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐
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Completing the Square- Ex:
𝑦 = 𝑥 2 + 4𝑥 + 5
𝑦 = (𝑥 2 + 4𝑥 + 4) + 5 − 4
𝑦 = (𝑥 + 2)2 + 1
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Polynomial Functions- Exponents only whole numbers𝑦 = 3𝑥 2 + 2𝑥 + 4 -All real numbers
Rational Expression- Negative Exponents-
Mr. Beck
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2
Review Sheet
PreCalc
2
𝑦 = 3𝑥 2 + 𝑥 + 4 - All real numbers except zero
Any Polynomial:
𝑦 = 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + 𝑎𝑛−2 𝑥 𝑛−2 … + 𝑎2 𝑥 2 + 𝑎1 𝑥1 + 𝑎0
Synthetic DivisionStep 1- Take Divider such as (X-2) and separate to just -2
Step 2- Change to positive 2, so that you can add instead of subtracting
Step 3- Set up and divide. Last number is remainder
Ex: 𝑥 3 + 3𝑥 2 − 4𝑥 + 7 ÷ (𝑥 − 2) = 𝑥 2 + 5𝑥 + 6 𝑅19
1
3
-4
7
1
5
6
R-19
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Remainder TheoremIf 𝑃(𝑥) ÷ (𝑥 − 𝑎) then P(a) is the remainder
Factor TheoremFor P(x), if P(a)=0, then x-a is a factor of the dividend.
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Cubic Equations
If Positive- 
If Negative-
Quartic Equations
If Positive- - W
If Negative-- M
If tangent at one point, one of the roots is double, ex: (𝑥 + 2)2
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Descartes’ Rule of Signs- The number of positive roots of P(x)=0 is either equal
to the number of variations in sign of P(x) or is less then this number by a positive
even integer
The number of negative roots of P(x) is equal to the number of variations in
sign of P(-x).
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Mr. Beck
Review Sheet
PreCalc
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Ex: P(x)=−2𝑥 4 + 𝑥 3 − 7𝑥 2 + 4𝑥 + 1
P(-x)= −2𝑥 4 − 𝑥 3 − 7𝑥 2 − 4𝑥 + 1
3 or 1 Possible Positive Values
1 Possible Negative Value
2 Possible Complex Values
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GLB and LUB
Step 1- Set up Synthetic Division
Step 2- Start with positive numbers, 1…2…3…, until you get all non-negative
values. That is the LUB.
Step3- Start with the negative numbers, -1….-2….-3….., until you get values
with alternating signs. That is the GLB. (zero doesn’t count as a positive
value).
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Polynomial Theorems1. The Fundamental Theorem of Algebra- For P(x) of degree n, P(x)=0 has
exactly n roots. (provided that double roots are counted twice, etc.)
ex: 2𝑥 4 − 𝑥 3 − 7𝑥 2 − 4𝑥 + 1 has 4 roots.
2. Complex Conjugates- If P(x) is polynomial with real coefficients and a+bi is
a complex root of P(x)=0 then a-bi is also a root.
3. Square Root Conjugates- If 𝑎 + √𝑏 is a root, then 𝑎 − √𝑏 is also a root.
4. Real Root- If P(x) is a polynomial of odd degree (n) with real coefficient,
then P(x)=0 has at least 1 real root. (Location Principle- If P(a)<0 and P(b)>0,
there must be at least 1 zero between a and b).
−𝑏
5. Sum and Product of Roots- for P(x)=𝑎𝑥 2 + 𝑏𝑥 + 𝑐, Sum of roots= 𝑎 ,
𝑐
Product of Roots= 𝑎.
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Rational Root TheoremFor 𝑎𝑛 𝑥 𝑛 + 𝑎𝑛−1 𝑥 𝑛−1 + ⋯ + 𝑎0 = 0 = (𝑞𝑥 − 𝑝)
𝑝
(Q must divide 𝑎𝑛 and p must divide 𝑎0 , 𝑡ℎ𝑒𝑛 𝑥 = 𝑞 𝑖𝑠 𝑎 𝑟𝑜𝑜𝑡
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And therefore any rational root must be a factor of 𝑎0
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Domain and RangeDomain- Possible values of the first element in an ordered pair.
Range- Possible values of the second element in an ordered pair.
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Extracting a Function from a CompositionWhen given a function and you need to find x.
Ex: 𝑓(𝑥 − 3) = 𝑥 2 − 11𝑥 + 24
𝐿𝑒𝑡 𝑈 = 𝑥 − 3
𝑥 =𝑢+3
𝑓(𝑥) = (𝑢 + 3)2 − 11(𝑢 + 3) + 24
𝑎
𝑛
Mr. Beck
Review Sheet
PreCalc
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𝑓(𝑥) = 𝑢2 + 6𝑢 + 9 − 11𝑢 − 33 + 24
𝑓(𝑥) = 𝑢2 − 5𝑢
𝑓(𝑥) = 𝑥 2 − 5𝑥
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Translation Functions
If you Replace y with –y- Reflect over x axis. Ex: 𝑓(𝑥) → −𝑓(𝑥)
If you Replace x with –x- Reflect over y axis. Ex: 𝑓(𝑥) → 𝑓(−𝑥)
If you Replace (x,y) with (-x,-y)- Symmetrical over the Origin.
If you Add to X move to Left, If you Subtract from X move to Right. Ex: 𝑓(𝑥) →
𝑓(𝑥 + 1)
𝑓(𝑥) → 𝑓(𝑥 − 1)
If you Add to Y move Up, If you subtract from Y move down.
𝑓(𝑥) → 𝑓(𝑥) + 1
𝑓(𝑥) → 𝑓(𝑥) − 1
Absolute Value- Reflect Negative Values over x axis.
Exchange x and y, reflection over line y=x
Multiply Y by 2 Stretch Vertically, Multiply X by 2 Shrink Horizontally
𝑓(𝑥) → 2𝑓(𝑥)
𝑓(𝑥) → 𝑓(2𝑥)
Sine Curve:
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Cosine Curve:
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Graphing of Trig Functions
𝑦 = 𝑎𝑠𝑖𝑛𝑏𝑥
a= amplitude, |a|
max 𝑣𝑎𝑙𝑢𝑒−min 𝑣𝑎𝑙𝑢𝑒
a=
2
b= Frequency
2𝜋
𝑃𝑒𝑟𝑖𝑜𝑑 =
𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
𝑦 = √𝑥
Mr. Beck
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Review Sheet
PreCalc
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Inverse of a Function:
Definition- 2 Functions, f and g are inverses of each other if for every point
(a,b) in f, is (b,a) in g.
To find inverse switch x and y, then solve for y.
Inverses are a reflection over the line y=x
𝐼𝑓 𝑦𝑜𝑢 𝑡𝑎𝑘𝑒 𝑓[𝑓 −1 (𝑥)] 𝑜𝑟 𝑓 −1 [𝑓(𝑥)] 𝑡ℎ𝑒 𝑟𝑒𝑠𝑢𝑙𝑡 𝑤𝑖𝑙𝑙 𝑎𝑙𝑤𝑎𝑦𝑠 𝑏𝑒 𝑥 𝑎𝑛𝑑 𝑡ℎ𝑒𝑦 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒𝑠.
𝐼𝑓 𝑡ℎ𝑒 𝑖𝑛𝑣𝑒𝑟𝑠𝑒 𝑜𝑓 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑎𝑙𝑠𝑜 𝑎 𝑓𝑢𝑛𝑡𝑖𝑜𝑛 𝑡ℎ𝑒𝑛 𝑤𝑒 𝑠𝑎𝑦 𝑖𝑡 𝑖𝑠 𝑎 𝑜𝑛𝑒 𝑡𝑜 𝑜𝑛𝑒 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛.
Line must pass both Horizontal and vertical line test for inverse to be a one to
one function.
Form FunctionLabel all knowns and unknowns
Create two equations and solve (use substitution)
Volume of Rectangular Shape=(𝑙𝑒𝑛𝑔𝑡ℎ)(𝑤𝑖𝑑𝑡ℎ)(ℎ𝑒𝑖𝑔ℎ𝑡)
Volume of Parallelepiped = (area of base)(height)
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Volume of Cone=
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Volume of Sphere= 3 𝜋𝑟 3
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𝜋𝑟 2 ℎ
3
4
A Parallelepiped is any figure with any shape based and walls coming up
Vertically.
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Factoring- (Anything can be factored)Ex:
𝑥 − 9 = (√𝑥 + 3)(√𝑥 − 3)
𝑥 2 + 6 = (𝑥 + 𝑖√6)(𝑥 − 𝑖√6)
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Factoring with negative exponentsIn negative exponents, factor out the less of the two numbers
Ex:
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(𝑥 2 − 25)2 + 2(𝑥 2 + 25)−2 − 8(𝑥 2 − 25)−2
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(𝑥 2 − 25)−2 [(𝑥 2 − 25)2 + 2(𝑥 2 + 25) − 8]
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𝐴(𝑡) = 𝐴0 (1 ± 𝑟)𝑡 (use only when talking about years)
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Compound Interest- 𝐴(𝑡) = 𝐴0 (1 + 𝑘)𝑘𝑡
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1
1
3
3
𝑟
A(t)= final amount
A0= initial amount
r= rate
k= number of times compounded per year.
t= number of years
Compounded continuously
Mr. Beck
Review Sheet
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𝐴(𝑡) = 𝐴0 𝑒 𝑟𝑡
A(t)= Final amount
A0= initial amount
R= rate
T= number of years
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Half Life Decay/Exponential Growth
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𝐴(𝑡) = 𝐴0 (2)𝑘
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1 𝑡
𝑘 = 𝐻𝑎𝑙𝑓𝑙𝑖𝑓𝑒 𝑑𝑒𝑐𝑎𝑦
𝑡
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𝐴(𝑡) = 𝐴0 (𝑟)𝑘
r=rate at which exponentially growths
k= how often it grows exponentially
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Exponential Functions
𝑓(𝑥) = (𝑎)(𝑏 𝑥 )
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The Definition of e
The irrational number e is defined as:
1
𝑒 = 𝑡ℎ𝑒 𝑙𝑖𝑚𝑖𝑡 𝑒 𝑎𝑠 𝑛 → ∞ (1 + 𝑛)𝑛
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𝑒 = lim (1 + 𝑛)
1 𝑛
𝑛→∞
Log Rules
𝐿𝑜𝑔𝑏 𝐴𝐵 = 𝐿𝑜𝑔𝑏 𝐴 + 𝐿𝑜𝑔𝑏 𝐵 (Product rule)
𝐴
𝐿𝑜𝑔𝑏 (𝐵) = 𝐿𝑜𝑔𝑏 𝐴 − 𝐿𝑜𝑔𝑏 𝐵 (Quotient Rule)
𝐿𝑜𝑔(𝑋 𝑛 ) = 𝑛𝑙𝑜𝑔(𝑥) (Power Rule)
𝐿𝑜𝑔𝑏 1 = 0
𝐿𝑜𝑔𝑏 𝑏 = 1
𝑙𝑛1 = 0
𝑙𝑛𝑒 = 1
𝑙𝑜𝑔𝑏𝑎𝑠𝑒 (𝑎𝑛𝑠𝑤𝑒𝑟) = 𝑝𝑜𝑤𝑒𝑟
log(term)=
defined when Term>0
undefined when Term≤0
𝑏 log𝑏 𝑥 = 𝑥
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Circle
(𝑥 − ℎ)2 + (𝑦 − 𝑘)2 = 𝑟 2
𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑦 + 𝐸𝑦 + 𝐹 = 0
In order to be a circle, A and C must be equal. F is negative
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Ellipse
PreCalc
Mr. Beck
Review Sheet
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𝑥2
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To find foci, 𝑐 2 = |𝑎2 − 𝑏 2 |
𝑎2
PreCalc
𝑦2
+ 𝑏2 = 1
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(𝑥−1)2
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New Center is (1,5)
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Hyperbola- The locus of all points, the difference of whose sums from 2 fixed
points (foci) is a constant.
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𝑥2
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To find foci, 𝑐 2 = 𝑎2 + 𝑏 2
To find asymptotes, use box methods. Go from the origin, the value of x and
up the value of y in all four directions. Draw a box and make the lines. Then
find equation using origin and one of the vertexes of the box.
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𝑎2
+
(𝑦−5)2
𝑏2
=1
𝑦2
− 𝑏2 = 1
𝑎2
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ParabolaA locus of all points equidistant from a line (directrix) and a given point
(focus).
Parabola has only one focus, which is in the curve.
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𝑦 = 4𝑝
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𝑦=
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Consider the equation 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0. If A, B, and C
are not all 0 and if graph is not degenerate, then:
1. The graph is the circle or an ellipse if 𝐵 2 − 4𝐴𝐶 < 0; (In a circle, B = 0 and
A = C.)
2. The graph is a parabola if 𝐵 2 − 4𝐴𝐶 = 0;
3. The graph is a hyperbola if 𝐵 2 − 4𝐴𝐶 > 0
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Trigonometry
𝒚
𝐬𝐢𝐧(𝜽) = 𝒓
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𝑥2
(𝑥−ℎ)2
4𝑝
+𝑘
𝒙
𝐜𝐨𝐬(𝜽) = 𝒓
𝒚
𝐭𝐚𝐧(𝜽) = 𝒙
𝒓
𝐜𝐬𝐜(𝜽) = 𝒚
𝒓
𝐬𝐞𝐜(𝜽) = 𝒙
𝒙
𝐜𝐨𝐭(𝜽) = 𝒚
Degrees-Minutes-Seconds
51° 28′ 57"
28
57
51 𝐷𝑒𝑔𝑟𝑒𝑒𝑠 + 60 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 + 3600 𝑠𝑒𝑐𝑜𝑛𝑑𝑠
Reference Angles
If In Quadrant I- It is the acute angle
If in Quadrant II- 180°-θ
If in Quadrant III- 180°+θ
If in Quadrant IV- 360°- θ
Mr. Beck
Review Sheet
PreCalc
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Radians-
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Radian- measurement of an angle and a number value.
Co-Terminal Angles- 2 angles whose terminal ray is in the same position.
Quadrantal Angles- 90, 180, 270, 360
10
10
= 1 radian
10
10
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Radian= central angle of intercepted arc is the same as the radius.
𝑠
𝜃=𝑟
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𝐴𝑛𝑔𝑙𝑒 𝑖𝑛 𝑟𝑎𝑑𝑖𝑎𝑛𝑠 =
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Converting between Degrees and Radians
𝐴𝑛𝑔𝑙𝑒 𝑖𝑛 𝑑𝑒𝑔𝑟𝑒𝑒𝑠
𝑅𝑎𝑑𝑖𝑎𝑛𝑠
= 𝜋
180°
𝑙𝑒𝑛𝑔𝑡ℎ 𝑜𝑓 𝑎𝑟𝑐
𝑟𝑎𝑑𝑖𝑢𝑠
90/-270
Students
Sin and Csc positive
All
All are positive
Quadrant II
Quadrant I
0/360
180/-180
Take
Tan and Cot are positive
Calculus
Cos and Sec are
positive
Quadrant III
Quadrant IV
270/-90
Special Right Triangles
1 45
√2
45
1
√3
30
2
1
60
Mr. Beck
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Review Sheet
PreCalc
High Leg- If Hypotenuse and Leg of right triangle congruent, therefore
triangles congruent.
CPCTC- Corresponding Parts of Congruent Triangles Congruent.
Inverse Trig FunctionsNote: If capital letter, then limited to Principal domain. (Only one
answer).
Note: If lowercase letter, than
there is no limit (more than one
Graphing of Inverse Trigonometric
answer).
Functions
Angle of Inclination
tan(𝛼) = 𝑚
Step 1: Get positive reference angle
Step 2: Determine whether obtuse
or acute angle. (If positive than
acute, If negative obtuse).
Step 3- Find supplement if needed.
Sine Curve
Cosine Curve
1. y=arc sin(x)
a. Domain- −1 ≤ 𝑥 ≤ 1
𝜋
𝜋
b. Range- − ≤ 𝑦 ≤
2
2
2. y=arc cos(x)
a. Domain- −1 ≤ 𝑥 ≤ 1
b. Range- 0 ≤ 𝑦 ≤ 𝜋
3. y=arc tan(x)
a. Domain- −∞ < 𝑥 < ∞
𝜋
𝜋
b. Range- − < 𝑦 <
2
2
4. y=arc csc(x)a. Domain- 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1
𝜋
𝜋
b. Range- − ≤ 𝑦 ≤ 𝑦 ≠ 0
2
2
5. y=arc sec(x)a. Domain- 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1
𝜋
b. Range- 0 ≤ 𝑦 ≤ 𝜋, 𝑦 ≠
2
6. y=arc cot(x)a. Domain- −∞ < 𝑥 < ∞
b. Range- 0 < 𝑦 < 𝜋
Tan Curve
Mr. Beck
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Review Sheet
Period- The number of radians needed for a complete cycle of a curve.
2𝜋
𝑃𝑒𝑟𝑖𝑜𝑑 = 𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦
Period of:
Sin Curve: 2𝜋
Cos Curve: 2𝜋
Tan Curve: 𝜋
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Frequency- The amount of full cycles with 2𝜋 radians.
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𝑦 = 2 sin(𝑥) (𝑎𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 𝑖𝑠 2)
𝑦 = sin(2𝑥) 𝑔𝑒𝑡𝑠 𝑠ℎ𝑟𝑢𝑛𝑘𝑒𝑛 (𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦 𝑖𝑠 2 𝑎𝑛𝑑 𝑝𝑒𝑟𝑖𝑜𝑑 𝑖𝑠 𝜋)



Identities
Reciprocal Identities
1
sin(𝜃) = csc(𝜃)

cos(𝜃) = sec(𝜃)

tan(𝜃) = cot(𝜃)


Quotient Identities
sin(𝜃)
tan(𝜃) = cos(𝜃)

cot(𝜃) =




Pythagorean Identities
𝑠𝑖𝑛2 (𝜃) + 𝑐𝑜𝑠 2 (𝜃) = 1
𝑡𝑎𝑛2 (𝜃) + 1 = 𝑠𝑒𝑐 2 (𝜃)
𝑐𝑜𝑡 2 (𝜃) + 1 = 𝑐𝑠𝑐 2 (𝜃)



sin(−𝜃) = − sin(𝜃)
cos(−𝜃) = cos(𝜃)
tan(−𝜃) = −tan(𝜃)
1
1
cos(𝜃)
sin(𝜃)
PreCalc
Mr. Beck



Review Sheet
PreCalc
Vectors
Vector- a directed distance. It has a magnitude and a direction.
Adding and Subtracting Vectors:
v
u
u+v
-v
u

u-v

Multiplying by a Scaler- It can change magnitude or turn it in the other
direction.
Absolute value of vector is its length.

Compass Directions0° N
270° W
90° E
180° S


Component form
Vector AB (2,3)



A(4,-6) and B(-2,-4) then AB (-6, 2)
To get vector= (𝑥2 − 𝑥1 ), (𝑦2 − 𝑦1 )
To get magnitude= √(𝑥2 − 𝑥1 )2 , (𝑦2 − 𝑦1 )2

Ex:
Mr. Beck
Review Sheet




V (2,1) W(1,1)
2W-3V
(4,2)-(6,3)= (-2,-1)
V+W= (3,2)




Vector and Parametric Equations of Lines
Vector Equation
⃗⃗⃗⃗⃗
𝑂𝑃 = ⃗⃗⃗⃗⃗
𝑂𝐴 + ⃗⃗⃗⃗⃗⃗⃗
𝑡𝐴𝐵
(𝑥, 𝑦) = (𝑥0 , 𝑦0 ) + 𝑡(𝑎, 𝑏)



Parametric Equation
𝑥 = 𝑥0 + 𝑡𝑎
𝑦 = 𝑦0 + 𝑡𝑏







Circle
𝑥 = 𝑟𝑐𝑜𝑠(𝑇) + ℎ
𝑦 = 𝑟𝑠𝑖𝑛(𝑇) + 𝑘
Ex:
(𝑥 + 1)2 + (𝑦 − 3)2 = 4
𝑥 = 2 cos(𝑇) − 1
𝑦 = 2 sin(𝑇) + 3



Ellipse
𝑥 = acos(𝑇) + ℎ
𝑦 = 𝑏𝑠𝑖𝑛(𝑇) + 𝑘

𝑥2

Hyperbola

If 𝑎2 − 𝑏2 = 1





𝑎2
𝑦2
+ 𝑏2 = 1
𝑥2
𝑦2
𝑥 = asec(𝑇) + ℎ
𝑦 = 𝑏𝑡𝑎𝑛(𝑇) + 𝑘
𝑦2
𝑥2
If 𝑎2 − 𝑏2 = 1
𝑥 = btan(𝑇) + ℎ
𝑦 = 𝑎𝑠𝑒𝑐(𝑇) + 𝑘




Height Problems
𝑆(𝑇) = −16𝑡 2 + 𝑉0 𝑡 + 𝑆0
𝑉0 − 𝑖𝑛𝑡𝑖𝑎𝑙 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦
𝑆0 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 ℎ𝑒𝑖𝑔ℎ𝑡

When asked for Maximum height
PreCalc
Mr. Beck
Review Sheet
PreCalc
−𝑏

( 2𝑎 , 𝑦)


When asked to find for a certain time, plug in for T.
When asked at what time does the object reach a certain height make the
height equal to the equation and solve.
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Polar Coordinate
(𝑟, 𝜃)
r= length of radius
𝜃= angle traveling on.
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

Converting to and from Polar form
(𝑟, 𝜃) = [𝑟𝑐𝑜𝑠(𝜃), 𝑟𝑠𝑖𝑛(𝜃)]
𝑥 = 𝑟𝑐𝑜𝑠(𝜃)
𝑦 = 𝑟𝑠𝑖𝑛(𝜃)
𝑥2 + 𝑦2 = 𝑟2
𝑦
𝑇𝑎𝑛(𝜃) = 𝑥



Polar Equations
r=1.5 (a circle)
𝜋
𝜃 = 3 (a line)


𝑟 = 𝑎sin(𝜃) (circle- starts from pole and goes upward. a is diameter)
𝑟 = acos(𝜃) (circle- starts from pole and goes towards the right)


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
Cardioid- makes a cusp
𝑟 = 𝑎 + asin(𝜃)
𝑟 = 𝑎 + acos(𝜃)
𝑵𝒐𝒕𝒆: 𝑾𝒉𝒆𝒏 𝒄𝒉𝒂𝒏𝒈𝒊𝒏𝒈 𝒕𝒉𝒆 𝒔𝒊𝒈𝒏 𝒐𝒇 𝒕𝒉𝒆 𝒇𝒊𝒓𝒔𝒕 𝒏𝒖𝒎𝒃𝒆𝒓, 𝒈𝒓𝒂𝒑𝒉 𝒅𝒐𝒆𝒔𝒏′ 𝒕 𝒄𝒉𝒂𝒏𝒈𝒆.
𝑾𝒉𝒆𝒏 𝒄𝒉𝒂𝒏𝒈𝒊𝒏𝒈 𝒔𝒊𝒈𝒏 𝒐𝒇 𝒄𝒐𝒔 𝒐𝒓 𝒔𝒊𝒏, 𝒕𝒉𝒆𝒏 𝒈𝒓𝒂𝒑𝒉 𝒄𝒉𝒂𝒏𝒈𝒆𝒔.


Limacon
𝑟 = 3 + 2 cos(𝜃) [𝑎 > 𝑏]


Looped Limacon
𝑟 = 1 + 2 cos(𝜃) [𝑎 < 𝑏]


Lemniscate
𝑟 2 = 4cos(2𝜃) (A leminscate looks like an eight)




Polar Rose
𝑟 = acos(𝑛𝜃)
𝑊ℎ𝑒𝑛 𝑛 𝑖𝑠 𝑎𝑛 𝑒𝑣𝑒𝑛 𝑛𝑢𝑚𝑏𝑒𝑟, 𝑡ℎ𝑒𝑟𝑒 𝑎𝑟𝑒 2𝑛 𝑟𝑜𝑠𝑒 𝑝𝑒𝑑𝑎𝑙𝑠
When n is an odd number there are n rose pedals.
Mr. Beck
Review Sheet
PreCalc
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




Points required for each graph
Circle- 2 points (endpoint and pole)
Cardioid- 4 points
Limacon- 4 points
Looped limacon- 5 points
Polar roses- Tips of the roses and pole.



Solving a Pair of Polar Equations
Either make them equal to each other, or use substitution.
When solving a pair of polar equations, always make a sketch to verify
answer.

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
Complex Numbers in Polar Form
𝑎 + 𝑏𝑖 = 𝑟𝑐𝑜𝑠(𝜃) + 𝑖𝑟𝑠𝑖𝑛(𝜃)
𝑟 = √𝑎2 + 𝑏 2
𝑟𝐶𝑖𝑠(𝜃) = 𝑟𝑐𝑜𝑠(𝜃) + 𝑖𝑠𝑖𝑛(𝜃)
𝑏
𝑇𝑎𝑛(𝜃) = 𝑎



Multiplying and Dividing by Complex Numbers
𝑟1 𝐶𝑖𝑠(𝜃1 ) × 𝑟2 𝐶𝑖𝑠(𝜃2 ) = 𝑟1 𝑟2 𝐶𝑖𝑠(𝜃1 + 𝜃2 )
𝑟1 𝐶𝑖𝑠(𝜃1 )
𝑟 𝐶𝑖𝑠(𝜃 −𝜃 )
= 1 𝑟1 2
𝑟 𝐶𝑖𝑠(𝜃 )


Complex Numbers- Raising/Taking Roots
𝑟 𝑛 𝐶𝑖𝑠(𝑛𝜃)

[𝑟𝐶𝑖𝑠(𝜃)]𝑛 = 𝑟 𝑛 𝐶𝑖𝑠( + 𝑘
𝑛









Finding the best fit Curve
R should be either close to 1 or -1.
Calculator:
Step 1- Make sure diagnostics is on
Step 2- Clear all lists
Step 3- Record all data in the table
Step 4- Graph points using ZoomStat and StatPlot
Step 5- Decide what is looks like (PowerReg, ExpReg, LineReg, QuadReg)
Step 6- Pick the Regression that fits best using the r to determine which is the
best fit.


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
Arithmetic and Geometric Sequences
Arithmetic- 𝑎𝑛 = 𝑎1 (𝑟)𝑛−1
Geometric- 𝑎𝑛 = 𝑎1 + (𝑛 − 1)𝑑
Sequence on Calculator-
2
2
2
1
1
𝜃
360
𝑛
)
Mr. Beck
Review Sheet

Seq(f(x), X, Start, End, Step)


Sum of Sequences/Series
𝑛(𝑡 +𝑡 )
Arithmetic- 𝑆𝑛 = 12 𝑛

Geometric- 𝑆𝑛 =

1
Infinite Geometric Sum- 𝑆 = 1−𝑟


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
Sequence of Partial Sums
If the sequence of partial sums approaches a Limit L, then the series
converges to L. (Must have a target to converge).
If r is between 1 and -1, then it converges. Otherwise it diverges.
To find the interval of convergence:
Ex:
1 + (𝑥 − 2)1 + (𝑥 − 2)2 + (𝑥 − 2)3 …
𝑟 = (𝑥 − 2)
−1 < 𝑥 − 2 < 1
1<𝑥<3




LimitsStrategyDivide by the x with biggest exponent in the denominator and solve.
6
𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑
0
𝑡1 (1−𝑟 𝑛 )
1−𝑟
𝑎

0





𝐿𝑖𝑚𝑖𝑡 𝑟𝑢𝑙𝑒𝑠:
𝐿𝑖𝑚(𝑓 − 𝑔) = lim(𝑓) − lim(𝑔)
𝐿𝑖𝑚(𝑓 + 𝑔) = lim(𝑓) + lim(𝑔)
𝐿𝑖𝑚(𝑓𝑔) = 𝐿𝑖𝑚(𝑓) × 𝐿𝑖𝑚(𝑔)
𝑓
lim(𝑓)
𝐿𝑖𝑚 (𝑔) = lim(𝑔)
0
𝑖𝑠 𝑢𝑛𝑑𝑒𝑓𝑖𝑛𝑒𝑑


𝝐 − 𝜹 𝑫𝒆𝒇𝒊𝒏𝒊𝒕𝒊𝒐𝒏 𝒐𝒇 𝒂 𝒍𝒊𝒎𝒊𝒕
If |𝑓(𝑥) − 𝐿| < 𝜖, 𝑡ℎ𝑒𝑛 0 < |𝑥 − 𝑎| < 𝛿


Sigma Notation
𝑛(𝑛+1)
∑𝑛𝑘=1 𝑘 =
2

∑𝑛𝑘=1 𝑘 2 =

∑𝑛𝑘=1 𝑘 3 = (



Mathematical Induction
Step 1- Test for 1
Step 2- Assume true that n=k.
𝑛(𝑛+1)(2𝑛+1)
6
𝑛(𝑛+1) 2
)
2
PreCalc
Mr. Beck
Review Sheet
PreCalc


Step 3- Change formula to Sk+1
Step 4- Add extra term to both sides and solve.





Binomial Theorem
Ex: 𝑛(𝑛2 + 5) is a multiple of 6
Step 1- Test for 1
Step 2- Assume n=k is true, and that 𝑛(𝑛2 + 5) is a multiple of 6.
Step 3- Make equation for Sk+1 and prove that it is multiple of 6.

Piecewise Function
−𝑥, 𝑥 < 0
Ex: 𝑓(𝑥) = {
𝑥, 𝑥 ≥ 0



Derivative
𝑓(𝑥+ℎ)−𝑓(𝑥)
Definition of Derivative- 𝑀𝑡𝑎𝑛 = lim
ℎ

Derivative for Specific Value- 𝐹


Power rule- 𝐼𝑓 𝑓(𝑥) = 𝑥 , 𝑡ℎ𝑒𝑛 𝑓
= 𝑛𝑥
𝑑
Product rule- 𝑑𝑥
(𝑓(𝑥)(𝑔(𝑥) = 𝑓(𝑥)𝑔′ (𝑥) + 𝑔(𝑥)𝑓 ′ (𝑥)=(first)(derivative of
sec)+(sec)(derivative of first).

Quotient rule- 𝑑𝑥 (𝑔) =



Continuity
Definition- A function is said to be continuous at a point x=c if:
1. lim 𝑓(𝑥) 𝑒𝑥𝑖𝑠𝑡𝑠


2. 𝑓(𝑐)exists
3. lim 𝑓(𝑥) = 𝑓(𝑐)


Differentiability
Differentiability over an interval implies continuity but continuity over an
interval DOESN’T imply differentiability.


Squeeze Theoremsin(ℎ)
lim
=1





ℎ→0
′ (𝑎)
= lim
2
𝑑
𝑓
′ (𝑥)
𝑔𝑓 ′ −𝑓𝑔′
𝑔2
=
𝑥→𝑎
𝑓(𝑥)−𝑓(𝑎)
𝑥−𝑎
𝑛−1
(den)(derivative of num)−(num)(derivative of den)
(𝑑𝑒𝑛)2
𝑥→𝑐
𝑥→𝑐
ℎ→0
lim
ℎ
ℎ
=1
ℎ→0 sin(ℎ)
sin(3𝑥)
3
lim sin(4𝑥) = 4
ℎ→0
lim
ℎ→0
lim
ℎ→0
1−cos(𝑥)
𝑥
cos(𝑥)−1
𝑥
=0
=0
Derivatives of Trig Functions
Mr. Beck






Review Sheet
𝑑 sin(𝑥)
𝑑𝑥
𝑑 cos(𝑥)
𝑑𝑥
𝑑 tan(𝑥)
𝑑𝑥
𝑑 cot(𝑥)
𝑑𝑥
𝑑 sec(𝑥)
𝑑𝑥
𝑑 csc(𝑥)
𝑑𝑥
= cos(𝑥)
= − sin(𝑥)
= sec 2 (𝑥)
= −𝑐𝑠𝑐 2 (𝑥)
= tan(𝑥) sec(𝑥)
= − csc(𝑥) cot(𝑥)



Linearization
𝐿(𝑥) = 𝑓(𝑎) + 𝑓′(𝑎)(𝑥 − 𝑎)
Pick a point near the one you are trying to estimate.


Differentials𝑑𝑦 = 𝑓′(𝑥)(𝑑𝑥)


Chain Rule
𝑑𝑦
𝑑𝑦 𝑑𝑢
= (𝑑𝑢)(𝑑𝑥 )
𝑑𝑥

PreCalc
When finding derivative other than x such as: 3cos(𝑥 2 ). Find derivative of the
whole term, and then the term on the inside (𝑥 2 ).



Implicit Differentiation
Finding the derivative when both xs and ys on one side of the equation.
2𝑥 2 𝑦 2 + 𝜋𝐶𝑜𝑠(𝑦) = 𝑥






Rate Problems
1. Relate the variables
2. Eliminate a variable
3. Find the derivative with respect to T
4. Plug in and solve
Similar Triangles

Graphing (Derivatives, Relative Min and Max, Concavity,
Increasing/Decreasing)
Step 1: When Give a regular function such as 𝑓(𝑥) = 𝑥 2 + cos(𝑥), check to
see where the derivative is undefined. (What value of x makes the function
undefined). Find roots of regular function. (Use these values when graphing).
Step 2: Check to see if there is a pinhole or an asymptote.




𝑥 2 −4
Pinhole: Where a point on the function is left blank. Ex: 𝑓(𝑥) = 𝑥−2 𝑥 ≠ 2.
There is a pinhole at x=2.
Asymptote: When the function approaches this line, it either goes to positive
or negative infinity. To find vertical asymptote, determine at which value of x,
Mr. Beck
Review Sheet
PreCalc
𝑥


the function will be undefined. Ex: 𝑓(𝑥) = 𝑥−4 𝐴𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒 𝑎𝑡 𝑥 = 4. To find
the horizontal asymptote make x approach infinity as if it were a limit and
𝑥
𝑋
the value you get for y is the asymptote. Ex: 𝑓(𝑥) = 𝑥−4 . 𝑋 =
1, 𝑡ℎ𝑒𝑟𝑒𝑓𝑜𝑟𝑒 𝑎𝑠𝑦𝑚𝑝𝑡𝑜𝑡𝑒 𝑎𝑡 𝑦 = 1.
Step 3: To find where the graph is increasing or decreasing, take the first
derivative of the function. Set the first derivative to zero and determine
critical values. Label a number line for the 1st derivative and make a number
line analysis with those critical values, to determine where the graph is
increasing or decreasing. Graph the critical values by plugging them into the
original equation. If for a critical value the first derivative goes from positive
to negative value, it is a relative maximum, and if for a critical value the first
derivative goes from a negative to a positive value, then it is a relative
minimum.
Step 4: To find concavity, find the second derivative; determine the critical
values by setting the 2nd derivative to zero. Then make a number line for the
second derivative. A positive value indicates concave up. A negative value
indicates concave down. A critical value that is involved in a sign change is an
inflection point, unless that value makes the function undefined. Note: Align
the two number lines one under the other, to make it easier to compare the
two when graphing. It is important to know both whether the graph is
decreasing or increasing and concavity.
Note: When noting intervals of concavity or increase/decrease, Concavity has only
parenthesis since concavity is never including the critical values.
Increasing/decreasing is always square brackets, since the critical values are also
increasing/decreasing. Positive/Negative infinity never have square brackets, since
it is not a “value” that you can technically include.
Test 1 Starts Here
 Absolute Min/Max
 Conditions:
 1. 𝑓 ′ (𝑥) = 0
 2. 𝑓 ′ (𝑥) = 𝑑𝑒𝑓𝑖𝑛𝑒𝑑
 3. Endpoints







To find an absolute min/max, take the function, get its derivative, set the
derivative to zero. Once solved, taker the answers and plug into the original
equation along with the endpoints. Ex:
2
𝑓(𝑥) = 3 𝑥 3 − 3𝑥 2 + 4𝑥 Interval of [0,3]
𝑓 ′ (𝑥) = 2𝑥 2 − 6𝑥 + 4
2𝑥 2 − 6𝑥 + 4 = 0
2(𝑥 2 − 3𝑥 + 2) = 0
(𝑥 2 − 3𝑥 + 2) = 0
(𝑥 − 2)(𝑥 − 1) = 0
Mr. Beck
Review Sheet


𝑥 = 2, 𝑥 = 1
5
𝑓(1) = 3

𝑓(2) = 3



PreCalc
4
𝑓(0) = 0 (absolute min)
𝑓(3) = 3 (absolute max)
*Stationary Points- are simply points where 𝑓 ′ (𝑥) = 0
 Applied Min/Max Questions
 1. Express Quantity As Need (Usually two equations)
 2. Optimize for one variable
 3. Find derivative
 4. Set=0, undefined
Test 2 Starts Here
 Rolle’s Theorem
 1. 𝑓(𝑎) = 𝑓(𝑏) = 0 (x-intercepts)
 2. Must be continuous on interval [a,b] (including)
 3. Differentiable on interval (a,b) (not including)
 Then there must be at least one value, c, such that 𝑎 ≤ 𝑐 ≤ 𝑏 and 𝑓 ′ (𝑐) = 0.
(At least one place where the derivative is zero).
 Extreme Value Theorem
 Every function that is continuous must have a maximum and minimum.





Mean Value Theorem
1. 𝑓(𝑥) 𝑖𝑠 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠 [𝑎, 𝑏]
2. 𝑓(𝑥)𝑖𝑠 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒 (𝑎, 𝑏)
𝑓(𝑏)−𝑓(𝑎)
3. 𝑓 ′ (𝑐) = 𝑏−𝑎
then there must be at least one C, 𝑎 ≤ 𝑐 ≤ 𝑏, such that the slope of
tangent=slope of secant








Rectilinear Motion—Motion on a Line
𝑠(𝑡) − 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑣(𝑡) − 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
𝑎(𝑡) − 𝑎𝑐𝑐𝑒𝑙𝑒𝑟𝑎𝑡𝑖𝑜𝑛 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛
Positive Velocity= moving to the right
Negative Velocity= moving to the left
Zero Velocity= at rest
To get from position to Velocity and from velocity to acceleration functions,
you simply take the derivative of the former.


Anti-derivative/Integral
Infinite Integral
Mr. Beck



Review Sheet
𝑥 𝑛+1
∫ 𝑥 𝑛 𝑑𝑥 = 𝑛+1 + 𝐶
𝐹(𝑥) + 𝑐 𝑖𝑠 𝑡ℎ𝑒 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 𝑡𝑜 𝑡ℎ𝑒 𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝑡ℎ𝑒 =
𝑑(𝐹(𝑥)
= 𝑓(𝑥)
𝑑𝑥
𝒀𝒐𝒖 𝒄𝒂𝒏 𝒂𝒍𝒘𝒂𝒚𝒔 𝒇𝒂𝒄𝒕𝒐𝒓 𝒐𝒖𝒕 𝒂 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕 𝒊𝒏 𝒇𝒓𝒐𝒏𝒕 𝒐𝒇 𝒂𝒏 𝒊𝒏𝒕𝒆𝒈𝒓𝒂𝒍






U Substitution in Integrals
Ex:
∫(𝑥 2 + 3)25 2𝑥 𝑑𝑥
𝐿𝑒𝑡 𝑢 = 𝑥 2 + 3
𝑑𝑢 = 2𝑥 𝑑𝑥
∫ 𝑢25 𝑑𝑢

∫


*When integrating functions like 𝐶𝑜𝑠(𝑛𝑥) it equals
𝑆𝑖𝑛(𝑛𝑥)
+𝑐
𝑛


Gravity Questions
𝑚
𝑓𝑡
𝑔 = 9.8 𝑠𝑒𝑐 2 𝑜𝑟 32 𝑠𝑒𝑐 2

PreCalc
(𝑥 2 +3)26
26
Quick Review
𝑑 sin(𝑥)
= cos(𝑥)
𝑑𝑥
𝑑 cos(𝑥)
= − sin(𝑥)
𝑑𝑥
𝑑 tan(𝑥)
= sec 2 (𝑥)
𝑑𝑥
𝑑 cot(𝑥)
= −𝑐𝑠𝑐 2 (𝑥)
𝑑𝑥
𝑑 sec(𝑥)
= tan(𝑥) sec(𝑥)
𝑑𝑥
𝑑 csc(𝑥)
= − csc(𝑥) cot(𝑥)
𝑑𝑥
+𝑐
𝑉(𝑡) = −𝑔𝑡 + 𝑉0
−𝑔𝑡 2

𝑆(𝑡) =


𝑊ℎ𝑒𝑛 𝑣𝑒𝑙𝑜𝑐𝑖𝑡𝑦 𝑖𝑠 𝑍𝑒𝑟𝑜, 𝑡ℎ𝑒𝑛 𝑖𝑠 ℎ𝑎𝑠 𝑟𝑒𝑎𝑐ℎ𝑒𝑑 𝑖𝑡𝑠 ℎ𝑖𝑔ℎ𝑒𝑠𝑡 𝑝𝑜𝑖𝑛𝑡 𝑜𝑟 𝑖𝑠 𝑎𝑡 𝑟𝑒𝑠𝑡
𝑊ℎ𝑒𝑛 𝑝𝑜𝑠𝑖𝑡𝑖𝑜𝑛 𝑖𝑠 𝑍𝑒𝑟𝑜, ℎ𝑖𝑡 𝑡ℎ𝑒 𝑔𝑟𝑜𝑢𝑛𝑑


Area under Graph by Limits of Integration
𝑨 = 𝐥𝐢𝐦 ∑𝒏𝒌=𝟏 𝑭(𝑿𝒌 ) 𝚫𝒙

𝚫𝒙 =


Ex: F(x) = 2 − 𝑥 3 [0,1]
1
Δ𝑥 = 𝑛

lim ∑𝑛𝑘=1(2 − [𝑛] ) 𝑛
2
+ 𝑉0 𝑡 + 𝑆0
𝒏→∞
𝒃−𝒂
𝒏
𝑘 3 1
𝑛→∞
1

(Since there are infinite amount of 2s, then there are 2n. Distribute 𝑛 in next
step)

2𝑛 1

2−(

2−


1
𝑘3
− ∑𝑛𝑘=1 [𝑛4 ]
𝑛
Quick Review
𝑛
𝑛(𝑛 + 1)
∑𝑘 =
2
𝑛(𝑛+1) 2 1
)
2
𝑛4
(𝑛2 +2𝑛+1) 1
4
1
𝑛2
1
1
lim 2 − (4 + 2𝑛 + 4𝑛2 )
𝑛→∞
7
𝐴=4
𝑘=1
𝑛
∑ 𝑘2 =
𝑘=1
𝑛
𝑛(𝑛 + 1)(2𝑛 + 1)
6
∑ 𝑘3 = (
𝑘=1
𝑛(𝑛 + 1) 2
)
2
Mr. Beck
Review Sheet
PreCalc

Area under Graph


𝐹(𝑥) = ∫𝑎 𝑓(𝑥)𝑑𝑥 (Area Function)
𝑇ℎ𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑎𝑟𝑒𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑖𝑠 𝑡ℎ𝑒 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛


Riemann Sums
𝒃
𝑨 = 𝐥𝐢𝐦 ∑𝒏𝒌=𝟏 𝑭(𝑪𝒌 ) 𝚫𝒙 = ∫𝒂 𝒇(𝒙)𝒅𝒙


The Definite Integral:
𝒃
∫𝒂 𝒇(𝒙)𝒅𝒙





𝐴𝑏𝑎 = 𝐹(𝑐)(𝑏 − 𝑎)
𝐴𝑐𝑎 + 𝐴𝑏𝑐 = 𝐴𝑏𝑎
𝐴𝑏𝑎 − 𝐴𝑐𝑎 = 𝐴𝑏𝑐
𝐴𝑏𝑎 + −𝐴𝑐𝑎 = 𝐴𝑏𝑐
𝐴𝑏𝑎 = −𝐴𝑎𝑏

𝑌𝑎𝑣 = lim ∑𝑛𝑘=1 𝐹(𝑋𝑘 ) (𝑏−𝑎)

𝑌𝑎𝑣 = (𝑏−𝑎) ∫𝑎 𝑓(𝑥)𝑑𝑥



𝑥
𝒎𝒂𝒙𝚫𝑿𝒌 →𝟎
1
𝑛→∞
1
𝑏
Average value of a function
𝑌𝑎𝑣 = 𝐹𝑎𝑣
𝑑𝑦
= 𝑖𝑛𝑠𝑡𝑎𝑛𝑡𝑎𝑛𝑒𝑜𝑢𝑠 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒
𝑑𝑥

∆𝑦

(𝑏−𝑎) ∫𝑎 𝑓(𝑥)𝑑𝑥 = 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑣𝑎𝑙𝑢𝑒


Second Fundamental Theorem of Calculus
Let f be continuous on an open Interval I, let A be any number in the interval
𝑥
I, if F is defined by 𝐹(𝑥) = ∫𝑎 𝑓(𝑥)𝑑𝑥.
Then 𝐹 ′ (𝑥) = 𝑓(𝑥)




∆𝑥
= 𝑎𝑣𝑒𝑟𝑎𝑔𝑒 𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒
1
𝑏
Rectangular Approximation MethodNote: Use Δ𝑋 as interval for heights. If
1
1
Δ𝑋 𝑖𝑠 4 , 𝑡ℎ𝑒𝑛 𝑤ℎ𝑒𝑛 𝑝𝑙𝑢𝑔𝑔𝑖𝑛𝑔 𝑖𝑛 𝑋𝑠 𝑓𝑜𝑟 ℎ𝑒𝑖𝑔ℎ𝑡𝑠, 𝑑𝑜 𝑖𝑛 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙𝑠 𝑜𝑓 4.

𝑳𝑹𝑨𝑴/𝑴𝑹𝑨𝑴/𝑹𝑹𝑨𝑴 𝒐𝒏 𝑪𝒂𝒍𝒖𝒍𝒂𝒕𝒐𝒓 =
(𝑺𝒖𝒎(𝑺𝒆𝒒(𝑭(𝒙), 𝒙, 𝑺𝒕𝒂𝒓𝒕 𝑷𝒐𝒊𝒏𝒕, 𝑬𝒏𝒅 𝑷𝒐𝒊𝒏𝒕, 𝑰𝒏𝒕𝒆𝒓𝒗𝒂𝒍/ 𝚫𝑿)
𝐿𝑅𝐴𝑀 = Δ𝑋(𝑆𝑢𝑚 𝑜𝑓 𝐻𝑒𝑖𝑔ℎ𝑡𝑠, 𝐿𝑒𝑓𝑡 𝑊𝑎𝑙𝑙 𝑜𝑓 𝐻𝑒𝑖𝑔ℎ𝑡𝑠)

Ex: 𝐿𝑅𝐴𝑀 = 4 ⌈𝑒 0 + 𝑒 −(4) + 𝑒 −(2) ⌉, where 𝑓(𝑥) = 𝑒 −𝑥

𝑀𝑅𝐴𝑀 = Δ𝑋(𝑆𝑢𝑚 𝑜𝑓 𝐻𝑒𝑖𝑔ℎ𝑡𝑠, 𝑀𝑖𝑑𝑑𝑙𝑒 𝑜𝑓 𝐻𝑒𝑖𝑔ℎ𝑡𝑠)
1
1 2
1 2
2
Mr. Beck
Review Sheet
1
1 2
3 2
5 2
PreCalc

Ex: 𝑀𝑅𝐴𝑀 = 4 ⌈𝑒 −(8) + 𝑒 −(8) + 𝑒 −(8) ⌉, where 𝑓(𝑥) = 𝑒 −𝑥

𝑅𝑅𝐴𝑀 = Δ𝑋(𝑆𝑢𝑚 𝑜𝑓 𝐻𝑒𝑖𝑔ℎ𝑡𝑠, 𝑅𝑖𝑔ℎ𝑡 𝑊𝑎𝑙𝑙 𝑜𝑓 𝐻𝑒𝑖𝑔ℎ𝑡𝑠)

Ex: 𝑅𝑅𝐴𝑀 = 4 ⌈𝑒 −(4) + 𝑒 −(2) + 𝑒 −(4) ⌉, where 𝑓(𝑥) = 𝑒 −𝑥
Test 3 Starts Here
Area Under Curve (Upper-Minus Lower/Right-Left)


1
1 2
1 2
3 2
2
2
f(x)
g(x)
𝑏


𝐴 = ∫𝑎 𝑓(𝑥) − 𝑔(𝑥) 𝑑𝑥
Upper – Lower Graphs (Vertical Lines=dx)



Right- Left (Horizontal Lines= dy)
Ex: 𝑦 = √𝑥, 𝑦 = −𝑥 + 6, 𝑦 = 1
2
𝐴 = ∫1 6 − 𝑦 − 𝑦 2 𝑑𝑦


Volume of Solid by Slices𝑏
𝑉 = ∫𝑎 𝐴(𝑥) 𝑑𝑥 (Depending on the Shape of the slices, the Area formula will
be different. For example, Square is 𝑆 2 .
𝐸𝑥:

Mr. Beck
Review Sheet




𝑥2 + 𝑦2 = 4
Cut Into Square slices
Side of Square is equal to 2y
2
𝑉 = ∫−2(2𝑦)2 𝑑𝑥

𝑉 = 4 ∫−2 𝑦 2 𝑑𝑥 (Need to replace y, because can’t solve unless in X)

𝑉 = 4 ∫−2 4 − 𝑥 2 𝑑𝑥





Volume of Revolutions
If rotating around x-axis, then dx
If rotation around y-axis, then dy
𝑏
𝑏
𝑉 = 𝜋 ∫𝑎 𝑟 2 𝑑𝑥 or 𝑉 = 𝜋 ∫𝑎 𝑟 2 𝑑𝑦
If hole in the disk then Bigger Radius-Smaller Radius.








Volume By Shells
If rotating around x-axis, then dy
If rotating around y-axis, then dx
𝑏
𝑉 = 2𝜋 ∫𝑎 𝑥 𝑓(𝑥) 𝑑𝑥
x= Radius
f(x)= height
dx= Thickness
Whenever doing volume by shells, it is helpful to draw in the cylinder.

Length of a Curve

𝑏
𝑑𝑥
𝑑𝑦
𝑠 = ∫𝑎 √[ 𝑑𝑡 ] + ⌈ 𝑑𝑡 ⌉ 𝑑𝑡 (Parametric)

𝑑𝑥
𝑠 = ∫𝑎 √[𝑑𝑦] + 1 𝑑𝑦

𝑏
𝑑𝑦
𝑠 = ∫𝑎 √[𝑑𝑥 ] + 1 𝑑𝑥






Surface Area of a Curve
𝑏
𝑆 = 2𝜋 ∫𝑎 𝑟 𝑑𝑠
Depending on the problem, you use one of the ds.
If they give you limits of integration for x, most likely you will use dx.
If they give you limits of integration for y, most likely you will use dy.
If they give you limits of integration for t, most likely you will use dt.


Sum of Terms
2
𝑘
Ex: lim ∑𝑛𝑘=1 𝑛 sin (𝜋 𝑛)
2
2
2
𝑏
2
2
2
𝑛→∞
PreCalc
Mr. Beck



Review Sheet
PreCalc
𝑘
𝑋𝑘 = 𝑛
1
lim ∑𝑛𝑘=1 (𝑛) 2 sin(𝜋𝑥)
𝑛→∞
1
∫0 2 sin(𝜋𝑥) 𝑑𝑥



To find limits of integration, find lowest possible x, and highest possible x.
Ex:
𝑘
𝑋𝑘 = 𝑛

Lowest= 𝑛=∞ = 0

Highest=𝑛=∞ = 1




Work
Two types of Work problems:
𝑊 = (𝑓)(𝑑)
𝑏
𝑊 = ∫𝑎 𝐹(𝑥)𝑑𝑥


𝑊 = ∫𝑎 (𝑑𝑒𝑛𝑠𝑖𝑡𝑦)(𝑉𝑜𝑙𝑢𝑚𝑒)(𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒)
Density/Specific Gravity of Water= 62.5



Hooke’s Law
𝐹(𝑥) = 𝑘𝑥
𝐹𝑜𝑟𝑐𝑒 = (𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡)(𝑑𝑖𝑠𝑡𝑎𝑛𝑡)


Trapezoidal Approximation
𝑌
𝑌
𝑇 = ∆𝑥( 21 + 𝑌2 + 𝑌3 + 𝑌4 + 25 )


New Derivatives and Integrals
𝑑(ln(𝑥))
1
=𝑥
𝑑𝑥

∫ 𝑥 𝑑𝑥 = ln(𝑥)

∫




𝑘=1
𝑘=∞
𝑏
1
𝑑𝑢
= 𝑙𝑛|𝑢| + 𝑐
𝑢
𝑥 𝑑𝑡
∫1 𝑡 = ln(𝑥)
1 𝑑𝑡
− ∫𝑥 = ln(𝑥)
𝑡
𝑑𝑎𝑥
= 𝑎 𝑥 (ln(𝑎))
𝑑𝑥
𝑎𝑥
𝑥
∫ 𝑎 = ln(𝑎) + 𝑐

𝑑𝑒 𝑥

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∫ 𝑒𝑥 = 𝑒𝑥 + 𝑐
𝑑𝑥
= 𝑒𝑥
Mr. Beck
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Strange Graphs
𝑦 = ln(𝑥)
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𝑦 = 𝑒𝑥

𝑦 = 𝑒 −𝑥
Review Sheet
PreCalc
Mr. Beck
Review Sheet
PreCalc
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Newton’s Method of Approximation
Used to estimate the root without calculus
𝑓(𝑥
)
𝑋𝑛 = 𝑋𝑛−1 − 𝑓′ (𝑋𝑛−1
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Using the Calculator
1. Type the function into Y1, ex= 𝑦 = 𝑥 3 − 𝑥 − 1
2. Type the derivative of the function into Y2, ex=𝑦 = 3𝑥 2 − 1
3. Use X value given in problem
4. Store this value as x, ex: 1 → 𝑥
𝑌
5. (𝑥 − 𝑌1 ) → 𝑥

𝑛−1)
2
Write the above into calculator and press answer to get approximation of
value. To get another iteration, simply repress enter.
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Lopital’s Rule
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If lim (𝑔(𝑥)) = 0 𝑜𝑟
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Derivative of the Inverse
1
(𝑓 −1 )′ = 𝑓′ (𝑓−1 (𝑥))
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𝐷𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑜𝑓 𝐼𝑛𝑣𝑒𝑟𝑠𝑒:
𝑇𝑎𝑘𝑒 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑦 𝑎𝑛𝑑 𝑡ℎ𝑒𝑛 𝑓𝑙𝑖𝑝 𝑡𝑜 𝑔𝑒𝑡 𝑑𝑒𝑟𝑖𝑣𝑎𝑡𝑖𝑣𝑒 𝑤𝑖𝑡ℎ 𝑟𝑒𝑠𝑝𝑒𝑐𝑡 𝑡𝑜 𝑥
ex: 𝑦 = 𝑥 5 + 7𝑥 3 + 4𝑥 + 1
Quick Review
Quick Review
𝑥 = 𝑦 5 + 7𝑦 3 + 4𝑦 + 1
1. y=arc sin(x)
Pythagorean Identities
𝑑𝑥
= 5𝑦 4 + 21𝑦 2 + 4
2 (𝜃)
2 (𝜃)
a. Domain- −1 ≤ 𝑥 ≤ 1
𝑑𝑦
𝑠𝑖𝑛
+ 𝑐𝑜𝑠
=1
𝑓(𝑥)
0
∞
∞

𝑑𝑦

Inverse Trig Functions

𝑑 sin−1 (𝑢)
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𝑑𝑥
1
𝑡𝑎𝑛2 (𝜃) + 1 = 𝑠𝑒𝑐 2 (𝜃)
𝑐𝑜𝑡 2 (𝜃) + 1 = 𝑐𝑠𝑐 2 (𝜃)
= 5𝑦 4 +21𝑦2 +4
𝑑𝑥
𝑑 cos−1 (𝑢)
𝑑𝑥
𝑑 tan−1 (𝑢)
𝑑𝑥
𝑑 cot−1 (𝑢)
𝑑𝑥
𝑑 sec−1 (𝑢)
𝑑𝑥
𝑑 csc−1 (𝑢)
𝑑𝑥
1
𝑑𝑢
= √1−𝑢2 𝑑𝑥
1
𝑑𝑢
= − √1−𝑢2 𝑑𝑥
1
𝑑𝑢
= 1+𝑢2 𝑑𝑥
1
𝑑𝑢
= − 1+𝑢2 𝑑𝑥
=
1
𝑑𝑢
𝑢√𝑢2 −1 𝑑𝑥
=−
𝑓 ′ (𝑥)
𝑜𝑟 0 ∗ ∞ 𝑡ℎ𝑎𝑛 𝑖𝑛 𝑒𝑞𝑢𝑎𝑙𝑠 lim(𝑔′ (𝑥))
1
𝑑𝑢
𝑢√𝑢2 −1 𝑑𝑥


Works the other way too
1
Ex: ∫ √1−𝑢2 = sin−1(𝑢)

Ex: ∫ 1+𝑢2 = tan−1(𝑢)
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Integration by Parts
1
𝜋
𝜋
b. Range- − 2 ≤ 𝑦 ≤ 2
2. y=arc cos(x)
a. Domain- −1 ≤ 𝑥 ≤ 1
b. Range- 0 ≤ 𝑦 ≤ 𝜋
3. y=arc tan(x)
a. Domain- −∞ < 𝑥 < ∞
𝜋
𝜋
b. Range- − 2 < 𝑦 < 2
4. y=arc csc(x)a. Domain- 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1
𝜋
𝜋
b. Range- − 2 ≤ 𝑦 ≤ 2 𝑦 ≠ 0
5. y=arc sec(x)a. Domain- 𝑥 ≤ −1 𝑜𝑟 𝑥 ≥ 1
𝜋
b. Range- 0 ≤ 𝑦 ≤ 𝜋, 𝑦 ≠ 2
6. y=arc cot(x)a. Domain- −∞ < 𝑥 < ∞
b. Range- 0 < 𝑦 < 𝜋
Mr. Beck
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Review Sheet
∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢
To solve, first you must determine what u and dv are. From that you figure
out what v and du are. Then you simply plug and solve. Keep in mind you
may have to repeat it more than once.
Ex:
∫ ln(𝑥) 𝑑𝑥
𝒖 = 𝐥𝐧(𝒙)
𝒅𝒙
𝒅𝒖 = 𝒙
𝒅𝒗 = 𝒅𝒙
𝒗=𝒙
𝑑𝑥
∫ ln(𝑥) 𝑑𝑥 = ln(𝑥) ∗ 𝑥 − ∫ 𝑥 𝑥
∫ ln(𝑥) 𝑑𝑥 = 𝑥𝑙𝑛(𝑥) − 𝑥 + 𝑐

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Trig Substitutions
1−cos(2𝑥)
𝑆𝑖𝑛2 (𝑥) =
2

𝐶𝑜𝑠 2 (𝑥) =
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∫ tan(𝑥) 𝑑𝑥 = −𝑙𝑛|cos(𝑥)| + 𝑐
∫ sec(𝑥) 𝑑𝑥 = 𝑙𝑛|sec(𝑥) + tan(𝑥)| + 𝑐
∫ csc(𝑥) 𝑑𝑥 = −𝑙𝑛|csc(𝑥) + cot(𝑥)| + 𝑐
Integration by Trig Substitution
By using identities such as:
Pythagorean Identities
𝑠𝑖𝑛2 (𝜃) + 𝑐𝑜𝑠 2 (𝜃) = 1
𝑡𝑎𝑛2 (𝜃) + 1 = 𝑠𝑒𝑐 2 (𝜃)
𝑐𝑜𝑡 2 (𝜃) + 1 = 𝑐𝑠𝑐 2 (𝜃)
You can solve integrals such as:
1
∫ √4−𝑥 2 𝑑𝑥
𝑙𝑒𝑡 𝑥 = 2 sin(𝑎)
𝑑𝑥 = 2 cos(𝑎) 𝑑𝑎
4 − 4𝑠𝑖𝑛2 (𝑎) = 4𝑐𝑜𝑠 2 (𝑎)
1
𝑑𝑥
∫
2

∫ √4𝑐𝑜𝑠2 ( ) 𝑑𝑥


2
𝑎
1
∫ 2 cos(𝑎) 𝑑𝑥

∫ 2cos(𝑎) 𝑑𝑥


1+cos(2𝑥)
√4−4𝑠𝑖𝑛 (𝐴)
1
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PreCalc
2cos(𝑎)
𝐴+𝑐
𝑥
𝐴 = 𝑆𝑖𝑛−1 (2)
To Define and integrate Improper Integrals
Anytime one of the limits of integration are infinity, it is an improper fraction.
Mr. Beck
Review Sheet
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Ex:
∞ 𝑑𝑥
𝑏 𝑑𝑥
∫1 𝑥 2 = lim ∫1 𝑥 2

lim − [𝑥] 1


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
𝑏→∞
1 𝑏
𝑏→∞
1
1
− [𝑏 − 1] = 1
Integration by Partial Fractions
Ex:
12𝑥+17
𝐴
𝐵
∫ (𝑥+1)(𝑥+2) = 𝑋+1 + 𝑋+2

=
𝐴𝑋+2𝐴+𝐵𝑋+𝐵
(𝑋+1)(𝑋+2)

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12𝑥 + 17 = (𝐴 + 𝐵)𝑥 + 2𝐴 + 𝐵
𝐴 + 𝐵 = 12
2𝐴 + 𝐵 = 17
A=5
B=7
5
7
+ (𝑥+2)
(𝑥+1)

∫ (𝑥+1) + (𝑥+2) = 5𝑙𝑛|𝑥 + 1| + 7𝑙𝑛|𝑥 + 2| + 𝑐

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Differential Equations
𝑑𝑦
Ex: 𝑑𝑥 = −4𝑥𝑦 2 Initial Condition (f(0)=1)
5
7

𝑑𝑦

∫ 𝑦 2 = ∫ −4𝑥 𝑑𝑥

− 𝑦 = −2𝑥 2 + 𝑐
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PreCalc
= −4𝑥 𝑑𝑥
𝑦2
𝑑𝑦
1
−1 = 0 + 𝑐
−1 = 𝑐
1
𝑦 = 2𝑥 2 +1
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Slope Fields
If horizontal row of slopes doesn’t change, than there is no x in the equation
If vertical column of slopes doesn’t change, than there is no y in the equation
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Euler’s Method
𝑑𝑦
∆𝑦 ≈ 𝑑𝑥 ∆𝑥