calculus chapter one prerequisites for...
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CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
SECTION 1-1 LINES
INCREMENTS:
Example: Find the coordinate increments. A(-3,2) to B(-1,-2)
SLOPE: Other names:
1. Vertical lines
2. Horizontal lines Write the equation of the vertical line through the point P(-3, 2)
3. Parallel lines
4. Perpendicular lines
Writing equations for lines:
2 Ingredients needed:
Point-Slope Formula:
Slope-intercept Formula:
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Example: Write the equation of the line that contains the point P(-2, 2) and is 1.) parallel to and
2.) normal to the line 2x + y = 4.
Example: For what value of k are the two lines 3x + ky = 9 and x + 2y = 4
(a) Parallel?
(b) Normal?
ASSIGNMENT: Page 7 – 9 #1 – 30 (multiples of 3), 32, 33, 37, 38, 43, 44
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CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
SECTION 1-2 FUNCTIONS AND GRAPHS
FUNCTION:
DOMAIN:
RANGE:
FAMILIES OF GRAPHS:
LINEAR QUADRATICS
D: R: D: R:
CUBICS QUARTICS
D: R: D: R:
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SQUARE ROOTS INVERSES
D: R: D: R:
ABSOLUTE VALUES HALF-CIRCLES
D: R: D: R:
Determining Domains and Ranges:
Rational Functions Square Roots
ASSIGNMENT: Page 17 Quick Review #1 - 12, Exercises #5 – 9, 14, 15
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CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
SECTION 1-2 FUNCTIONS AND GRAPHS (Day 2)
Given the function: y = f(x)
Graph: y = - f(x) y = f(-x)
y = |f(x)| y = f|x|
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Odd vs. Even Functions:
Even Functions:
Odd Functions:
Example: 𝒚 = 𝟏
𝒙𝟐− 𝟏
Piecewise Functions:
𝟒 − 𝒙𝟐 𝒙 < 1
𝒇(𝒙) = 𝟑
𝟐𝒙 +
𝟑
𝟐 𝟏 ≤ 𝒙 ≤ 𝟑
𝟑 𝒙 < 3
ASSIGNMENT: Page 17 – 18 #20, 22, 24, 25, 31, 34, Handout (1st half)
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CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
SECTION 1-2 FUNCTIONS AND GRAPHS (Day 3)
Piecewise Functions (going backwards)
Compositions: f{g(x)} or (f g)(x)
Example: Given f(x) = x + 5 and g(x) = x2 - x - 3
Find a.) f{g(x)} b.) g{f(x)}
Finding Vertical Asymptotes:
Finding Horizontal Asymptotes:
Finding Holes: 𝒚 = 𝒙−𝟐
𝒙𝟐+ 𝒙−𝟔
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What does the following mean graphically?
f(x) > 0 f(x) < 0
f(x) = 0 f(x) > g(x)
ASSIGNMENT: Finish Handout, then page 18, #41, 42, 47, 49, 50, 63 (this is going backwards)
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CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
SECTION 1-3 EXPONENTIAL FUNCTIONS (Day 1)
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Graph: y = 2x y = 2-x
y = 𝟏
𝟐x The Exponential: y = ex
Exponential Growth and Decay:
Growth: y = k ∗ ax (if k is positive and a > 1) Decay: y = k * ax (if 0 < a < 1 or x < 0)
Continuous Compounding: A = Pert or y = y0ekt
#29 on page 25
Compound Interest: A = P(1 + 𝒓
𝒏) nt (n is the number of times per year)
#28 on page 25
Rules for Exponents:
Multiplying same bases:
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Dividing same bases:
Exponents raised to exponents:
Different bases raised to same power: ax ∗ bx
Quotients raised to same power: (𝒂
𝒃)x
ASSIGNMENT: Page 24 – 25 Quick Review #7-10, Exercises #1-7, 11 – 14, 26, 30, 32, 33, 34, 36
CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
SECTION 1-4 Parametric Functions
t becomes a third variable: 2 benefits:
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Example: Graph the parametric equations described by: x = t2 and y = √𝟒 − 𝒕𝟐 from 𝟎 ≤ 𝒕 ≤ 𝟐
Initial Point:
Terminal Point:
Cartesian Equation:
Sine Graph: y = sin x y = cos x
y = tan x
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Graph the parametric equations described by x = 4 cos t and y = 2 sin t from 𝟎 ≤ 𝒕 ≤ 𝟐𝝅
Initial Point: Terminal Point:
Cartesian: Use sin2x + cos2x = 1
ASSIGNMENT: Page 30 Quick Review #4, 5, 7, 8 Exercises: #1 - 4, 7, 13, 15, 21
CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
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SECTION 1-5 Inverses and Functions
Function:
Testing a function:
One-to-one:
Testing One-to-one:
Inverses:
Denoted:
2 Step Process for Finding an Inverse:
Checking:
Coordinates: If f(3) = 8, then f-1(8) =
Compositions:
Graphing
Example: Given: f(x) = x2 – 3; Find f-1(x)
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Checking:
Famous Inverses:
Squares: Cubes:
Exponentials: y = 2x Trigonometric: y = cos x
The Exponential y = ex ASSIGNMENT: Page 39 #1 – 6, 13 – 18 (skip 17), 24
CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
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SECTION 1-5 Logarithms (Day 2)
Graph y = ex y = ln x
Natural Logarithm facts:
A logarithmic no-no: ln (3x + 5) ≠
Punching a logarithm into a calculator: 𝐥𝐨𝐠𝒂 𝒙 =
Properties of Natural Logarithms:
Addition: ln x + ln y =
Subtraction: ln x - ln y =
Exponents: n ln x =
Simplify: ln 𝟗
𝟖 ln √𝟐𝟓
𝟑
Solve the equation:
2 ln x + 3 ln 2 = 2 e3x + 2 = 3 10 = 5 e3x
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Breaking down equations using natural logarithms:
𝒚 = 𝒍𝒏 (𝒙 √𝒙 + 𝟓
(𝒙 − 𝟏)𝟑)
ASSIGNMENT: HANDOUT
CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
SECTION 1-6 Trigonometry (Day One)
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Converting from Degree to Radians: Radians to Degrees:
Trigonometric Functions:
sin 𝜽 = cos 𝜽 =
csc 𝜽 = sec 𝜽 =
tan 𝜽 = ctn 𝜽 =
Graphing Trig Functions: 𝒚 = 𝒂 (𝒕𝒓𝒊𝒈 𝒇𝒖𝒏𝒄𝒕𝒊𝒐𝒏)[𝒌𝒙 ± 𝒄] + 𝒅
Amplitude Periodicity
Horizontal shift Vertical Shift
Graph: y = 2 cos 3x + 4 y = - 3 sin (𝝅
𝟑 x – 𝝅)
Finding exact values for Trig Angles:
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A. Find cos 𝟐𝝅
𝟑 B. Find tan
𝟕𝝅
𝟒
C. Find sec ( - 𝟓𝝅
𝟔 ) D. Find sin
𝟑𝝅
𝟐
ASSIGNMENT: HANDOUT also Page 48-49 #11-16
CALCULUS CHAPTER ONE – PREREQUISITES FOR CALCULUS
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SECTION 1-6 Trigonometry (Day Two)
Inverse Trig Functions:
y = Cos -1 x or Arccos x y = Sin -1 x or Arcsin x
y = Tan -1 x or Arctan x
Finding values for Inverse Trig functions:
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y = Sin -1 (√𝟐
𝟐) y = Arctan (−
𝟏
√𝟑)
y = Arccos 0 y = Cos-1 (−𝟏
𝟐)
Compound Trig Functions:
y = ctn (Sin -1 𝟏
𝟐) y = Cos -1 (𝒔𝒊𝒏
𝝅
𝟔)
y = cos [𝑨𝒓𝒄𝒕𝒂𝒏 (𝟏
√𝟑) − 𝑻𝒂𝒏−𝟏 √𝟑]
ASSIGNMENT: HANDOUT
TRIGONOMETRIC IDENTITIES
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Reciprocal:
sin 𝜽 = cos 𝜽 = tan 𝜽 =
csc 𝜽 = sec 𝜽 = ctn 𝜽 =
Quotient:
tan 𝜽 = ctn 𝜽 =
Pythagorean:
sin 2 x + cos 2 x = 1 + tan 2 x = 1 + ctn 2 x =
Sum Formulas:
sin (A ± B) = cos (A ± B) = tan (A ± B) =
Double Angle:
sin 2x = cos 2x = tan 2x =
Half-angle:
sin 𝟏
𝟐 x = cos
𝟏
𝟐 x = tan
𝟏
𝟐 x =
Two new ones you will need to know:
sin 2 x = cos 2x =