calculus chapter one prerequisites for...

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:

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


SECTION 1-2 FUNCTIONS AND GRAPHS

FUNCTION:

DOMAIN:

RANGE:

FAMILIES OF GRAPHS:

LINEAR QUADRATICS

D: R: D: R:

CUBICS QUARTICS

D: R: D: R:

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


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|

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)


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

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)


SECTION 1-3 EXPONENTIAL FUNCTIONS (Day 1)

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:

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


SECTION 1-4 Parametric Functions

t becomes a third variable: 2 benefits:

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

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


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)

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


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

Breaking down equations using natural logarithms:

𝒚 = 𝒍𝒏 (𝒙 √𝒙 + 𝟓

(𝒙 − 𝟏)𝟑)

ASSIGNMENT: HANDOUT


SECTION 1-6 Trigonometry (Day One)

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:

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


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:

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

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 =

calculus chapter one prerequisites for...

Documents