Jason Gallicchio’s AP Calculus Outline

Jason Gallicchio’s AP Calculus Outline
I. Review of Algebra and Trigonometry

A. Polynomials

1. Special Products and Factors

a. Difference of two Squares:

b. Perfect Squares: and

c. Sum of two Cubes:

d. Difference of Two Cubes:

2. Quadratic Formula:

3. Synthetic Div. Ex. ®

a. Form: Bring Down the three and "Multiply by 2 and Add".

B. Basic Trigonometry

1. Special Angles:

2. Law of Cosines: Law of Sines:

3. Inverse Trig:

C. 32 Fundamental Trigonometric Identities

Pythagorean

Sum + Difference Identities

Negative Angle Identities

Quotient

Double Angle Identities

Product and Factor Identities

Reciprocal

Half Angle Identities

Reduction Identities

f(x) is one of the 6 trig functions.

Examples of Reduction:

II. Functions and Graphs
A. Properties of Functions

1. Domain and Range

a. Algebraic

(1) Domain: Solve for and do analysis of signs.

(2) Range: Solve for and do analysis of signs.

b. Non-Algebraic: Know D + R for trig, logorhythmic, and Exponential.

2. Sum, Product, quotient, and composition

a. Sum:

b. Product:

c. Composition:

3. Inverse Functions (Must be one to one 1-1)

a. To find, solve for and switch vars.

c. The graph of and have symmetry w.r.t. the line

4. Even and Odd

a. Even: and stays the same (Sym. w.r.t. -axis)

b. Odd: and (Sym. w.r.t. origin)

5. Kohne Equation: ;

6. Zeros of a Function

a. A "Zero is where the graph crosses the -axis

b. To find, set and solve for

c. Newton’s Method:

7. Parametric Equations

a. and where = parameter

b. To "Eliminate Parameter" solve for and substitute.

8. Polar and Rectangular coordinates. .

B. Properties of Graphs

1. Intercepts

a. To get an -intercept, substitute 0 in for and solve.

b. To get a -intercept, substitute 0 in for and solve.

2. Symmetry

a. Sym. w.r.t. y-axis iff is even ( and y stays the same).

b. Sym. w.r.t. origin iff is odd ( and ).

3. Asymptotes

a. Vert.: solve for and see what coordinates of x make undefined.

b. Horiz.: solve for and see where is undivided, or .

4. Translations and other movements. (Ex. )

a. Vertex at

b. Equation for axis of symmetry: .

c. : Graph opens up. : Graph opens down.

d. : Graph widens. : Graph narrows.

e. and have periods of . is .

C. Conic Sections

1. Line: .

a. is slope

b. is intercept.

2. Circle:

a. radius is .

b. Center at .

3. Parabola:

a. Vertex at

b. Equation for axis of symmetry: .

c. : Graph opens up. : Graph opens down.

d. : Graph widens. : Graph narrows.

4. Ellipse: ("It’s just a squished circle" - Ed)

a. Intercepts: .

b. Long axis is the major axis, short one is minor axis.

c. Center at .

5. Hyperbola:

a. Horizontal: Vertical:

b. Slant Asymptote:

c. Center at .

D. Polar Graphs

1. Rose Curves:

a. Number or Petals

i. if is odd.

ii. if is even.

b. Interval for graph:

i. if is odd.

ii. if is even.

c. Petal length = |a|.

d. Location of first petal:

i. : 0°.

ii. : where is largest.

2. Limacon:

a. : Cardioid.

i. : Vertical.

ii. : Horizontal.

b. : Inner Loop.

c. : "Near Cardioid".

3. Lemniscate (figure 8) { }

a. : symmetry w.r.t. polar axis.

b. : symmetry at 45° tilt.

4. Spirals: q in radians.

a. (Archamedies’s Spiral)

b. (reciprocal Spiral)

5. Special Graphs:

a. : Circle.

b. : Line (not only a ray)

III. Limits and Continuity

A. Finite Limits

1. "Lucky Seven" (K is a constant)

2. Single Sided Limits

a. (From Rt. Side)

b. (From Lt. Side)

3. Limits at Infinity

4. e - d Definition of Limits

a. is a point on a number line.

d. Iff: for every positive e there is a positive d so:

e. whenever

f. e = distance btw. and on -axis

d = distance btw. and a on -axis

5. Frequent Limits

6. Hierarchy of Growth:

B. Nonexistent Limits

1. Types of nonexistence

c. "Weird Case" Ex-

2. Infinite Limits Ex-

C. Continuity

1. Definition:

2. Graphical Interpretation of continuity — all one line

3. Absolute Max. and Min: If is continuous on then:

a. has an absolute max.

b. has an absolute min

4. Intermediate Value Theorem

a. If is continuous on

b. Then it takes on every value of btw. and .

5. Polynomial Functions are continuous because D =

IV. Differential Calculus

A. The Derivative

1. Definition:

2. Derivative Formulas

1.	Constant Rule	5.	Product Rule
2.	Constant Times Function	6.	Quotient Rule
3.	Sum Rule	7.	General Power Rule
4.	Power Rule

3. Derivatives of Composite Functions (The Chain Rule)

If: and

Then: or

4. Implicit Derivatives

If:

Then:

5. Derivatives of Higher Order

: Function

: 1st Derivative

: 2nd Derivative =

: 3rd Derivative = ....

6. Differentials of Inverse Functions

7. Differentials of Logarithmic and Exponential Functions

8. Differentials of Trigonometric Functions

9. Derivatives of Parametrically Defined Functions (use Chain Rule)

B. Applications of Theorems About Derivatives

1. must be continuous for to exist

2. The Mean Value theorem:

a. If: is continuous and differentiable on

b. Then: There is one number: on such that:

3. l’Hôpital’s Rule-indeterminate forms:

a. IF: or

THEN:

C. Applications of the Derivative

1. Geometric Applications

a. Tangent Line and Normal Line

ii.

b. Increasing and Decreasing

(1) If = (+) then is increasing at

(2) If = (-) then is decreasing at

(3) If = 0 then there is a max. or min. at

c. Critical #’s

n is a critical # iff = 0

d. Concavity

(1) If = (+) then is concave up at

(2) If = (-) then is concave down at

(3) If = 0 then there is an inflection point at

2. Optimization Problems

a. 1st Derivative test for relative maximum and minimum

(1) Set and solve for (At these values, )

(2) Do Analysis of signs on those critical values:

(a) If: <Ä Ä Ä Ä (+)Ä Ä Ä Ä CÄ Ä Ä Ä (-)Ä Ä Ä Ä > Then: Rel. Max. @

(a) If: <Ä Ä Ä Ä (-)Ä Ä Ä Ä CÄ Ä Ä Ä (+)Ä Ä Ä Ä > Then: Rel. Min. @

b. 2nd Derivative test for relative maximum and minimum

(1) Find Critical values from

(a) If = (+) Then: has Rel. Min. at

(b) If = (-) Then: has Rel. Max. at

(2) Mr. Kohne doesn’t like the 2nd derivative test

(a) is hard to calculate.

(b) Test may be inconclusive.

c. Extreme value problems (usually on an interval)

(1) Always write the equation first!

(2) Find Rel. Max. or Min. and also check the ends of the interval.

(3) Literals (Constants)

(a) If: Something is given that doesn’t change with .

(b) Then: Treat it as a constant when taking derivative.

3. Rates of Change

a. Average and Instantaneous Rates of Change. (on )

(1) Average =

(2) Instantaneous =

b. Velocity and Acceleration in linear motion.

(1) = position.

(2) = velocity.

(3) = acceleration.

c. Procedure for Related Rates.

(1) Draw Figure.

(2) WE KNOW and WE WANT at (as derivatives).

(3) Write an Equation.

(4) DO NOT substitute in for variables that change with time.

(5) Take differential of both sides and divide by .

(6) Substitute in given data.

(7) Solve for what you want.

V. Integral Calculus

A. The Indefinite Integral (The Anti-derivative)

1. Techniques of Integration

a. 20 Basic Integration Formulas

1.		11.
2.		12.
3.		13.
4.		14.
5.		15.
6.		16.
7.		17.
8.		18.
9.		19.
10.		20.

b. Divide and Conquer (Degree of Numerator ≥ Degree of Denominator)

Ex:

c. Break into Fractions (Separate into Several Integrals).

Ex:

d. Expand and Simplify.

Ex:

e. Complete the Square.

Ex:

f. Parts.

(1) Formula for Parts:

(2) let u = LIATE

g. Trig Substitution.

(1) sin/cos

(a) At least one odd power: Break off one and use identity.

(b) All even powers: use half angle identities:

(2) Real Trig Substitution.

Expression

Substitution

Identity

h. Partial Fractions.

(1) Case 1: Linear, non-repeating factors.

Ex:

(2) Case 2: Linear, repeating factors.

Ex:

(3) Case 3: Quadratic, non-repeating factors.

Ex:

(4) Case 4: Quadratic, repeating factors.

Ex:

2. Applications of anti-derivatives.

a. Initial Conditions: Substitute them in and solve for .

b. Distance, Velocity, and Acceleration.

(1) = position.

(2) = velocity.

(3) = acceleration.

c. Separable differential equations:

Isolate vars. and take anti-derivative of both sides.

d. Growth and Decay:

B. The Definite Integral

1. Definition of Definite Integral:

2. Properties:

If on ,

Then:

3. Approximations to the definite Integral:

Rectangles

Trapezoids

Parabolas (Simpson)

4. Fundamental Theorems:

5. Improper integrals (limits of Definite Integrals)

Unbounded Integrals

Bounded Integrals

6. Applications of the Definite Integral on :

a. Area under and btw. curves. Be careful of continuity and crossing.

Area Under a Curve

Area Btw. two Curves

Area of Polar Curves

Area Btw. Polars

b. Volumes of Revolution (Disk Method)

Volume w.r.t x-axis

Volume w.r.t y-axis

Volume Geometrically

c. Volumes of Revolution (Shell Method)

Volume w.r.t x-axis

Volume w.r.t y-axis

Volume Geometrically

d. Arc Length

Arc Length w.r.t x-axis

Arc Length w.r.t y-axis

Arc Length-Parametric

e. Surface Areas of Rotation (= The Radius)

Surface Area w.r.t x-Axis

Surface Area w.r.t y-Axis

Surface Area-Parametric

f. Average Value of a Function

Average Value (y) of a Curve

Average Value (x) of a Curves

g. Volumes By Slicing:

Volumes By Slicing w.r.t y-axis

Volumes By Slicing w.r.t x-axis

h. Distance

Relative Distance

Total Distance

i. Work and Hooke’s Law

Work

Work (Generally)

Hooke’s Law

VI. Sequences and Series

A. Sequences of Real Numbers; Convergence

1. Take limit as , if it exists, sequence converges.

2. Fist finitely many terms can be deleted without affecting convergence.

3. If , then

4. Use L’Hopital’s Rule

B. Series of Real Numbers Convergence

1. Write Some Terms.

2. If terms tend to zero, series diverges.

3. Is it infinite geometric or p-series? (Use partial fractions)

a. Infinite Geometric: if

b. P-Series: if p>1.

4. Are all terms positive?

a. Comparison Test:

I. If converges and , converges.

II. If diverges and , diverges.

b. Integral Test: converges if converges.

c. Ratio Test:

i. Converges if <1

ii. Diverges if >1

iii. Either if =1

d. Root Test:

i. Converges if <1

ii. Diverges if >1

iii. Either if =1

5. If absolute value of all terms converges, converges absolutely.

6. Alternating? Leibniz’s Theorem for conditionally:

a. is all positive

c. ( decreases)

d. is off by no more than .

C. Power Series

1. Definition: ( constant.)

2. Interval of convergence: where makes the series converge.

3. Radius of convergence: distance from center of interval to side.

4. You can do anything to a series by doing it to all the terms. (Integration)

5. Taylor series. (Brooks Taylor-The blues guitarist.)

a. Definition:

b. Properties.

i. Pick an a close to the . (It’s just faster that way.)

ii. "About " means that .

c. Maclaurin Series. (). Here are the main ones:

d. Error approximation: for on