Stuff to memorize for the AP Calculus Test

Y=f(x) must be continuous at each:

Y=f(x) must be continuous at each:

-Critical Point or undefined and endpoints

Local Minimum

Goes (-,0,+) or (-, und, +)

Local Maximum

Goes (+,0,-) or (+, und, -)

Point of Inflection

• Concavity Changes

• (+,0,-) or (-,0,+)

• (+,und,-) or (-,und,+)

D/dx(sinx)

Cosx

D/dx(cosx)

-sinx

D/dx(tanx)

Sec²x

D/dx(cotx)

-csc²x

D/dx(secx)

Secxtanx

D/dx(cscx)

-cscxcotx

D/dx(lnx)

1/x

D/dx(ln(n))

1/n

D/dx(eⁿ)

Eⁿ

∫Cosx

Sinx

∫-sinx

Cosx

∫Sec²x

Tanx

∫-csc²x

Cotx

∫Secxtanx

Secx

∫-cscxcotx

Cscx

∫1/n

Ln(n)

∫Eⁿ

Eⁿ

When doing integrals never forget

Constant (+c)

∫Axⁿ

A/n+1(xⁿ⁺¹)+C

∫Tanx

• Ln|secx|+c

• Ln|cosx|+c

∫Secx

Ln|secx+tanx|+c

D/dx(sin⁻¹u)

1/√1-u²

D/dx(cos⁻¹x)

-1/√1-x²

D/dx(tan⁻¹x)

1/1+x²

D/dx(cot⁻¹x)

-1/1+x²

With derivative inverses

You plug in the number of the trigonometric function into x

D/dx(sec⁻¹x)

1/|x|√x²-1

D/dx(csc⁻¹x)

-1/|x|√x²-1

D/dx(aⁿ)

Aⁿln(a)

D/dx(Logₙx)

1/xln(a)

Chain Rule

• Take derivative of outside of parenthesis

• Take derivative of inside parenthesis and keep the original of what was in the parenthesis

• For example, sin(x²+1)→ 2xcos(x²+1)

Product Rule

d/dx first times second + first times d/dx second

Quotient Rule

LoDHi-HiDLo/LoLo

Fundamental Theorem of Calculus

• ∫(a to b) f(x) dx = F(b) - F(a)

• Basically saying that F’(x)=f(x)

f relative max→f ‘ goes from

Positive to negative

f relative min→f ‘ goes from

Negative to positive

Intermediate Value Theorem

If I pick an X value that is included on a continuous function, I will get a Y value, within a certain range, to go with it.

Mean Value Theorem

If function is continuous on [a,b] and first derivative exist on interval (a,b), then there is at least one number (c) such that f’(c)=f(b)-f(a)/b-a

Rolle’s Theorem with Mean Value Theorem

If function is continuous on [a,b] and first derivative exist on interval (a,b), and f(a)=f(b) then there is at least one number x=c such that f’(c)=0

Trapezoidal Rule

The average of left hand point method and right hand point method

Theorem of mean value and average value

1/b-a ∫f(x)dx which is the average value

Disk Method

V=π∫[R(x)]²dx

Washer Method

π∫(R(x))²-(r(x))²dx

General volume equation

V=∫area(x)dx

-If no shape is given, then you just take the integral of the function

Distance, Velocity, and Acceleration

Velocity is d/dx of position

Acceleration is d/dx of velocity

Displacement

∫(Velocity)dt

Distance

∫|Velocity|dt

Speed

The absolute value of velocity

Average Velocity

Final Position-Initial Position/Total time

cos(0)

1

sin(0)

0

Pie circle starts with

π/6

tan(π/3)

√3

tan(π/6)

√3/3

tan(90⁰)

Undefined

Double Argument

sin2x=2sinxcosx

cos2x=cos²x-sin²x=1-2sin²x

cos²x=1/2(1+cos2x)

sin²x=1/2(1-cos2x)

Pythagorean Identity

sin²x+cos²x=1

L’Hopital Rule

If limit equals 0/0 or ∞/∞, then you can take the derivative