CALC - Memorize for the AP Calculus Test

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 × derivative of x

D/dx(eⁿ)

Eⁿ derivative of n

∫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⁻¹x)

1/√1-x²

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