Calculus Flashcards

(a+b)²

a²+2ab+b²

a²-b²

(a+b)(a-b)

sqrt(2)

1.4

sqrt(3)

1.7

π

3.14

e

2.718

ln 1

0

ln e

1

#/0

undefined

0/0

indeterminate form

0/#

0

2³

8

3³

27

4³

64

5³

125

2⁴

16

3⁴

81

2⁵

32

rewrite 1/x^a

x^-a

rewrite b)rt(a)

x^a/b

Area of a trapezoid

1/2(b₁+b₂)h

sin(pi/3)

sqrt(3)/2

cos(pi/2)

0

sin(pi/2)

1

limf(x)_x→a exists

limit from the left equals limit from the right

limf(x)x→ -∞ and ∞=L

Horizontal asymptotes

limf(x)_x→c

Plug in C, if 0/0 simplify and plug in again.

limf(x)_x→=∞or-∞

Vertical Asymptotes

cos(0)

1

sin(0)

0

cos(pi/2)

0

sin(pi/2)

1

cos(pi)

-1

sin(pi)

0

csc

1/sin

sec

1/cos

tan

sin/cos

cot

cos/sin

cos(3pi/2)

0

sin(3pi/2)

-1

cos(2pi)

1

sin(2pi)

0

Average Rate of Change

f(b)-f(a)/b-a

Instantaneous Rate of change

Derivative

Condition for IVT

f(x) is continuous on the interval [a,b]

Definition of continuity

lim x→a exists, f(a) exists, x→a=f(a)

Analysis for IVT

There is a c, a<c<b such that f(a)<f(C)<f(b) or f(b)<f(C)<f(a)

Derivative fails to exist

sharp turn or cusp, discontinuity, tangent line

Critical Numbers

f’(a)=0 or undefined

Vertical tangent line

Derivative is undefined

Horizontal tangent line

f’(a)=0

f is increasing

f’>0

f is decreasing

f’<0

f is concave up 

f’’>0 , f’ is increasing

f is concave down

f’’<0, f’ is decreasing

f has relative maximum

f’ = 0 or undef. changes from + to -

f has relative minimum

f’=0 or undef. changes from - to +

f has an inflection point

f’’=0 and changes signs. f’ has a relative min or max

limit definition of the derivative (h->>0 vers)

lim x→0(f(x+h)-f(x)/h)

limit definition of the derivative (x→a version)

lim x→a(f(x)-f(a)/x-a)

formula for the equation of a tangent line

y=f(a)+f’(a)(x-a)

d/dx c

0

d/dx (xⁿ⁾

n\cdot x^{n-1}

d/dx (f x g)

f’g+g’f

d/dx (f/g)

f’g-g’f/g²

d/dx e^x

e^x

d/dx ln x

1/x

d/dx cosx

-sinx

d/dx tan x

sec²x

d/dx cot x

-csc²x

d/dx sec x

sec x tan x

d/dx csc x 

-csc x cot x

integral of the prime

f(b)-f(a) just subtract b and a 

average value

1/b-a int f(x)dx