Calculus Ab Formula Sheet2

Calculus AB Formula Sheet Definition of a derivative:

f '( x ) = lim

h→ 0

f (x + h) − f (x) f (x ) − f (c) h or l i m called the difference quotient x→ c x − c

When to take a derivative: When the problem asks for the instantaneous rate of change, the slope of the tangent line or the slope of a curve at a point: units: units of y over units of x. f '( x ) = d y / d x f ( n ) ( x ) represents the nth derivative

Note: average rate of change from a to b =

f (b ) − f (a ) b − a

, but use this to approximate the derivative from a table.

Derivatives of inverses:

Switch x and y and take the derivative implicitly or use d y / d x 1 / dd xy a t f ( a )

of f

−1

(x )a t (x = a ) =

Limits: - Indeterminate forms:

0 / 0 , ∞ / ∞ , ∞ − ∞ , 0 ⋅ ∞ , 0 0 ,1 ∞

, indeterminate forms require you to manipulate them

algebraically or use L’Hopital’s rule. (If they are not indeterminate, just plug in the numbers and you are good to go) __LaHopitals must be 0/0 or ∞ / ∞ - Infinite limits will get - a / 0 (where a can be any number except 0) DO a sign test on both sides to determine if positive or negative infinity. - Limits as

x → ∞

-Take highest powers , know that

e

x

> x

n

> ln ( x )

- Limits you need to know:

lim

sin x = 1 x

lim

lim +

x = 1 |x|

l i m−+

x→ 0

x→ 0

x→ ∞

x→ 0

sin x = 0 x x = −1 |x|

lim

x→ 0

x = d .n .e. |x|

-For a double sided limit to exist xl →i ma + f ( x ) = xl →i ma − f ( x )

Continuity: For a function to be continuous at x=a,

lim f ( x ) = f ( a ) x→ a

Types of discontinuities: jumps, holes, vertical asymptote

(the value of f(a) is irrelevant).

Derivative formulas you must know:

(DON’T forget the chain rule)

d dx

(k ) = 0

d dx

( f ( x ) ) n = n ( f ( x ) ) n −1 . f ' ( x )

d dx

( f ⋅ g ) = f '⋅ g + g '⋅ f f f 'g + g' f = g g2

d dx

d dx

sin ( f ( x )) = c o s( f ( x )) ⋅ f '( x )

d dx

c o s( f ( x )) = − sin ( f ( x )) ⋅ f '( x )

d dx

ta n ( f ( x )) = se c 2 ( f ( x )) ⋅ f '( x )

d dx

c o t( f ( x ) = − c sc 2 ( f ( x ) ) ⋅ f '( x )

d dx

se c ( f ( x )) = se c ( f ( x )) ta n ( f ( x )) ⋅ f '( x )

d dx

c sc ( f ( x )) = − c sc ( f ( x )) ⋅ c o t( f ( x )) ⋅ f '( x ) 1 ln ( f ( x )) = ⋅ f '( x ) f (x) 1 lo g b ( f ( x )) = ⋅ f '( x ) f ( x ) ln b

d dx

d dx

e

f (x)

= e

d dx

a

f (x)

= a

d dx

sin

−1

d dx

ta n

−1

( f ( x )) =

d dx

se c

−1

( f ( x )) =

d dx

f (x)

⋅ f '( x )

f (x)

( f ( x )) =

⋅ ln a ⋅ f '( x ) f '( x )

d dx

1 − f (x)2 f '( x ) 1 + f (x)2 f '( x ) | f ( x )|

d dx

c o s −1 ( f ( x ) ) = −

c o s −1 ( f ( x ) ) = −

f (x)2 − 1

d dx

c sc

−1

f '( x ) 1 − f (x)2

f '( x ) 1 + f (x)2

( f ( x )) = −

f '( x ) | f ( x )|

f (x)2 − 1

Chain Rule:

( f ( g ( x ) )' = df df du = ⋅ dx du dx

f '( g ( x )) ⋅ g '( x )

the derivative of the outside X thederivative of the inside

Smooth and differentiable: The derivative exists everywhere and is always the same from the left and the right of any point (piecewise functions) i.e. y= |

x|

is continuous everywhere but not differentiable at x=0.

Tangent lines: The equation of the tangent line at x=a of f(x) is

y − f (a ) = m (x − a )

(Local linear approx. , estimating functions ( Normal line: same as tangent except

10

where

m = f '(a )

.

) .. use tangent lines)

m = − 1 / f '(a )

Critical Value: is anywhere the derivative is undefined or zero. Maximums/ minimums First derivative test: Find critical points and do a sign test. If sign changes from + to – then it’s a max, if sign changes from – to + then it is a minimum. No sign change not a relative extremum

f ' ' is neg. - cc down and therefore a maximum, If f ' ' is positive- cc up and therefore a minimum If f ' ' is 0 – no conclusion can be drawn

Second derivative test: Find critical pts, plug them into the second derivative, it

Inflection points: where the rate of change is a maximum, where concavity changes sign..

f ''> 0 f ''< 0

then f(x) is cc up then f(x) is cc down

Increasing/Decreasing

f '( x ) > 0 f '( x ) < 0

increasing decreasing

Absolute Min/Max: Make a table of critical values in an interval, include the end points as possibilities, watch out for vertical asymptotes. (A function that goes up to infinity has no abs. max, or if it goes down to negative infinity it has not absolute min) Error:

d y = f '( x )d x

Related Rates: Differentiate implicitly with respect to t. Riemann Sums: LHS, RHS, MID, TRAPEZOIDS, INSCRIBED rectangles , and CIRCUMSCRIBED rectangles 1

Area of a trapezoid: 2

(b1 + b 2 ) ⋅ h

Estimate compared to actual area: IF f(x) is increasing then LHS under estimate, RHS over estimate If f(x) is decreasing then LHS over RHS under IF f(x) is cc up cc down

Trapezoid – over Trapezoid –under

midpt. --under midpt. – over

Definition of an integral: b

∫

n

f ( x ) d x = lim

n→ ∞

a

∑

f (a + ∆ x ) ⋅ ∆ x

where

k =1

∆x =

b − a n

A definite integral represents the net change in a function from a to b, also the signed area under a curve, units are units of f(x) times units of x. Integration formulas you need to know:

∫(f ∫ ∫

± g )d x =

∫

fdx ±

∫ gdx

x n +1 x dx = n + 1 1 d x = ln | x | x n

∫ c o s x d x = sin x ∫ sin x d x = − c o s x ∫ ta n x d x = ln |se c x |o r − ln |c o s x | ∫ se c x d x = ta n x ∫ se c x d x = ln |se c x + ta n x | ∫ e dx = e ∫ a dx = 2

ax

1 a

x

a x ln a

1

∫

a

∫

x

∫

x

2

− x 1

d x = sin

=1 x a

dx =

1 a

se c

ta n

=1 x a

x − a 1 dx = + a 2 2

2

2

ax

2

1 a

=1 x a

Fundamental theorem of Calculus: Part 1: If f(x) is continuous on [a,b] and

d F = f dx

b

then

∫

f ( x )d x = F (b ) − F (a )

a

x

d Part II: : If f(x) is continuous on [a,b] then d x

∫

f (t)d t = f (x )

a

d dx

h(x)

∫

g (x)

f (t)d t = f (h ( x )) ⋅ h '( x ) − f ( g ( x )) ⋅ g '( x )

or alternatively

Properties of definite integrals: a

∫

∫

f (x )d x = 0

a

a

b

f (x )dx = −

a

f (x )d x

b

a

a

∫

E v e n fu n c tio n

∫

f (x )d x = 2

−a

∫

f (x )d x

0

a

O d d fu n c tio n

∫

f (x )d x = 0

−a

Areas between two curves

∫ ∫

f (x ) − g (x )d x

f is on top and g is on bottom

f ( y )d y − g ( y )d y

f is further to the right.

Volumes:

πr Around x axis:

2

∫π

f (x)2

∫π

f (x)2dx − π g (x)2 dx

disks (solid)

washers….

2πrh

∫ 2π yf ( y )d y

shells

About other horizontal lines, find r in terms of x for disks, washers, for shells find r and h in terms of y

Around y axis:

∫π

f ( y)2dy

∫π

f ( y)2dx − π g ( y)2 dy

disks (solid)

washers….

2πrh

∫ 2π xf (x )dx

shells

About other vertical lines, find r in terms of y for disks, washers, for shells find r and h in terms of x

∫

b a

c

fd x =

∫ a

b

fd x +

∫ c

fd x

Differential equations: Salt problems: dy/dt = rate in – rate out If Rate of change is proportional to the amount there: dy/dy=ky … solution

y = ae

kt

y '= ky ( L − y ) Logistical growth

y =

L (1 + b e − k t )

Theorems you may have forgotten: INTERMEDIATE VALUE THEOREM: IF f is continuous on a closed interval [a,b] then f must take on every value inbetween f(a) and f(b). (useful if f(a)is positive and f(b) is negative, then a zero must occur between a and b) MEAN VALUE THEOREM FOR DERIVATIVES: If f is continuous on [a,b] and differentiable on (a,b) then there exists at least one number c in (a,b) such that

f '(c ) = values outside of that range.

f (b ) − f (a ) b − a

; be able to do this from a graph, REMEMBER c has to lie between a and b.. don’t use

ROLLE’S THEOREM: (Just a special case of MVT): If f is continuous on [a,b] and differentiable on (a,b) and f(a)=f(b), then there exist at lest one c between a and b such that f’(c)=0 EXTREME VALUE THEOREM: If f is continuous on [a,b], then f(x) has both a maximum and a minimum on [a,b]

Average Value of a function: The average of a function over [a,b] is

f ave

1 = b − a

b

∫

f (x )d x

a

Position, Velocity, Acceleration : s(t) v(t) a(t) speed = |velocity|

v (t) = s'(t) a (t) = s''(t) = v '(t) b

Total distance traveled from a to b =

∫

v (t)d t

a

Verses: b

Displacement = change in position from a to b = =

∫ v ( t )d t a

Signs: Position

negative: to the left of start Positive to the right of start

Velocity: negative moving to left positive moving to right

If velocity and acceleration have opposite signs then the object is slowing down It they have the same signs it is speeding up

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