AP Calculus BC SUMNOCO Limits Curve Sketching and Analysis More Derivatives

Limits
Notation for:
Limit from the left of 𝑓(𝑥) as 𝑥 → 𝑐
AP Calculus BC SUMNOCO
Curve Sketching and Analysis
Critical Points:
Global Min:
Limit from the right of 𝑓(𝑥) as 𝑥 → 𝑐
More Derivatives
Where 𝑢 is a function of 𝑥
and a is a constant
𝑑
(𝑥 𝑛 ) =
𝑑𝑥
𝑑
𝑑𝑥
Global Max:
𝑑
𝑑𝑥
𝑑
Definition of Continuity
A function is continuous at the point
𝑥 = 𝑐 if and only if:
Point of Inflection:
1.
Derivatives
𝑑𝑥
𝑑
𝑑𝑥
𝑑
𝑑𝑥
Definition of Derivative
d
 f ( x)  
dx
2.
3.
𝑑
𝑑𝑥
𝑑
Situations in which limits fail to
exist:
𝑑𝑥
𝑑
Alternate Form of Def. of Derivative
d
 f ( x)  at x  c
dx
𝑑𝑥
𝑑
𝑑𝑥
𝑑
Situations in which derivatives fail
to exist:
𝑑𝑥
Chain Rule
d
[ f (u )] 
dx
𝑑
𝑑𝑥
𝑑
𝑑𝑥
Point-Slope Form:
ln 1 =
Product Rule
d
(uv) 
dx
𝑑
𝑑𝑥
ln 𝑒 =
Intermediate Value Theorem
𝑑
𝑑𝑥
Quotient Rule
d u
 
dx  v 
Where 𝑢 and 𝑣 are functions of 𝑥
Solution to 𝒅𝒚/𝒅𝒙 = 𝒌𝒚
(sin 𝑢) =
(cos 𝑢) =
(tan 𝑢) =
(cot 𝑢) =
(sec 𝑢) =
(csc 𝑢) =
(ln 𝑢) =
(𝑒 𝑢 ) =
(𝑢−1 ) =
(sin−1 𝑢) =
(cos −1 𝑢) =
(tan−1 𝑢) =
(𝑎 𝑢 ) =
(log 𝑎 𝑢) =
Extreme Value Theorem
The Mean Value Theorem
(derivatives)
Rolle’s Theorem
Distance, Velocity, and
Acceleration
𝑠(𝑡) is the position function,
 x(t ), y (t )  is the position in
parametric
velocity =
Parametric Equations
dy

dx
d2y

dx 2
acceleration =
Arc length =
velocity vector =
The Fundamental Theorem of
Calculus
acceleration vector =
Polar Curves
4 conversions
speed (rectangular and parametric) =
Area =
2nd FTC
d g ( x)
f (t )dt 
dx a
displacement =
Slope =
distance (rectangular and parametric)
=
Taylor Series
Area Under The Curve
(Trapezoids)
average velocity =
Maclaurin Series
l'Hôpital's Rule (Bernoulli’s Rule)
Sum of infinite geometric
Mean Value Theorem for Integrals
(Average Value)
ex 
Euler’s Method
Solids of Revolution and Friends
Disk Method
cos x 
sin x 
Washer Method
1

1 x
General volume equation
Integration by Parts
ln( x  1) 
Arc Length (rectangular)
Logistics
dP

dt
Series Tests/Error Bound