AP Calculus BC Formulas | Quizlet

L'Hopital's Rule

When is a function continuous?

Limit Definition of a Derivative

Intermediate Value Theorem (IVT)

If f(x) is continuous on [a, b], then all y-values between f(a) & f(b) exist at some point of c

Mean Value Theorem (MVT)

If f(x) is differentiable, a slope of the ARC exists at some point of c

Extreme Value Theorem (EVT)

If f(x) is continuous on a closed interval of [a, b], f(x) must have a maximum and minimum

1st Fundamental Theorem of Calculus

2nd Fundamental Theorem of Calculus

Product Rule

Quotient Rule

Chain Rule

Power Rule

dy/dx sin(x)

= cosx

dy/dx cos(x)

= -sin(x)

dy/dx tan(x)

dy/dx cot(x)

dy/dx sec(x)

= sec(x)tan(x)

dy/dx csc(x)

= csc(x)cot(x)

dy/dx arcsin(x)

dy/dx arctan(x)

Critical Value

If f'(a)=0 or is undefined, a is a critical value

First Derivative Test

When the value x=a is a critical value:

1) If f'(x) switches from (+)→(-), f(c) is a relative maximum

2) If f'(x) switches from (-)→(+), f(c) is a relative minimum

When does the Second Derivative Test indicate f(x) is a relative minimum?

f'(x)=0 and f''(x)>0

When does the Second Derivative Test indicate f(x) is a relative maximum?

f'(x)=0 and f''(x)<0

What indicates that f(x) is concave up?

f'(x) is decreasing or f''(x) > 0

What indicates that f(x) is concave down?

f'(x) is increasing or f''(x) < 0

Points of Inflection

occur when:

1) if f''(a)=0/dne AND switches signs

2) if f''(a)=0/dne AND f'(a) changes from increasing to decreasing or decreasing to increasing

Reimann Sums

for a

Trapazoidal rule

for a

Average Value

Euler's Method

new-y = old-y + (dy/dx)(new-x - old-x)

Integration by Parts

Exponential Growth & Decay

y=Ce^kt

Logistic Differential Equations

Logistic Growth Fomula

P = M/(1+Ae^-kt)

What is the formula for the area under a curve in terms of x?

A = ∫[a, b] (upper - lower)^2 dx

What is the formula for the area under a curve in terms of y?

A = ∫[a, b] (right - left)^2 dy

What is the formula for the volume using the Disk Method with a horizontal axis of rotation?

V = π ∫[a, b] (upper - lower)^2 dx

What is the formula for the volume using the Disk Method with a vertical axis of rotation?

V = π ∫[a, b] (right - left)^2 dy

What is the formula for the Washer Method when using a horizontal axis of rotation?

V = π ∫[a, b] (upper)^2 - (lower)^2 dx

What is the formula for the Washer Method when using a vertical axis of rotation?

V = π ∫[a, b] (right)^2 - (left)^2 dy

Length of a Curve / Arc Length

Cross Section Perpendicular to the x-axis Formula

Cross Section Perpendicular to the y-axis Formula

Velocity

x'(t)

Speed

|v(t)| or |x'(t)|

Acceleration

v'(t) or x''(t)

Displacement (Change in Position)

∫[a, b] v(t) dt

Finding position with displacement

Position = Initial Position + Displacement

Total Distance Traveled

Parametric First Derivative

Parametric Second Derivative

Parametric Arc Length

Position Vector

< x(t), y(t) >

Velocity Vector

Acceleration Vector

Speed of a Parametric

Displacement of a Parametric

Distance Traveled (Arc Length) for a Parametric

x in polar terms

x = rcosθ

y in polar terms

y = rsinθ

Area inside a polar curve

Area between 2 polar curves

Taylor Polynomial Formula

Lagrange Error Bound Formula

Alternating Series Remainder Formula