AP Calculus BC Formulas and Theorems

A comprehensive set of all formulas and theorems (plus extra polar review) for the AP Calculus BC Exam.

Squeeze (Sandwich) Theorem

DNE

∞

a

Criteria for continuity

Types of discontinuities

Removable - a hole

Jump - abrupt change to another value

Infinite - an asymptote

Oscillating - function oscillates indefinitely approaching a point

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Definition of a derivative at c

Definition of a general derivative

Product rule

Quotient rule

6 trig function derivatives

Chain rule

Types of non-differentiable functions

Cusp - derivative immediately reverses sign on opposite sides of point

Vertical tangent - slope is “infinity”

Corner - immediate change in slope from a piecewise function

Critical point

1st derivative test

If c is a critical point and f’(c) changes sign at x=c, then c is a local extrema

Inflection point

If f(x) is continuous at x=c and the concavity changes at x=c, then f(x) has an inflection point at x=c.

Extreme Value Theorem (EVT)

L'Hôpital's rule

Intermediate Value Theorem (IVT)

Rolle’s Theorem

Mean Value Theorem (MVT)

Newton’s Method

6 accidental integrals

Riemann sum

Definite integral definition

Fundamental Theorem of Calculus

Average value of a function

Reciprocal and natural log integrals

6 trig function integrals

Exponential / log derivative and integral

Inverse trig derivatives

(work in reverse for integrals)

Euler’s method

Areaa between two curves

Disc method

No gap, perpendicular rectangle

Washer method

Gap, perpendicular rectangle

Shell method

May or may not have gap, parallel rectangle

2nd Fundamental Theorem of Calculus

2nd FTC for an argument other than x

Length of a curve

Lateral (surface) area

Squared trig integrals

Integration by parts

use DETAIL backwards ((D), exponential, trig, algebraic, inverse, natural log(ln))

Partial fractions

Logistic growth

Improper integral evaluation

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1

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nth term test

Integral test

Direct comparison test

Limit comparison test

Absolute convergence theorem

Alternating series test

Absolute convergence definition

Conditional convergence definition

A series that converges but does not converge absolutely

Alternating p-series convergence

Alternating series estimation theorem

Ratio test

(nth) Root test

Power series definition

Solving for interval of convergence

1. Perform ratio test

2. Set x term less than 1 for absolute convergence, greater than one for divergence

3. Plug in endpoints to original sum

Maclaurin series

Taylor polynomial

Any partial sum of a Maclaurin series

Taylor series

Graphical estimation of the range of values of x for which a Taylor polynomial estimates a function within a certain value.

Lagrange form of the remainder (general formula)

Lagrange form of the remainder (error formula)

Parametric arc length

Parametric surface area

Polar arc length

Polar surface area

Polar area

Position vector

Velocity vector

Speed

Acceleration vector

Net change in position

Total distance traveled

Polar circle graph and equation

a is the diameter of the circle

Polar limaçon graph and equation

Furthest point from origin is a+b

Origin to perpendicular axis intercept is a

If a<b: length of inner loop is b-a

If a>b: distance from origin to dimple is a-b

Rose curve graph and equation

a is the length of each petal

there are n petals if n is odd, otherwise 2n petals if n is even

Lemniscate graph and equation

a is the distance from the origin to the furthest point on the loop

Spiral graph and equation

Any point at a certain theta value is exactly a * theta away from the origin