AP Calculus AB/BC Comprehensive Vocabulary Flashcards

Vocabulary flashcards covering essential AP Calculus AB & BC concepts including limits, derivatives, integrals, motion, differential equations, area/volume, parametric, polar, vectors, and series.

Zero of a Function

A value of x that makes f(x)=0; found by factoring, quadratic formula, or graphing.

Intersection of Two Functions

A point where f(x)=g(x); solve algebraically or find on a graphing calculator.

Even Function

Function with y-axis symmetry; satisfies f(−x)=f(x).

Odd Function

Function with origin symmetry; satisfies f(−x)=−f(x).

Domain of a Polynomial

All real numbers (−∞,∞).

Domain Restriction—Denominator

Values that make denominator 0 are excluded from domain.

Domain Restriction—Square Root

Radicand must be ≥0 to remain in the real numbers.

Domain Restriction—Logarithm

Argument of log or ln must be >0.

Vertical Asymptote

x-value where denominator of simplified fraction equals 0; written x=a.

Horizontal Asymptote Rule 1

If numerator degree < denominator degree, y=0 is the asymptote.

Horizontal Asymptote Rule 2

If numerator degree = denominator degree, HA = ratio of leading coefficients.

Horizontal Asymptote Rule 3

If numerator degree > denominator degree, no horizontal asymptote.

Intermediate Value Theorem (IVT)

A continuous function on [a,b] takes every y-value between f(a) and f(b).

Existence of Root via IVT

If f(a) and f(b) have opposite signs, a zero exists in (a,b).

Limit—Direct Substitution

First step in evaluating a limit; plug in the approaching value.

Indeterminate Form 0/0

Expression requiring algebraic manipulation or L’Hospital’s Rule to evaluate limit.

Graphical Limit

The y-value a function approaches as x approaches a from left/right.

Limit Does Not Exist (DNE)

Occurs at jump discontinuity or vertical asymptote where one-sided limits disagree or diverge.

Continuity—Formal Definition

Function continuous at x=a if limit exists, f(a) exists, and both are equal.

L’Hospital’s Rule

Evaluates limits of 0/0 or ∞/∞ forms by taking derivatives of numerator and denominator.

L’Hospital’s Other Forms

Products, differences, or powers transformed to quotient before applying the rule.

Limit Definition of Derivative

f '(x)=limₕ→0 [f(x+h)−f(x)]/h.

Pointwise Derivative

f '(c)=limₓ→c [f(x)−f(c)]/(x−c).

Differentiable Function

Continuous and has equal left/right derivatives; graph is smooth without corners or vertical tangents.

Tangent Line Equation

y−f(c)=f '(c)(x−c); matches function value and slope at x=c.

Normal Line

Line perpendicular to tangent; slope = −1 / f '(c).

Secant Line

Line through two points on a curve; slope = [f(b)−f(a)]/(b−a).

Linear Approximation

Using tangent line to estimate function values near point of tangency.

Overestimate via Tangent

Occurs when function is concave down and tangent lies above curve.

Underestimate via Tangent

Occurs when function is concave up and tangent lies below curve.

Average Rate of Change

[f(b)−f(a)]/(b−a); slope of secant line on interval [a,b].

Approximating Derivative from Table

Use slope of two points close to desired x-value.

Euler’s Method Formula

y{new}=y{old}+h(dy/dx); numerical solution step.

Euler’s Method Over/Under

Compare signs of dy/dx and d²y/dx² to decide if approximation is high or low.

Critical Value

x where f '(x)=0 or undefined; potential extremum.

Increasing Function (Analytic)

f '(x)>0 on interval.

Decreasing Function (Analytic)

f '(x)<0 on interval.

Relative (Local) Extrema

Occurs at critical value with derivative sign change.

Absolute Extrema on Closed Interval

Largest/smallest y among critical points and endpoints.

Inflection Point

x where f ''(x)=0/undefined AND concavity changes.

Concave Up

f ''(x)>0; tangent lines lie below curve; tangent underestimates.

Concave Down

f ''(x)<0; tangent lines lie above curve; tangent overestimates.

Riemann Sum

Finite sum of rectangle areas approximating an integral.

Left Riemann Sum

Height of each rectangle from function value at left endpoint.

Right Riemann Sum

Height from function value at right endpoint.

Midpoint Riemann Sum

Height from function value at midpoint of subinterval.

Inscribed Sum

Rectangles lie completely under curve; gives lower estimate.

Circumscribed Sum

Rectangles cover curve; gives upper estimate.

Fixed Rectangle Width

Δx=(b−a)/n for n equal subintervals.

Trapezoidal Rule

Approximation using trapezoids; coefficients 1,2,2,…,2,1 with Δx/2 factor.

Fundamental Theorem of Calculus I

∫_a^b f(x)dx = F(b)−F(a) where F' = f.

Fundamental Theorem of Calculus II

d/dx ∫_a^{g(x)} f(t)dt = f(g(x))·g'(x).

Average Value of Function

(1/(b−a))∫_a^b f(x)dx.

Mean Value Theorem for Integrals

There exists c in [a,b] where f(c) equals average value.

Improper Integral

Integral with infinite limit or discontinuous integrand; requires limit notation.

Partial Fraction Decomposition

Express rational function as sum of simpler fractions for integration.

Integration by Parts

∫u dv = u·v − ∫v du; choose u via LIPET hierarchy.

Arc Length Formula (Cartesian)

∫_a^b √[1+(dy/dx)²] dx for smooth curve y=f(x).

Position Function

Original displacement s(t) or x(t); units of length.

Velocity

First derivative of position; speed with direction; units length/time.

Speed

Absolute value of velocity; scalar magnitude.

Acceleration

Derivative of velocity or second derivative of position; units length/time².

Instantaneous Velocity

v(a)=s '(a); slope of tangent to position curve at t=a.

Average Velocity (Position Data)

[s(b)−s(a)]/(b−a).

Speeding Up

Velocity and acceleration have same sign on interval.

Slowing Down

Velocity and acceleration have opposite signs.

Displacement

∫_a^b v(t)dt; net change in position (signed).

Total Distance Traveled

∫_a^b |v(t)| dt; absolute path length.

Greatest Distance from Start

Max |s(t)−s(0)| at turning points and endpoints.

Slope Field

Graph of short line segments representing dy/dx at (x,y) points.

Separable Differential Equation

Can be written g(y)dy = f(x)dx and integrated both sides.

Exponential Growth/Decay Solution

y=Ce^{kt}; k>0 growth, k<0 decay.

Newton’s Law of Cooling

dT/dt = k(T−TR); solution T = TR + Ce^{kt}.

Logistic Growth Model

P= C / (1 + Ae^{−k t}); approaches carrying capacity C.

Disc Method

Volume of revolution: V=π∫_a^b [R(x)]² dx about x-axis.

Washer Method

Volume: V=π∫_a^b (R²−r²) dx; outer minus inner radius.

Shell Method (about y-axis)

V=2π∫_a^b x·f(x) dx; cylindrical shells.

Cross-Section Volume (x-axis)

V=∫_a^b A(x) dx where A(x) is area of cross section.

Area of Square Cross Section

A = (side)², where side comes from region width.

Area of Equilateral Triangle Cross Section

A = (√3/4)·b², where b is base length.

Area of Semi-circle Cross Section

A = (π/8)·d² where d is diameter (region width).

Parametric First Derivative

dy/dx = (dy/dt)/(dx/dt).

Parametric Second Derivative

d²y/dx² = [d/dt(dy/dx)] / (dx/dt).

Arc Length—Parametric

∫ √[(dx/dt)²+(dy/dt)²] dt over interval.

Polar to Cartesian x

x = r cos θ.

Polar to Cartesian y

y = r sin θ.

Polar Slope dy/dx

[r' sinθ + r cosθ] / [r' cosθ − r sinθ].

Polar Area Formula

(1/2)∫ r² dθ between angles α and β.

Polar Arc Length

∫ √[r² + (dr/dθ)²] dθ.

Vector Magnitude

‖⟨a,b⟩‖ = √(a² + b²).

Dot Product

⟨a,b⟩·⟨c,d⟩ = ac + bd; zero dot product implies perpendicular vectors.

Particle Stops (Vector)

When velocity vector = ⟨0,0⟩.

Vector Velocity

Derivative of position vector: v(t)=⟨x'(t), y'(t)⟩.

Vector Acceleration

Derivative of velocity vector: a(t)=⟨x''(t), y''(t)⟩.

Total Distance (Vector)

∫ ‖v(t)‖ dt over interval.

Geometric Series Sum

S = a₀/(1−r) when |r|<1 and index starts at n=0.

Taylor Polynomial

Finite sum Σ (f⁽ⁿ⁾(c)/n!)(x−c)ⁿ approximating function near x=c.

Maclaurin Polynomial

Taylor polynomial centered at c=0.

Lagrange Error Bound

|R_n(x)| ≤ (M|x−c|^{n+1})/( (n+1)! ); M is max of |f⁽ⁿ⁺¹⁾| on interval.

Mean Value Theorem (Derivatives)

If f continuous on [a,b] and differentiable (a,b), ∃c with f '(c)=[f(b)−f(a)]/(b−a).