ULTIMATE CALCULUS AB SAVE + STUDY
THE ONE-PAGE AP EXAM FORMULA + CONCEPT GUIDE
01 LIMITS + CONTINUITY 04 THEOREMS + MOTION 07 AREA + VOLUME
lim x→a f(x) = L: left = right = L Rolle: continuous [a,b], differentiable (a,b), Area = ∫ₐᵇ (top−bottom) dx
Continuous at a: f(a) exists, limit exists, and f(a)=f(b) ⇒ some f′(c)=0. Area = ∫ (right−left) dy
limit = f(a). MVT: f′(c)=[f(b)−f(a)]/(b−a) Disk: V = π∫ₐᵇ [R(x)]² dx
0/0? Factor, rationalize, or simplify. MVT needs continuity on [a,b] and Washer: V = π∫ₐᵇ (R²−r²) dx
lim x→0 sin(x)/x = 1 differentiability on (a,b). Known cross sections: V = ∫ A(x) dx
lim x→0 (1−cos x)/x = 0 v(t)=s′(t); a(t)=v′(t)=s″(t) Radius = distance to axis of rotation. Sketch
IVT: continuous on [a,b] ⇒ every value Speed = |v|. Speed increases if v and a before integrating.
between f(a) and f(b) is attained. have same sign.
Displacement = ∫ v(t) dt
Total distance = ∫ |v(t)| dt
08 DIFFERENTIAL EQUATIONS
02 DERIVATIVE RULES
dy/dx = ky ⇒ y = Ceᵏˣ
f′(x) = lim h→0 [f(x+h)−f(x)]/h 05 ANTIDERIVATIVES dy/dx = f(x)g(y): separate variables
d/dx [xⁿ] = nxⁿ⁻¹ Integrate both sides, include C, then apply
(cf ± g)′ = cf′ ± g′ ∫ xⁿ dx = xⁿ⁺¹/(n+1)+C, n≠−1 initial condition.
(fg)′ = f′g + fg′ ∫ 1/x dx = ln|x|+C Euler: yₙ₊₁ = yₙ + f(xₙ,yₙ)Δx
(f/g)′ = (gf′−fg′)/g² ∫ eˣ dx = eˣ+C Slope fields: short line segments show dy/dx
Chain: [f(g(x))]′ = f′(g(x))g′(x) ∫ aˣ dx = aˣ/ln(a)+C at each point.
(sin x)′=cos x; (cos x)′=−sin x ∫ sin x dx = −cos x+C Logistic growth is not required in AP Calc
(tan x)′=sec²x ∫ cos x dx = sin x+C AB.
(eˣ)′=eˣ; (ln|x|)′=1/x ∫ sec²x dx = tan x+C
(aˣ)′=aˣ ln a ∫ 1/(1+x²) dx = arctan x+C
(arcsin x)′=1/√(1−x²) ∫ 1/√(1−x²) dx = arcsin x+C
09 NUMERICAL INTEGRATION
(arctan x)′=1/(1+x²) u-sub: choose u = inside function; replace
du = u′ dx. Δx = (b−a)/n
Left: Lₙ = Δx Σ f(xᵢ), i=0…n−1
Right: Rₙ = Δx Σ f(xᵢ), i=1…n
03 DERIVATIVE APPLICATIONS Trapezoid: Tₙ = (Lₙ+Rₙ)/2
06 DEFINITE INTEGRALS + FTC
Tangent line: y−f(a)=f′(a)(x−a) Increasing f: left under, right over.
∫ₐᵇ f(x)dx = F(b)−F(a), F′=f Decreasing f: left over, right under.
Increasing: f′>0. Decreasing: f′<0.
d/dx ∫ₐˣ f(t)dt = f(x) Concave up: trapezoid over; concave down:
Concave up: f″>0. Concave down: f″<0.
d/dx ∫ₐᵍ⁽ˣ⁾ f(t)dt = f(g(x))g′(x) under.
Critical numbers: f′=0 or undefined (f
defined). ∫ₐᵇ f = −∫ᵦᵃ f; ∫ₐᵃ f = 0
Local max: f′ changes +→−; min: −→+. ∫ₐᶜ f + ∫ ᵇ f = ∫ₐᵇ f
Inflection: concavity changes. Average value = [1/(b−a)]∫ₐᵇ f(x)dx 10 EXAM-DAY CHECKLIST
Absolute extrema on [a,b]: check critical Signed area: below x-axis counts negative.
points AND endpoints. ✓ Write units and interpret answers in
context.
✓ Justify with derivative signs or theorem
12 INTEGRAL APPLICATIONS conditions.
11 RELATED RATES + LINEARIZATION ✓ For rates: distinguish value, slope, and
Net change = ∫ₐᵇ rate(t) dt
accumulation.
Related rates: differentiate w.r.t. Final amount = initial + net change ✓ For FRQs: show setup before calculator
time t. Accumulation units = (rate units) × (input results.
Draw a diagram, write an equation, units). ✓ Round only at the end; follow AP
differentiate implicitly, substitute values. For a graph of f′: f rises where f′>0; extrema directions.
Linearization: L(x)=f(a)+f′(a)(x−a) of f occur at sign changes of f′. ✓ Check domain, endpoints, and signs.
Differentials: dy=f′(x)dx
13 QUICK GRAPH CONNECTIONS
f′ graph above axis ⇒ f increasing.
f′ graph increasing ⇒ f concave up.
f″ changes sign ⇒ possible inflection of f.
Area under f′ from a to b = f(b)−f(a).
KNOW THE RULE • SET UP THE EQUATION • JUSTIFY THE RESULT
Original independent study resource | Not affiliated with or endorsed by College Board
THE ONE-PAGE AP EXAM FORMULA + CONCEPT GUIDE
01 LIMITS + CONTINUITY 04 THEOREMS + MOTION 07 AREA + VOLUME
lim x→a f(x) = L: left = right = L Rolle: continuous [a,b], differentiable (a,b), Area = ∫ₐᵇ (top−bottom) dx
Continuous at a: f(a) exists, limit exists, and f(a)=f(b) ⇒ some f′(c)=0. Area = ∫ (right−left) dy
limit = f(a). MVT: f′(c)=[f(b)−f(a)]/(b−a) Disk: V = π∫ₐᵇ [R(x)]² dx
0/0? Factor, rationalize, or simplify. MVT needs continuity on [a,b] and Washer: V = π∫ₐᵇ (R²−r²) dx
lim x→0 sin(x)/x = 1 differentiability on (a,b). Known cross sections: V = ∫ A(x) dx
lim x→0 (1−cos x)/x = 0 v(t)=s′(t); a(t)=v′(t)=s″(t) Radius = distance to axis of rotation. Sketch
IVT: continuous on [a,b] ⇒ every value Speed = |v|. Speed increases if v and a before integrating.
between f(a) and f(b) is attained. have same sign.
Displacement = ∫ v(t) dt
Total distance = ∫ |v(t)| dt
08 DIFFERENTIAL EQUATIONS
02 DERIVATIVE RULES
dy/dx = ky ⇒ y = Ceᵏˣ
f′(x) = lim h→0 [f(x+h)−f(x)]/h 05 ANTIDERIVATIVES dy/dx = f(x)g(y): separate variables
d/dx [xⁿ] = nxⁿ⁻¹ Integrate both sides, include C, then apply
(cf ± g)′ = cf′ ± g′ ∫ xⁿ dx = xⁿ⁺¹/(n+1)+C, n≠−1 initial condition.
(fg)′ = f′g + fg′ ∫ 1/x dx = ln|x|+C Euler: yₙ₊₁ = yₙ + f(xₙ,yₙ)Δx
(f/g)′ = (gf′−fg′)/g² ∫ eˣ dx = eˣ+C Slope fields: short line segments show dy/dx
Chain: [f(g(x))]′ = f′(g(x))g′(x) ∫ aˣ dx = aˣ/ln(a)+C at each point.
(sin x)′=cos x; (cos x)′=−sin x ∫ sin x dx = −cos x+C Logistic growth is not required in AP Calc
(tan x)′=sec²x ∫ cos x dx = sin x+C AB.
(eˣ)′=eˣ; (ln|x|)′=1/x ∫ sec²x dx = tan x+C
(aˣ)′=aˣ ln a ∫ 1/(1+x²) dx = arctan x+C
(arcsin x)′=1/√(1−x²) ∫ 1/√(1−x²) dx = arcsin x+C
09 NUMERICAL INTEGRATION
(arctan x)′=1/(1+x²) u-sub: choose u = inside function; replace
du = u′ dx. Δx = (b−a)/n
Left: Lₙ = Δx Σ f(xᵢ), i=0…n−1
Right: Rₙ = Δx Σ f(xᵢ), i=1…n
03 DERIVATIVE APPLICATIONS Trapezoid: Tₙ = (Lₙ+Rₙ)/2
06 DEFINITE INTEGRALS + FTC
Tangent line: y−f(a)=f′(a)(x−a) Increasing f: left under, right over.
∫ₐᵇ f(x)dx = F(b)−F(a), F′=f Decreasing f: left over, right under.
Increasing: f′>0. Decreasing: f′<0.
d/dx ∫ₐˣ f(t)dt = f(x) Concave up: trapezoid over; concave down:
Concave up: f″>0. Concave down: f″<0.
d/dx ∫ₐᵍ⁽ˣ⁾ f(t)dt = f(g(x))g′(x) under.
Critical numbers: f′=0 or undefined (f
defined). ∫ₐᵇ f = −∫ᵦᵃ f; ∫ₐᵃ f = 0
Local max: f′ changes +→−; min: −→+. ∫ₐᶜ f + ∫ ᵇ f = ∫ₐᵇ f
Inflection: concavity changes. Average value = [1/(b−a)]∫ₐᵇ f(x)dx 10 EXAM-DAY CHECKLIST
Absolute extrema on [a,b]: check critical Signed area: below x-axis counts negative.
points AND endpoints. ✓ Write units and interpret answers in
context.
✓ Justify with derivative signs or theorem
12 INTEGRAL APPLICATIONS conditions.
11 RELATED RATES + LINEARIZATION ✓ For rates: distinguish value, slope, and
Net change = ∫ₐᵇ rate(t) dt
accumulation.
Related rates: differentiate w.r.t. Final amount = initial + net change ✓ For FRQs: show setup before calculator
time t. Accumulation units = (rate units) × (input results.
Draw a diagram, write an equation, units). ✓ Round only at the end; follow AP
differentiate implicitly, substitute values. For a graph of f′: f rises where f′>0; extrema directions.
Linearization: L(x)=f(a)+f′(a)(x−a) of f occur at sign changes of f′. ✓ Check domain, endpoints, and signs.
Differentials: dy=f′(x)dx
13 QUICK GRAPH CONNECTIONS
f′ graph above axis ⇒ f increasing.
f′ graph increasing ⇒ f concave up.
f″ changes sign ⇒ possible inflection of f.
Area under f′ from a to b = f(b)−f(a).
KNOW THE RULE • SET UP THE EQUATION • JUSTIFY THE RESULT
Original independent study resource | Not affiliated with or endorsed by College Board
