AP Calculus BC Cheat Sheet ©Dr. Ricky Ng @ Meritus Academy Preliminaries: All elementary functions and operations, limit and continuity from graph.
Limit (Concept and Evaluation) Graphing (Refine Graphs with Calculus) Mean Value Theorem for Integral
* Be familiar with concepts and graphs. 1st Derivative & 1st Derivative Test If f is continuous on [a, b], then there exists c ∈ [a, b] s.t.
Squeeze / Sandwitch Theorem Inc/Dec: f ′ ≥ 0 ⇒ f increasing ↗; f ′ ≤ 0 ⇒ f decreasing ↘ Z b
def 1
Suppose g(x) ≤ f (x) ≤ h(x) for all x near c, x ̸= c, Critical Point: f cts. at c with f ′ (c) = 0 or f ′ (c) DNE. f (c) = favg = f (x) dx. (11)
b−a a
lim g(x) = lim h(x) = L ⇒ lim f (x) = L (1) 1st Der. Test: f : ↘↗ at c; l. min. f : ↗↘ at c; l. max.
x→c x→c x→c favg is called the average value of f on [a, b].
2nd Derivative & 2nd Derivative Test
L’Hôpital’s Rule
f ′′ tells if f is above or below tangent lines.
f (x) 0 ∞ f (x) f ′ (x) Integration Techniques (u-Sub and More)
lim = or ⇒ lim = lim ′ , (2) Concavity: f ′′ > 0, concave up; f ′′ < 0, concave down.
x→c g(x) 0 ∞ x→c g(x) x→c g (x)
Integration By Parts (IBP)
if latter exists. For other indeterminate forms, rewrite as Inflection Point (POI): where f ′′ changes sign. Z Z
quotient or apply ln first. 2nd Der. Test: For f ′ (c) = 0, f ′′ (c) = + or −, l. min or l. u dv = uv − v du (12)
max.
* Check endpoints for abs. max/min in optimization Get u from ILATE if possible. Tabular method if u is ATE.
Derivative (Slope of Tangent, Rate of Change)
problem. Partial Fraction Decomposition (PFD)
def f (x + h) − f (x) dy ∆y
f ′ (x) = lim , ≈ , ∆x small (3)
h→0 h dx ∆x * Perform polynomial division if necessary.
* Rewrite for linear approximation: xnew = x + ∆x, Riemann Integral (Net Area, Accumulation)
Basic Rule: For deg(h) < deg(f g),
Given continuous f on [a, b],
f (xnew ) ≈ f (x) + f ′ (x)∆x (4) Z b n h(x) A(x) B(x)
def X = + , (13)
Derivative Formulas (Know The Rules) f (x)dx = lim Ai , (7) f (x)g(x) f (x) g(x)
a n→∞
i=1
[sin x]′ = cos x [xn ]′ = nxn−1 where deg(A) = deg(f ) − 1 and deg(B) = deg(g) − 1.
Ai is area of region on each partition of [a, b]. Region could
[cos x]′ = − sin x [ex ]′ = ex be rectangle or trapezoid. Repeated Linear Factors: For deg(p) < n,
[tan x]′ = sec2 x ′
[ln |x|] =
1
Fundamental Theorem of Calculus p(x) A1 An
[sec x]′ = sec x tan x x = + ··· + , (14)
FTC 1: If f is continuous on [a, b], then the area function (ax − b)n (ax − b) (ax − b)n
[ax ]′ = ln(a) · ax
[csc x]′ = − csc x cot x given by
1
Z x where A1 , . . . , An ∈ R.
[cot x]′ = − csc2 x [loga x]′ = F (x) = f (t) dt (8)
x ln(a) a Improper Integral (Use Proper Limit Notation)
is differentiable on (a, b), and F ′ (x) = f (x).
• Watch for t → a+ or t → b− .
Other Techniques (Applications of Chain Rule) FTC 2: When G′ (x) = f (x), then
Z b • Two-sided ∞ or discontinuity, split and evaluate.
Log-Diff.: y = [f (x)]g(x) . Take ln and differentiate.
f (x) dx = G(b) − G(a). (9)
d a
Implicit Diff.: Write and use chain rule carefully.
dx Chain Rule: If u is differentiable and f is continuous, Position, Velocity, Acceleration
Related Rates: Set up eqns. and variables; implicit diff. "Z
u(x)
#
d Motion in One Dimension
Inverse: If y = f −1 (x), rewrite as f (y) = x, implicit diff., f (t) dt = f (u(x))u′ (x). (10)
dx a • v(t) = s′ (t), a(t) = v ′ (t) = s′′ (t).
′ dy dy 1
f (y) · =1⇒ = ′ −1 (5) Integral Formulas (+C) • Direction: va > 0 ⇒ away; va < 0 ⇒ toward.
dx dx f (f (x))
* Reverse Derivative Formulas. Only list some to memorize: Z b
* Test yourself on y = arccos x, y = arctan x. Z • Speed = |v(t)| ≥ 0; Distance = |v(t)| dt.
sec x dx = ln | sec x + tan x| + C [u-sub] a
• Apply principles of Calculus to motion.
Theorems By Continuity (Theory) Z
csc x dx = − ln | csc x + cot x| + C [u-sub]
Given f : [a, b] → R continuous, Motion in 3D as Vector-Valued Functions
Z
IVT: f takes all values between f (a) and f (b). r : R → R3 ,
Position Vector: ⃗
tan x dx = − ln | cos x| + C [u-sub]
EVT: f attains abs. max and abs. min on [a, b]. Z
1 1 x r(t) = (x(t), y(t), z(t)) = x(t)⃗i + y(t)⃗j + z(t)⃗k
⃗
MVT: If f is also diff. on (a, b), there exists c ∈ (a, b) s.t. dx = arctan +C [trig-sub]
a2 + x2 a a r′ or
Calculus: Get ⃗
R
⃗
r dt coordinate-wise w.r.t. time.
′ f (b) − f (a) Z
1 x
f (c) = . (6) √ dx = arcsin +C [trig-sub]
b−a Physics: Generalize displacement, velocity, acceleration,
a2 − x2 a
speed, distance (arc length) to involve 3 variables.
Limit (Concept and Evaluation) Graphing (Refine Graphs with Calculus) Mean Value Theorem for Integral
* Be familiar with concepts and graphs. 1st Derivative & 1st Derivative Test If f is continuous on [a, b], then there exists c ∈ [a, b] s.t.
Squeeze / Sandwitch Theorem Inc/Dec: f ′ ≥ 0 ⇒ f increasing ↗; f ′ ≤ 0 ⇒ f decreasing ↘ Z b
def 1
Suppose g(x) ≤ f (x) ≤ h(x) for all x near c, x ̸= c, Critical Point: f cts. at c with f ′ (c) = 0 or f ′ (c) DNE. f (c) = favg = f (x) dx. (11)
b−a a
lim g(x) = lim h(x) = L ⇒ lim f (x) = L (1) 1st Der. Test: f : ↘↗ at c; l. min. f : ↗↘ at c; l. max.
x→c x→c x→c favg is called the average value of f on [a, b].
2nd Derivative & 2nd Derivative Test
L’Hôpital’s Rule
f ′′ tells if f is above or below tangent lines.
f (x) 0 ∞ f (x) f ′ (x) Integration Techniques (u-Sub and More)
lim = or ⇒ lim = lim ′ , (2) Concavity: f ′′ > 0, concave up; f ′′ < 0, concave down.
x→c g(x) 0 ∞ x→c g(x) x→c g (x)
Integration By Parts (IBP)
if latter exists. For other indeterminate forms, rewrite as Inflection Point (POI): where f ′′ changes sign. Z Z
quotient or apply ln first. 2nd Der. Test: For f ′ (c) = 0, f ′′ (c) = + or −, l. min or l. u dv = uv − v du (12)
max.
* Check endpoints for abs. max/min in optimization Get u from ILATE if possible. Tabular method if u is ATE.
Derivative (Slope of Tangent, Rate of Change)
problem. Partial Fraction Decomposition (PFD)
def f (x + h) − f (x) dy ∆y
f ′ (x) = lim , ≈ , ∆x small (3)
h→0 h dx ∆x * Perform polynomial division if necessary.
* Rewrite for linear approximation: xnew = x + ∆x, Riemann Integral (Net Area, Accumulation)
Basic Rule: For deg(h) < deg(f g),
Given continuous f on [a, b],
f (xnew ) ≈ f (x) + f ′ (x)∆x (4) Z b n h(x) A(x) B(x)
def X = + , (13)
Derivative Formulas (Know The Rules) f (x)dx = lim Ai , (7) f (x)g(x) f (x) g(x)
a n→∞
i=1
[sin x]′ = cos x [xn ]′ = nxn−1 where deg(A) = deg(f ) − 1 and deg(B) = deg(g) − 1.
Ai is area of region on each partition of [a, b]. Region could
[cos x]′ = − sin x [ex ]′ = ex be rectangle or trapezoid. Repeated Linear Factors: For deg(p) < n,
[tan x]′ = sec2 x ′
[ln |x|] =
1
Fundamental Theorem of Calculus p(x) A1 An
[sec x]′ = sec x tan x x = + ··· + , (14)
FTC 1: If f is continuous on [a, b], then the area function (ax − b)n (ax − b) (ax − b)n
[ax ]′ = ln(a) · ax
[csc x]′ = − csc x cot x given by
1
Z x where A1 , . . . , An ∈ R.
[cot x]′ = − csc2 x [loga x]′ = F (x) = f (t) dt (8)
x ln(a) a Improper Integral (Use Proper Limit Notation)
is differentiable on (a, b), and F ′ (x) = f (x).
• Watch for t → a+ or t → b− .
Other Techniques (Applications of Chain Rule) FTC 2: When G′ (x) = f (x), then
Z b • Two-sided ∞ or discontinuity, split and evaluate.
Log-Diff.: y = [f (x)]g(x) . Take ln and differentiate.
f (x) dx = G(b) − G(a). (9)
d a
Implicit Diff.: Write and use chain rule carefully.
dx Chain Rule: If u is differentiable and f is continuous, Position, Velocity, Acceleration
Related Rates: Set up eqns. and variables; implicit diff. "Z
u(x)
#
d Motion in One Dimension
Inverse: If y = f −1 (x), rewrite as f (y) = x, implicit diff., f (t) dt = f (u(x))u′ (x). (10)
dx a • v(t) = s′ (t), a(t) = v ′ (t) = s′′ (t).
′ dy dy 1
f (y) · =1⇒ = ′ −1 (5) Integral Formulas (+C) • Direction: va > 0 ⇒ away; va < 0 ⇒ toward.
dx dx f (f (x))
* Reverse Derivative Formulas. Only list some to memorize: Z b
* Test yourself on y = arccos x, y = arctan x. Z • Speed = |v(t)| ≥ 0; Distance = |v(t)| dt.
sec x dx = ln | sec x + tan x| + C [u-sub] a
• Apply principles of Calculus to motion.
Theorems By Continuity (Theory) Z
csc x dx = − ln | csc x + cot x| + C [u-sub]
Given f : [a, b] → R continuous, Motion in 3D as Vector-Valued Functions
Z
IVT: f takes all values between f (a) and f (b). r : R → R3 ,
Position Vector: ⃗
tan x dx = − ln | cos x| + C [u-sub]
EVT: f attains abs. max and abs. min on [a, b]. Z
1 1 x r(t) = (x(t), y(t), z(t)) = x(t)⃗i + y(t)⃗j + z(t)⃗k
⃗
MVT: If f is also diff. on (a, b), there exists c ∈ (a, b) s.t. dx = arctan +C [trig-sub]
a2 + x2 a a r′ or
Calculus: Get ⃗
R
⃗
r dt coordinate-wise w.r.t. time.
′ f (b) − f (a) Z
1 x
f (c) = . (6) √ dx = arcsin +C [trig-sub]
b−a Physics: Generalize displacement, velocity, acceleration,
a2 − x2 a
speed, distance (arc length) to involve 3 variables.
