AP Calculus BC Formula List Flashcards

Flashcards for AP Calculus BC exam review, focusing on formulas and theorems.

cos² x + sin² x = _

1

sin(2x) = _

2 sin x cos x

cos (2x)= _

cos² x-sin² x, 1-2 sin² x, 2 cos² x-1

1+ tan² x = _

sec² x

_ = csc² x

1+cot² x

cos² x = _

(1+cos(2x)) / 2

sin² x= _

(1-cos(2x)) / 2

e = _

lim (1 + (1/x))^x as x approaches infinity

f'(x) = _

lim (f(x+h) - f(x)) / h as h approaches 0

f'(a) = _

lim (f(a+h) - f(a)) / h as h approaches 0

|x| = _

x if x≥0, -x if x <0

f'(c) = _

lim (f(x) - f(c)) / (x - c) as x approaches c

f is continuous at x = c if and only if _

1) lim f(x) exists as x->c; 2) f(c) is defined; 3) lim f(x)= f(c) as x->c

Average rate of change of f(x) on [a, b] = _

(f(b) - f(a)) / (b - a)

Intermediate Value Theorem: _

If ƒ is continuous on [a, b] and k is any number between f(a) and ƒ (b), then there is at least one number c between a and b such that ƒ (c) = k.

d/dx [f(x)g(x)]= _

f(x)g'(x) + g(x)f'(x)

d/dx [f(g(x))]= _

f'(g(x)) * g'(x)

d/dx [x^n] = _

n*x^(n-1)

d/dx [f(x)/g(x)] = _

(g(x)f'(x) - f(x)g'(x)) / (g(x))^2

d/dx [sin u] = _

cos(u) * du/dx

d/dx [cos u] = _

-sin(u) * du/dx

d/dx [tan u] = _

sec²(u) * du/dx

d/dx [cot u] = _

-csc²(u) * du/dx

d/dx [sec u] = _

sec(u)tan(u) * du/dx

d/dx [csc u] = _

-csc(u)cot(u) * du/dx

Rolle's Theorem: _

If ƒ is continuous on [a, b] and differentiable on (a, b) and if ƒ (a) = ƒ (b), then there is at least one number c on (a, b) such that f'(c) = 0.

Mean Value Theorem: _

If ƒ is continuous on [a, b] and differentiable on (a, b), then there exists a number c on (a, b) such that f'(c) = (f(b)-f(a))/(b-a).

Definition of a Critical Number: _

Let f be defined at c. If f'(c)=0 or if f' is undefined at c, then c is a critical number of f.

Definition of Increasing Functions: _

A function f is increasing on an interval if for any two numbers x₁ and x₂ in the interval, if x₁<x₂, then f(x₁) < f(x₂).

Definition of Decreasing Functions: _

A function f is decreasing on an interval if for any two numbers x₁ and x₂ in the interval, if x₁

Test for Increasing Functions: _

If f'(x)>0 for all x in (a,b), then ƒ is increasing on [a, b].

Test for Decreasing Functions: _

If f'(x)<0 for all x in (a,b), then ƒ is decreasing on [a, b].

First Derivative Test (Minimum): _

If f'(x) changes from negative to positive at x= c, then (c, f(c)) is a relative or local minimum of f.

First Derivative Test (Maximum): _

If f'(x) changes from positive to negative at x=c, then (c, f(c)) is a relative or local maximum of f.

Second Derivative Test (Minimum): _

If f'(c)=0 and f''(c)>0, then (c, f(c)) is a relative or local minimum of f.

Second Derivative Test (Maximum): _

If f'(c)=0 and f''(c)<0, then (c, f(c)) is a relative or local maximum of f.

Definition of Concavity: _

The graph of ƒ is concave upward on I if f' is increasing on the interval and concave downward on I if f' is decreasing on the interval.

Test for Concavity (Upward): _

If ƒ''(x)>0 for all x in the interval I, then the graph of ƒ is concave upward in I.

Test for Concavity (Downward): _

If ƒ''(x)<0 for all x in the interval I, then the graph of ƒ is concave downward in I.

Definition of an Inflection Point: _

A function ƒ has an inflection point at (c, ƒ (c)) if f''(c)=0 or f''(c) does not exist and if f'' changes sign from positive to negative or negative to positive at x=c.

Definition of a definite integral: _

∫ from a to b of f(x)dx = lim Σ f(xk) * (Δxk) as Δx->0

∫ x^n dx = _ (n != -1)

(x^(n+1))/(n+1) + C

∫ cos(u) du = _

sin(u) + C

∫ sec²(u) du = _

tan(u) + C

∫ sec(u)tan(u) du = _

sec(u) + C

∫ sin(u) du = _

-cos(u) + C

∫ csc²(u) du = _

-cot(u) + C

∫ csc(u)cot(u) du = _

-csc(u) + C

First Fundamental Theorem of Calculus: ∫ from a to b of f'(x) dx = _

f(b) - f(a)

d/dx [∫ from a to x of f(t) dt] = _

f(x)

d/dx [∫ from a to g(x) of f(t) dt] = _

f(g(x)) * g'(x)

Average value of f(x) on [a,b] = _

(1/(b-a)) * (∫ from a to b of f(x) dx)

d/dx [ln u] = _

(1/u)*(du/dx)

d/dx [e^u] = _

e^u * (du/dx)

d/dx [loga(u)] =

(1/(uln(a)))(du/dx)

d/dx [a^u] = _

a^u * ln(a) * (du/dx)

(f⁻¹)'(a) = _

1 / f'(f⁻¹(a))

∫ tan(u) du = _

-ln|cos(u)| + C

∫ sec(u) du = _

ln|sec(u) + tan(u)| + C

∫ e^u du = _

e^u + C

∫ (1/u) du = _

ln|u| + C

∫ cot(u) du = _

ln|sin(u)| + C

∫ csc(u) du = _

-ln|csc(u) + cot(u)| + C

∫ a^u du = _

(a^u) / ln(a) + C

Volume by cross sections taken perpendicular to the x-axis: V = _

∫ from a to b of A(x) dx

Volume around a horizontal axis by discs: V = _

π ∫ from a to b of (r(x))^2 dx

Volume around a horizontal axis by washers: V = _

π ∫ from a to b of ((R(x))^2 - (r(x))^2) dx

Velocity is _

v(t) = s'(t)

Acceleration is _

a(t) = v'(t) = s''(t)

Speed = _

|v(t)|

Displacement = _

∫ from a to b of v(t) dt

Total Distance = _

∫ from a to b of |v(t)| dt

The speed of the object is increasing when _

When velocity and acceleration have the same sign.

The speed of the object is decreasing when _

When velocity and acceleration have opposite signs.

d/dx [arcsin u] = _

(1 / √(1 - u^2)) * (du/dx)

d/dx [arctan u] = _

(1 / (1 + u^2)) * (du/dx)

d/dx [arcsec u] = _

(1 / (|u|√(u^2 - 1))) * (du/dx)

d/dx [arccos u] = _

-(1 / √(1 - u^2)) * (du/dx)

d/dx [arccot u] = _

-(1 / (1 + u^2)) * (du/dx)

d/dx [arccsc u] = _

-(1 / (|u|√(u^2 - 1))) * (du/dx)

∫ du/√(a^2 - u^2) = _

(1/a) * arcsin(u/a) + C

∫ du/(u√(u^2 - a^2)) = _

(1/a) * arcsec(|u|/a) + C

∫ du/(a^2 + u^2) = _

(1/a) * arctan(u/a) + C

dP/dt = _

kP(L-P), where L = lim P(t)

Integration by parts: _

∫ u dv = uv - ∫ v du

Length of arc for functions: s = _

∫ from a to b = √(1 + [f'(x)]^2) dx

Velocity vector = _

(x'(t), y'(t))

Acceleration vector = _

(x''(t), y''(t))

Speed (or magnitude of velocity vector) = |v(t)| = _

√((dx/dt)^2 + (dy/dt)^2)

Distance traveled from t = a to t = b (or length of arc) is s= _

∫ from a to b √( (dx/dt)^2 + (dy/dt)^2 ) dt

Slope of polar curve: dy/dx = _

(rcosθ + r'sinθ) / (-rsinθ + r'cosθ)

Area within a polar curve: A = _

(1/2) ∫ from a to b r^2 dθ

Pn(x)= _

f(c) + f'(c)(x-c) + (f''(c)/2!)(x-c)^2 + (f'''(c)/3!)(x-c)^3 + … + (f^n(c)/n!)(x-c)^n

Taylor's Theorem Remainder: _

R_n(x) = (f^(n+1)(z) / (n+1)!) * (x-c)^(n+1)

Maclaurin series for e^x: _

e^x = 1 + x + (x^2/2!) + (x^3/3!) + … + (x^n/n!) + …

Maclaurin series for cos x: _

cos x = 1 - (x^2/2!) + (x^4/4!) - (x^6/6!) + … + ((-1)^n) * (x^(2n)/(2n)!) + …

Maclaurin series for sin x: _

sin x = x - (x^3/3!) + (x^5/5!) - (x^7/7!) + … + ((-1)^n) * (x^(2n+1)/(2n+1)!) + …