AP Calculus BC

This deck contains all the necessary mathematical formulas and theorems required to get a 5 on AP Calculus BC.

"What is the Logarithm Product Rule?"

"ln(ab) = ln(a) + ln(b)"

"What is the Logarithm Quotient Rule?"

"ln(a/b) = ln(a) - ln(b)"

"What is the Logarithm Power Rule?"

"ln(a^b) = b * ln(a)"

"What is the Logarithm Change of Base Formula?"

"log_a(x) = ln(x) / ln(a)"

"What is the Inverse Property of e and ln (e raised to ln)?"

"e^(ln x) = x"

"What is the Inverse Property of ln and e (ln of e raised to)?"

"ln(e^x) = x"

"What is the value of ln(1)?"

"0"

"What is the value of ln(e)?"

"1"

"What is the Exponent Multiplication Rule (x^a * x^b)?"

"x^(a+b)"

"What is the Exponent Division Rule (x^a / x^b)?"

"x^(a-b)"

"What is the Power of a Power Rule ((x^a)^b)?"

"x^(ab)"

"What is the Negative Exponent Rule (x^-n)?"

"1 / (x^n)"

"What is the Fractional Exponent Rule (x^(p/q))?"

"q-th root of (x^p) OR (q-th root of x)^p"

"What is the algebraic definition of an Even Function?"

"f(-x) = f(x)"

"What is the algebraic definition of an Odd Function?"

"f(-x) = -f(x)"

"What is the geometric symmetry of an Even Function?"

"y-axis symmetry"

"What is the geometric symmetry of an Odd Function?"

"Origin symmetry"

"What is the Integral Property of an Even Function from -a to a?"

"2 * [Integral from 0 to a of f(x) dx]"

"What is the Integral Property of an Odd Function from -a to a?"

"0"

"What is the piecewise definition of the Absolute Value function |x|?"

"x if x >= 0; -x if x < 0"

"What is the slope-intercept form of a line?"

"y = mx + b"

"What is the point-slope form of a line (Leibniz-friendly)?"

"y - y1 = m(x - x1)"

"What is the formula for the slope (m) between two points?"

"m = (y2 - y1) / (x2 - x1)"

"What is the Quadratic Formula?"

"x = [-b ± sqrt(b^2 - 4ac)] / 2a"

"What is the Binomial Expansion of (a + b)^2?"

"a^2 + 2ab + b^2"

"What is the Binomial Expansion of (a - b)^2?"

"a^2 - 2ab + b^2"

"What is the Difference of Squares factorization?"

"a^2 - b^2 = (a - b)(a + b)"

"What is the Sum of Cubes factorization?"

"a^3 + b^3 = (a + b)(a^2 - ab + b^2)"

"What is the Difference of Cubes factorization?"

"a^3 - b^3 = (a - b)(a^2 + ab + b^2)"

"What is the quotient identity for tan(x)?"

"sin(x) / cos(x)"

"What is the quotient identity for cot(x)?"

"cos(x) / sin(x)"

"What is the reciprocal identity for sec(x)?"

"1 / cos(x)"

"What is the reciprocal identity for csc(x)?"

"1 / sin(x)"

"What is the fundamental Pythagorean identity?"

"sin^2(x) + cos^2(x) = 1"

"What is the Pythagorean identity involving tan(x)?"

"1 + tan^2(x) = sec^2(x)"

"What is the Pythagorean identity involving cot(x)?"

"1 + cot^2(x) = csc^2(x)"

"What is the double angle formula for sin(2x)?"

"2sin(x)cos(x)"

"What is the primary form of the double angle formula for cos(2x)?"

"cos^2(x) - sin^2(x)"

"What is the double angle formula for cos(2x) in terms of cosine only?"

"2cos^2(x) - 1"

"What is the double angle formula for cos(2x) in terms of sine only?"

"1 - 2sin^2(x)"

"What is the power reduction formula for sin^2(x)?"

"(1 - cos(2x)) / 2"

"What is the power reduction formula for cos^2(x)?"

"(1 + cos(2x)) / 2"

"What is the odd/even symmetry property for sin(-x)?"

"-sin(x) (Odd function)"

"What is the odd/even symmetry property for cos(-x)?"

"cos(x) (Even function)"

"What is the odd/even symmetry property for tan(-x)?"

"-tan(x) (Odd function)"

"What are the (cos, sin) coordinates and tan value for 0 radians?"

• (cos, sin) coordinates: (1, 0)

• tan value: 0

"What are the (cos, sin) coordinates and tan value for pi/6 radians?"

• (cos, sin) coordinates: (√3/2, 1/2)

• tan value: 1/√3

"What are the (cos, sin) coordinates and tan value for pi/4 radians?"

• (cos, sin) coordinates: (√2/2, √2/2)

• tan value: 1

"What are the (cos, sin) coordinates and tan value for pi/3 radians?"

• (cos, sin) coordinates: (1/2, √3/2)

• tan value: √3

"What are the (cos, sin) coordinates and tan value for pi/2 radians?"

• (cos, sin) coordinates: (0, 1)

• tan value: undefined

"What is the period of the sine and cosine functions?"

"2*pi"

"What is the period of the tangent and cotangent functions?"

"pi"

"What are the criteria for the existence of a limit lim_{x -> c} f(x)?"

"The left-hand limit must equal the right-hand limit: lim_{x -> c^-} f(x) = lim_{x -> c^+} f(x) = L."

"What are the three criteria for a function f(x) to be continuous at a point x = c?"

"1. f(c) is defined. 2. lim_{x -> c} f(x) exists. 3. lim_{x -> c} f(x) = f(c)."

"What is the definition of a Vertical Asymptote at x = a?"

"A vertical asymptote exists at x = a if lim_{x -> a^-} f(x) = ±∞ or lim_{x -> a^+} f(x) = ±∞."

"What is the definition of a Horizontal Asymptote at y = L?"

"A horizontal asymptote exists at y = L if lim_{x -> ∞} f(x) = L or lim_{x -> -∞} f(x) = L."

"What are the two indeterminate forms required to apply L'Hôpital's Rule?"

"The limit must produce the indeterminate form 0/0 or ±∞/±∞."

"State L'Hôpital's Rule in both Lagrange and Leibniz notation."

"Lagrange: lim_{x -> c} [f(x)/g(x)] = lim_{x -> c} [f'(x)/g'(x)]. Leibniz: lim_{x -> c} [f(x)/g(x)] = lim_{x -> c} [(d/dx f(x))/(d/dx g(x))]."

"What is the value of the special trigonometric limit lim_{x -> 0} [sin(x)/x]?"

"1"

"What is the value of the special trigonometric limit lim_{x -> 0} [(1 - cos(x))/x]?"

"0"

"What is the limit definition of the constant 'e'?"

"lim_{x -> ∞} (1 + 1/x)^x = e or lim_{x -> 0} (1 + x)^(1/x) = e."

"What is the value of the special limit lim_{n -> ∞} [nth-root of n]?"

"1"

"State the conditions and result of the Squeeze (Sandwich) Theorem."

"Conditions: g(x) ≤ f(x) ≤ h(x) near c and lim_{x -> c} g(x) = lim_{x -> c} h(x) = L. Result: lim_{x -> c} f(x) = L."

"State the conditions and result of the Intermediate Value Theorem (IVT)."

"Condition: f(x) is continuous on the closed interval [a

“What is the theorem regarding Differentiability and Continuity?”

Theorem: If f is differentiable at x = c, then f is continuous at x = c. (Note: The converse is FALSE; continuity does not guarantee differentiability).

"What are the three common types of discontinuities?"

"1. Removable (Hole). 2. Jump (Step). 3. Infinite (Vertical Asymptote)."

"How do you remove a 'Removable Discontinuity' at x = c?"

"Redefine f(c) such that f(c) = lim_{x -> c} f(x)."

"What is the limit definition of the derivative (h-form) in Lagrange and Leibniz notation?"

"Lagrange: f'(x) = lim (h->0) [f(x+h) - f(x)] / h | Leibniz: dy/dx = lim (Δx->0) Δy/Δx"

"What is the alternative limit definition of the derivative at a point 'a'?"

"f'(a) = lim (x->a) [f(x) - f(a)] / (x - a)"

"What is the Constant Rule for differentiation?"

"Lagrange: (c)' = 0 | Leibniz: d/dx(c) = 0"

"What is the Power Rule for differentiation?"

"Lagrange: (x^n)' = nx^(n-1) | Leibniz: d/dx(x^n) = nx^(n-1)"

"What is the Sum and Difference Rule for differentiation?"

"Lagrange: (u ± v)' = u' ± v' | Leibniz: d/dx(u ± v) = du/dx ± dv/dx"

"What is the Product Rule for differentiation?"

"Lagrange: (uv)' = u'v + uv' | Leibniz: d/dx(uv) = (du/dx)v + u(dv/dx)"

"What is the Quotient Rule for differentiation?"

"Lagrange: (u/v)' = (u'v - uv') / v^2 | Leibniz: d/dx(u/v) = [v(du/dx) - u(dv/dx)] / v^2"

"What is the Chain Rule for composite functions?"

"Lagrange: [f(g(x))]' = f'(g(x)) * g'(x) | Leibniz: dy/dx = (dy/du) * (du/dx)"

"What is the derivative of sin(u) using the Chain Rule?"

"Lagrange: [sin(u)]' = cos(u) * u' | Leibniz: d/dx(sin u) = cos(u) * du/dx"

"What is the derivative of cos(u) using the Chain Rule?"

"Lagrange: [cos(u)]' = -sin(u) * u' | Leibniz: d/dx(cos u) = -sin(u) * du/dx"

"What is the derivative of tan(u) using the Chain Rule?"

"Lagrange: [tan(u)]' = sec^2(u) * u' | Leibniz: d/dx(tan u) = sec^2(u) * du/dx"

"What is the derivative of cot(u) using the Chain Rule?"

"Lagrange: [cot(u)]' = -csc^2(u) * u' | Leibniz: d/dx(cot u) = -csc^2(u) * du/dx"

"What is the derivative of sec(u) using the Chain Rule?"

"Lagrange: [sec(u)]' = sec(u)tan(u) * u' | Leibniz: d/dx(sec u) = sec(u)tan(u) * du/dx"

"What is the derivative of csc(u) using the Chain Rule?"

"Lagrange: [csc(u)]' = -csc(u)cot(u) * u' | Leibniz: d/dx(csc u) = -csc(u)cot(u) * du/dx"

"What is the derivative of e^u using the Chain Rule?"

"Lagrange: [e^u]' = e^u * u' | Leibniz: d/dx(e^u) = e^u * du/dx"

"What is the derivative of a^u using the Chain Rule?"

"Lagrange: [a^u]' = a^u * ln(a) * u' | Leibniz: d/dx(a^u) = a^u * ln(a) * du/dx"

"What is the derivative of ln(u) using the Chain Rule?"

"Lagrange: [ln(u)]' = (1/u) * u' | Leibniz: d/dx(ln u) = (1/u) * du/dx"

"What is the derivative of log_a(u) using the Chain Rule?"

"Lagrange: [log_a(u)]' = [1 / (u * ln a)] * u' | Leibniz: d/dx(log_a u) = [1 / (u * ln a)] * du/dx"

"What is the derivative of the absolute value |u|?"

"Lagrange: [|u|]' = (u / |u|) * u' | Leibniz: d/dx(|u|) = (u / |u|) * du/dx (for u ≠ 0)"

"What is the derivative of an inverse function g(x) = f^-1(x)?"

"Lagrange: g'(x) = 1 / f'(g(x)) | Leibniz: dy/dx = 1 / (dx/dy)"

"What is the derivative of arcsin(u) using the Chain Rule?"

"Lagrange: [sin^-1(u)]' = u' / sqrt(1 - u^2) | Leibniz: d/dx(sin^-1 u) = (du/dx) / sqrt(1 - u^2)"

"What is the derivative of arccos(u) using the Chain Rule?"

"Lagrange: [cos^-1(u)]' = -u' / sqrt(1 - u^2) | Leibniz: d/dx(cos^-1 u) = -(du/dx) / sqrt(1 - u^2)"

"What is the derivative of arctan(u) using the Chain Rule?"

"Lagrange: [tan^-1(u)]' = u' / (1 + u^2) | Leibniz: d/dx(tan^-1 u) = (du/dx) / (1 + u^2)"

"What is the derivative of arccot(u) using the Chain Rule?"

"Lagrange: [cot^-1(u)]' = -u' / (1 + u^2) | Leibniz: d/dx(cot^-1 u) = -(du/dx) / (1 + u^2)"

"What is the derivative of arcsec(u) using the Chain Rule?"

"Lagrange: [sec^-1(u)]' = u' / (|u| * sqrt(u^2 - 1)) | Leibniz: d/dx(sec^-1 u) = (du/dx) / (|u| * sqrt(u^2 - 1))"

"What is the derivative of arccsc(u) using the Chain Rule?"

"Lagrange: [csc^-1(u)]' = -u' / (|u| * sqrt(u^2 - 1)) | Leibniz: d/dx(csc^-1 u) = -(du/dx) / (|u| * sqrt(u^2 - 1))"

"How do you find the slope dy/dx of a parametric curve?"

"Leibniz: dy/dx = (dy/dt) / (dx/dt)"

"How do you find the second derivative d^2y/dx^2 of a parametric curve?"

"Leibniz: d^2y/dx^2 = [d/dt (dy/dx)] / (dx/dt)"

"What is the formula for the slope dy/dx of a polar curve r = f(θ)?"

"Leibniz: dy/dx = (r'sinθ + rcosθ) / (r'cosθ - rsinθ)"

"What are the exact conditions and statement of the Extreme Value Theorem (EVT)?"

CONDITIONS:

f is continuous on the closed interval [a, b]

STATEMENT (Extreme Value Theorem):
If f is continuous on the closed interval [a, b], then f attains both an absolute maximum value and an absolute minimum value on [a, b].
That is, there exist points c and d in [a, b] such that
f(c) ≥ f(x) for all x in [a, b] and
f(d) ≤ f(x) for all x in [a, b].

"What are the exact conditions and statement of the Mean Value Theorem (MVT) for Derivatives?"

CONDITIONS:
f is continuous on the closed interval [a, b]
f is differentiable on the open interval (a, b)

STATEMENT (Mean Value Theorem):
If f is continuous on [a, b] and differentiable on (a, b), then there exists at least one number c in (a, b) such that
f′(c) = (f(b) − f(a)) / (b − a).

"What are the exact conditions and statement of the Mean Value Theorem (MVT) for Integrals?"

CONDITIONS:
f is continuous on the closed interval [a, b]

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STATEMENT (Area / Rectangle Form):
If f is continuous on [a, b], then there exists at least one number c in (a, b) such that
the integral from a to b of f(x) dx equals f(c) multiplied by (b − a).

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STATEMENT (Average Value Form):
If f is continuous on [a, b], then there exists at least one number c in (a, b) such that
f(c) equals (1 / (b − a)) multiplied by the integral from a to b of f(x) dx.

"What are the exact conditions and statement of Rolle's Theorem?"

CONDITIONS:
f is continuous on the closed interval [a, b]
f is differentiable on the open interval (a, b)
f(a) = f(b)

STATEMENT (Rolle’s Theorem):
If f satisfies the above conditions, then there exists at least one number c in (a, b) such that
f′(c) = 0.