AP Calculus BC Exam with complete
solutions 2024/2025
Squeeze Theorem - ANSWER-If h(x) ≤ ƒ(x) ≤ g(x) for all x in an open
interval containing c, except possibly at c itself, and lim x→c h(x) =
lim x→c g(x) = L, then lim x→c ƒ(x) exists and equals L.
Continuity - ANSWER-A function ƒ is continuous at c if:
a. ƒ(c) is defined
b. lim x→c ƒ(x) exists
c. lim x→c ƒ(x) = ƒ(c)
Intermediate Value Theorem - ANSWER-If ƒ is continuous on an interval [a,b] and
k is any number between ƒ(a) and ƒ(b), there is at least one number c in [a,b] such
that ƒ(c) = k.
Definition of the derivative - ANSWER-ƒ'(x) = lim h→0 (ƒ(x+h) - ƒ(x))/h
Power rule - ANSWER-d/dx xⁿ = nx^(n-1)
Product rule - ANSWER-d/dx ƒ(x)g(x) = ƒ'(x)g(x) + ƒ(x)g'(x)
Quotient rule - ANSWER-d/dx ƒ(x)/g(x) = (g(x)ƒ'(x) - ƒ(x)g'(x))/g²(x)
Chain rule - ANSWER-d/dx ƒ(g(x)) = ƒ'(g(x))g'(x)
Implicit differentiation - ANSWER-d/dx ƒ(y) = ƒ'(y)(dy/dx)
Definition of a minimum - ANSWER-ƒ(c) is the minimum of ƒ on an
interval if ƒ(c) ≤ f(x) for all x contained in the interval.
, Definition of a maximum - ANSWER-ƒ(c) is the maximum of ƒ on an
interval if ƒ(c) ≥ f(x) for all x contained in the interval.
Extreme Value Theorem - ANSWER-If ƒ is continuous on a closed interval, then ƒ
has both a minimum and maximum on the interval.
Critical number - ANSWER-A number c is called critical if ƒ'(c) = 0 or ƒ is not
differentiable at c.
Points tested for minima/maxima - ANSWER-Critical points and enpoints.
Local minimum/maximum - ANSWER-A point where ƒ' changes from positive to
negative or vice versa.
Inflection point - ANSWER-A point where ƒ" changes from positive to negative or
vice versa.
Antiderivative - ANSWER-A function F(x) that satisfies F'(x) = ƒ(x)
Definition of the indefinite integral - ANSWER-The collection of all
antiderivatives of a function, e.g. ∫ƒ(x)dx = F(x) + C, since all
functions of the form F(x) + C satisfy (F(x) + C)' = ƒ(x).
Differential equation - ANSWER-An equation involving the derivative(s) of a
function.
Definition of the definite integral - ANSWER-∑ƒ(ci)∆xi, as ∆x→0
FUNDAMENTAL THEOREM OF CALCULUS TURN UP DRANK FADED
MILLI BILLI BEAN - ANSWER-∫ƒ(x)dx from a to b = F(b) - F(a), where
F is an antiderivative of ƒ
Mean Value Theorem / Average value of a function - ANSWER-∫ƒ(x)dx
from a to b = ƒ(c)(b-a)
or
(∫ƒ(x)dx)/(b-a) = ƒ(c)
Second part of the Fundamental Theorem of Calculus - ANSWER-d/dx
(∫ƒ(t)dt from a to x) = f(x)
solutions 2024/2025
Squeeze Theorem - ANSWER-If h(x) ≤ ƒ(x) ≤ g(x) for all x in an open
interval containing c, except possibly at c itself, and lim x→c h(x) =
lim x→c g(x) = L, then lim x→c ƒ(x) exists and equals L.
Continuity - ANSWER-A function ƒ is continuous at c if:
a. ƒ(c) is defined
b. lim x→c ƒ(x) exists
c. lim x→c ƒ(x) = ƒ(c)
Intermediate Value Theorem - ANSWER-If ƒ is continuous on an interval [a,b] and
k is any number between ƒ(a) and ƒ(b), there is at least one number c in [a,b] such
that ƒ(c) = k.
Definition of the derivative - ANSWER-ƒ'(x) = lim h→0 (ƒ(x+h) - ƒ(x))/h
Power rule - ANSWER-d/dx xⁿ = nx^(n-1)
Product rule - ANSWER-d/dx ƒ(x)g(x) = ƒ'(x)g(x) + ƒ(x)g'(x)
Quotient rule - ANSWER-d/dx ƒ(x)/g(x) = (g(x)ƒ'(x) - ƒ(x)g'(x))/g²(x)
Chain rule - ANSWER-d/dx ƒ(g(x)) = ƒ'(g(x))g'(x)
Implicit differentiation - ANSWER-d/dx ƒ(y) = ƒ'(y)(dy/dx)
Definition of a minimum - ANSWER-ƒ(c) is the minimum of ƒ on an
interval if ƒ(c) ≤ f(x) for all x contained in the interval.
, Definition of a maximum - ANSWER-ƒ(c) is the maximum of ƒ on an
interval if ƒ(c) ≥ f(x) for all x contained in the interval.
Extreme Value Theorem - ANSWER-If ƒ is continuous on a closed interval, then ƒ
has both a minimum and maximum on the interval.
Critical number - ANSWER-A number c is called critical if ƒ'(c) = 0 or ƒ is not
differentiable at c.
Points tested for minima/maxima - ANSWER-Critical points and enpoints.
Local minimum/maximum - ANSWER-A point where ƒ' changes from positive to
negative or vice versa.
Inflection point - ANSWER-A point where ƒ" changes from positive to negative or
vice versa.
Antiderivative - ANSWER-A function F(x) that satisfies F'(x) = ƒ(x)
Definition of the indefinite integral - ANSWER-The collection of all
antiderivatives of a function, e.g. ∫ƒ(x)dx = F(x) + C, since all
functions of the form F(x) + C satisfy (F(x) + C)' = ƒ(x).
Differential equation - ANSWER-An equation involving the derivative(s) of a
function.
Definition of the definite integral - ANSWER-∑ƒ(ci)∆xi, as ∆x→0
FUNDAMENTAL THEOREM OF CALCULUS TURN UP DRANK FADED
MILLI BILLI BEAN - ANSWER-∫ƒ(x)dx from a to b = F(b) - F(a), where
F is an antiderivative of ƒ
Mean Value Theorem / Average value of a function - ANSWER-∫ƒ(x)dx
from a to b = ƒ(c)(b-a)
or
(∫ƒ(x)dx)/(b-a) = ƒ(c)
Second part of the Fundamental Theorem of Calculus - ANSWER-d/dx
(∫ƒ(t)dt from a to x) = f(x)
