AP Calculus BC:
Study Guide
AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this
product.
,Key Exam Details
The AP® Calculus BC exam is a 3-hour 15-minute, end-of-course test comprised of 45 multiple-
choice questions (50% of the exam) and 6 free-response questions (50% of the exam).
The exam covers the following course content categories:
• Limits and Continuity: 4–7% of test questions
• Differentiation: Definition and Fundamental Properties: 4–7% of test questions
• Differentiation: Composite, Implicit, and Inverse Functions 4–7% of test questions
• Contextual Applications of Differentiation: 6–9% of test questions
• Analytical Applications of Differentiation: 8–11% of test questions
• Integration and Accumulation of Change: 17–20% of test questions
• Differential Equations: 6–9% of test questions
• Applications of Integration: 6–9% of test questions
• Parametric Equations, Polar Coordinates, and Vector-Valued Functions: 11–12% of test
questions
• Infinite Sequences and Series: 17–18% of test questions
This guide offers an overview of the main tested subjects, along with sample AP multiple-choice
questions that look like the questions you’ll see on test day.
Limits and Continuity
About 4–7% of the questions on your exam will cover Limits and Continuity.
Limits
The limit of a function f as x approaches c is L if the value of f can be made arbitrarily close to L
by taking x sufficiently close to c (but not equal to c). If such a value exists, this is denoted
lim f ( x) = L . If no such value exists, we say that the limit does not exist, abbreviated DNE.
x →c
Limits can be found using tables, graphs, and algebra.
Important algebraic techniques for finding limits include factoring and rationalizing radical
expressions. Other helpful tools are given by the following properties.
Suppose lim f ( x) = L , lim g ( x) = M , lim h( x) = N , and a is any real number.
x →c x →c x→ L
1
,Then:
• lim f ( x) + g ( x) = L + M
x →c
• lim f ( x) − g ( x) = L − M
x →c
• lim af ( x) = aL
x →c
f ( x) L
• lim = , as long as M 0
x →c g ( x) M
• lim h ( f ( x) ) = N
x →c
For many common functions, evaluating limits requires nothing more than evaluating the
function at the point c (assuming the function is defined at the point). These include polynomial,
rational, exponential, logarithmic, and trigonometric functions.
sin x 1 − cos x
Two special limits that are important in calculus are lim = 1 and lim =0.
x →0 x x → 0 x
One-Sided Limits
Sometimes we are interested in the value that a function f approaches as x approaches c from
only a single direction. If the values of f get arbitrarily close to L as x approaches c while taking
on values greater than c, we say lim+ f ( x) = L . Similarly, if x is taking on values less than c, we
x →c
write lim− f ( x) = L .
x →c
We can now characterize limits by saying that lim f ( x) exists if and only if both lim+ f ( x) and
x →c x →c
lim f ( x) exist and have the same value. A limit, then, can fail to exist in a few ways:
x →c −
• lim f ( x) does not exist
x →c +
• lim f ( x) does not exist
x →c −
• Both of the one-sided limits exist, but have different values
2
, Example
The function shown has the following limits:
• lim f ( x) = −1
x →−2−
• lim f ( x) = 1
x →−2+
• lim f ( x) DNE
x →−2
• lim f ( x) = 4
x →1−
• lim f ( x) = 4
x →1+
• lim f ( x) = 4
x →1
Note that f(1) = 3, but this is irrelevant to the value of the limit.
Infinite Limits, Limits at Infinity, and Asymptotes
When a function has a vertical asymptote at x = c, the behavior of the function can be described
using infinite limits. If the function values increase as they approach the asymptote, we say the
limit is ∞, whereas if the values decrease as they approach the asymptote, the limit is -∞. It is
important to realize that these limits do not exist in the same sense that we described earlier;
rather, saying that a limit is is simply a convenient way to describe the behavior of the
function approaching the point.
We can also extend limits by considering how the function behaves as x → . If such a limit
exists, it means that the function approaches a horizontal line as x increases or decreases without
bound. In other words, if lim f ( x) = L , then f has a horizontal asymptote y = L . It is possible for
x →
a function to have two horizontal asymptotes since it can have different limits as x → and
x → − .
3
Study Guide
AP is a registered trademark of the College Board, which was not involved in the production of, and does not endorse, this
product.
,Key Exam Details
The AP® Calculus BC exam is a 3-hour 15-minute, end-of-course test comprised of 45 multiple-
choice questions (50% of the exam) and 6 free-response questions (50% of the exam).
The exam covers the following course content categories:
• Limits and Continuity: 4–7% of test questions
• Differentiation: Definition and Fundamental Properties: 4–7% of test questions
• Differentiation: Composite, Implicit, and Inverse Functions 4–7% of test questions
• Contextual Applications of Differentiation: 6–9% of test questions
• Analytical Applications of Differentiation: 8–11% of test questions
• Integration and Accumulation of Change: 17–20% of test questions
• Differential Equations: 6–9% of test questions
• Applications of Integration: 6–9% of test questions
• Parametric Equations, Polar Coordinates, and Vector-Valued Functions: 11–12% of test
questions
• Infinite Sequences and Series: 17–18% of test questions
This guide offers an overview of the main tested subjects, along with sample AP multiple-choice
questions that look like the questions you’ll see on test day.
Limits and Continuity
About 4–7% of the questions on your exam will cover Limits and Continuity.
Limits
The limit of a function f as x approaches c is L if the value of f can be made arbitrarily close to L
by taking x sufficiently close to c (but not equal to c). If such a value exists, this is denoted
lim f ( x) = L . If no such value exists, we say that the limit does not exist, abbreviated DNE.
x →c
Limits can be found using tables, graphs, and algebra.
Important algebraic techniques for finding limits include factoring and rationalizing radical
expressions. Other helpful tools are given by the following properties.
Suppose lim f ( x) = L , lim g ( x) = M , lim h( x) = N , and a is any real number.
x →c x →c x→ L
1
,Then:
• lim f ( x) + g ( x) = L + M
x →c
• lim f ( x) − g ( x) = L − M
x →c
• lim af ( x) = aL
x →c
f ( x) L
• lim = , as long as M 0
x →c g ( x) M
• lim h ( f ( x) ) = N
x →c
For many common functions, evaluating limits requires nothing more than evaluating the
function at the point c (assuming the function is defined at the point). These include polynomial,
rational, exponential, logarithmic, and trigonometric functions.
sin x 1 − cos x
Two special limits that are important in calculus are lim = 1 and lim =0.
x →0 x x → 0 x
One-Sided Limits
Sometimes we are interested in the value that a function f approaches as x approaches c from
only a single direction. If the values of f get arbitrarily close to L as x approaches c while taking
on values greater than c, we say lim+ f ( x) = L . Similarly, if x is taking on values less than c, we
x →c
write lim− f ( x) = L .
x →c
We can now characterize limits by saying that lim f ( x) exists if and only if both lim+ f ( x) and
x →c x →c
lim f ( x) exist and have the same value. A limit, then, can fail to exist in a few ways:
x →c −
• lim f ( x) does not exist
x →c +
• lim f ( x) does not exist
x →c −
• Both of the one-sided limits exist, but have different values
2
, Example
The function shown has the following limits:
• lim f ( x) = −1
x →−2−
• lim f ( x) = 1
x →−2+
• lim f ( x) DNE
x →−2
• lim f ( x) = 4
x →1−
• lim f ( x) = 4
x →1+
• lim f ( x) = 4
x →1
Note that f(1) = 3, but this is irrelevant to the value of the limit.
Infinite Limits, Limits at Infinity, and Asymptotes
When a function has a vertical asymptote at x = c, the behavior of the function can be described
using infinite limits. If the function values increase as they approach the asymptote, we say the
limit is ∞, whereas if the values decrease as they approach the asymptote, the limit is -∞. It is
important to realize that these limits do not exist in the same sense that we described earlier;
rather, saying that a limit is is simply a convenient way to describe the behavior of the
function approaching the point.
We can also extend limits by considering how the function behaves as x → . If such a limit
exists, it means that the function approaches a horizontal line as x increases or decreases without
bound. In other words, if lim f ( x) = L , then f has a horizontal asymptote y = L . It is possible for
x →
a function to have two horizontal asymptotes since it can have different limits as x → and
x → − .
3
