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AP Calculus BC Comprehensive Syllabus

AP Calculus BC Comprehensive Syllabus

Unit 1: Limits and Continuity

Subtopic Subtopic Number AP Points to understand
Introducing Calculus: Can change occur at an instant? 1.1 Change of a function over an interval is the slope of the line that passes through the two endpoints of the interval. 

When this interval is taken to be infinitesimally small, we get the instantaneous rate of change at a particular point, which is the derivative of the function at that point. It is the slope of the tangent line to the graph of the function at that point.

Limits: A limit is a value that a function approaches as the input variable approaches a certain value. 

Defining Limits and Using Limit Notation 1.2 The limit of a function F at a point x, denoted as xaf(x), is the value that the function approaches as x gets closer and closer to a, but not necessarily equal to a.

There are various methods to represent limits, including algebraic manipulation, factoring, rationalizing, and applying trigonometric or logarithmic identities.

Estimating Limit Values from Graphs 1.3 Graphical information about a function:

  • Domain: The set of all possible input values for a function.
  • Range: The set of all possible output values for a function.
  • Intercepts: The x-intercept is where the graph of a function crosses the x-axis, while the y-intercept is where the graph crosses the y-axis.
  • Symmetry: A function is symmetric if its graph can be reflected over a certain line and still look the same.
  • Continuity: A function is continuous if there are no abrupt breaks or holes in its graph. This means that the function can be drawn without lifting your pencil from the paper.
  • Points of discontinuity: Points where a function is not continuous, either because the function has a hole or a jump.
  • Local extrema: Points on a function where the function has a maximum or minimum value in a small interval around that point.
  • Concavity: A function is concave up if its graph curves upward, and concave down if its graph curves downward. The point where the concavity changes is called an inflection point.

Limits do not exist at a point when the function approaches different values from the left and right sides of that point, or when the function approaches infinity or negative infinity at that point.

Estimating Limit Values from Tables 1.4 Estimating limit values from tables involves using the values in a table of x and f(x) to approximate the limit of a function as x approaches a certain value. Here are some key terms to understand:

  1. X values: The values of x in the table, which approach the limit value.
  2. f(x) values: The corresponding function values for each x-value in the table.
Determining Limits using Algebraic properties of Limits 1.5 One-sided limits refer to the limit of a function as the input approaches a certain value from only one direction (either from the left or from the right).

Limit theorems are rules that help us evaluate limits of functions based on the behavior of the function at nearby points. 

Using limit theorems, we can find the limits of sums, differences, products, quotients, and composite functions. These theorems include: 

  • Squeeze Theorem
  • Algebraic Limit Theorem
  • Limit of a Composite Function Theorem
Determining Limits using Algebraic Manipulation 1.6 Equivalent expressions for a function are expressions that have the same value as the original function for all values of x.

To use equivalent expressions to evaluate a limit, you can manipulate the original function algebraically to create an equivalent expression that is easier to evaluate. 

Some common techniques for simplifying functions include

  • Factoring
  • Rationalizing the denominator
  • Using algebraic identities.
Selecting Procedures for determining Limits 1.7 Procedures to determine limits:

  1. Direct substitution: This method involves substituting the value at which the function is approaching directly into the function to determine the limit. 
  2. Factorization: This method involves factoring the numerator and denominator of the function and canceling out common factors to simplify the function. 
  3. Conjugate method: This method involves multiplying the numerator and denominator of the function by the conjugate of the denominator to eliminate square roots or radicals in the denominator. 
  4. L’Hopital’s Rule: This method involves taking the derivative of both the numerator and denominator of the function and evaluating the limit of the resulting ratio. If the resulting limit is of the form “0/0” or “infinity/infinity”, then this method can be used to evaluate the limit.
  5. Squeeze theorem: This method involves sandwiching the function between two other functions that have the same limit. If the limits of the two other functions are equal, then the limit of the original function is also equal to this limit.
Determining Limits using Squeeze Theorem 1.8 The squeeze theorem, also known as the sandwich theorem, states that if two functions, g(x) and h(x), approach the same limit L as x approaches a certain value, and another function f(x) is always between g(x) and h(x) near that value, then f(x) also approaches L as x approaches that value. 
Connecting Multiple representations of Limits 1.9 Translating mathematical information from a single representation or across multiple representations.
Exploring types of Discontinuities 1.10 Discontinuity: A point where a function fails to be continuous.

Types of discontinuities:

  1. Removable discontinuity: A removable discontinuity occurs when a function has a hole at a certain point. The function is undefined at that point, but it can be made continuous by defining the function at that point.
  2. Jump discontinuity: A jump discontinuity occurs when the function has a sudden jump in value at a certain point. The left-hand and right-hand limits exist, but they are not equal.
  3. Infinite discontinuity: An infinite discontinuity occurs when the function approaches infinity or negative infinity at a certain point. The limit of the function does not exist at that point.
  4. Oscillating discontinuity: An oscillating discontinuity occurs when the function oscillates between two values infinitely many times as it approaches a certain point. The limit of the function does not exist at that point.
Defining continuity at a point 1.11 Continuity of a point: A function f is said to be continuous at a point x = c if the limit of f(x) as x approaches c exists and is equal to f(c), meaning that the function has no holes or jumps at that point.
Confirming Continuity over an Interval 1.12 A function is said to be continuous at a point x = a if the following conditions hold:

  1. The function is defined at x = a.
  2. The limit of the function as x approaches a from both the left and the right exists.
  3. The value of the function at x = a is equal to the limit.
Removing Discontinuities 1.13 To remove the discontinuity of a function:

  1. Taking the limit of the function as it approaches the point of discontinuity
  2. Simplifying the function by factoring or canceling out common factors
  3. Re-defining the function at the point of discontinuity with a new value that makes the function continuous
  4. Applying a piecewise function to the original function, with different rules for the left and right limits at the point of discontinuity

To determine the values of x that make a discontinuous function continuous:

  1. Identify the point(s) of discontinuity of the function
  2. Determine the type of discontinuity at each point (e.g. removable, jump, infinite)
  3. Apply one of the techniques mentioned above to remove the discontinuity at each point
  4. Check if the function is now continuous at the previously discontinuous point(s)
  5. If the function is still discontinuous, repeat the process until it becomes continuous at all points
Connecting Infinite Limits and Vertical Asymptotes 1.14 If a function grows or decays without bound, has a horizontal asymptote, or oscillates.

Infinite limits: An infinite limit occurs when the limit of a function as the input variable approaches a certain value (usually infinity or negative infinity) does not exist because the function grows or decays without bound. 

Asymptotic behavior: Asymptotic behavior refers to the behavior of a function as the input variable approaches infinity or negative infinity. A function can have a horizontal asymptote, which is a straight line that the function approaches but never touches. 

Unbounded behavior: Unbounded behavior refers to the behavior of a function that grows or decays without bound as the input variable approaches a certain value (usually infinity or negative infinity). 

Connecting Limits at Infinity and Horizontal Asymptotes 1.15 Limit end behavior refers to the behavior of a function as its input values approach the endpoints of its domain. For example, a function may have a different limit at x=0 than it does at x=1, and the way the function behaves as x approaches each of these values can provide important information about the function’s behavior.

When comparing the relative magnitudes of functions and their rates of change, we can use limits to determine which function grows faster or slower than another function. 

Working with the Intermediate Value Theorem (IVT) 1.16 The Intermediate Value Theorem is a fundamental theorem in calculus that states that if a function is continuous on a closed interval [a, b], and if f(a) and f(b) have opposite signs (i.e., f(a) < 0 and f(b) > 0, or vice versa), then there exists at least one point c in the interval (a, b) where the function takes on the value 0 (i.e., f(c) = 0).

The behavior of a function on an interval refers to how the function behaves or changes within that interval. 

Unit 2: Differentiation: Definition and Fundamental Properties

Subtopic Subtopic Number AP Points to understand
Defining Average and Instantaneous Rates of Change at a Point 2.1 Average rates of change using difference quotients refer to the average rate of change of a function over a given interval. 

The difference quotients f(a+h) – f(a)h and f(x) – f(a)x-a express the average rate of change of a function over an interval.

The instantaneous rate of change refers to the rate of change of a function at a specific point. It is the limit of the average rate of change as the interval around the point shrinks to zero.

It is expressed by: h0f(a+h) – f(a)h or xaf(x) – f(a)x-a provided the limit exists.

Defining the Derivative of a Function and Using Derivative Notation 2.2 Derivative of f is the function whose value at x is h0f(x+h) – f(x)h

For y = f(x), the derivative is dydx, f'(x) and y’

The derivative of a function at a point is the slope of the tangent line to the graph of the function at that point. 

Estimating Derivatives of a Function at a Point 2.3 The concept of derivatives enables us to determine instantaneous rates of change by utilizing limits to extrapolate information about rates of change over small intervals.
Connecting Differentiability and Continuity: Determining when Derivatives do and do not exist 2.4 The relationship between differentiability and continuity is that a function must be continuous in order to be differentiable at a point, but it is possible for a function to be continuous without being differentiable at that point.

There are several reasons why a function may not be differentiable at a point, including:

  • The function has a sharp corner or cusp at that point.
  • The function has a vertical tangent (i.e., the slope of the tangent line approaches infinity or negative infinity) at that point.
  • The function has a discontinuity (either a jump or an infinite oscillation) at that point.
Applying the Power Rule 2.5 The power rule is a differentiation rule that applies to functions of the form f(x) = xn, where n is any real number. The power rule states that the derivative of such a function is given by: f'(x) = nx(n-1)
Derivative Rules: Constant, Sum, Difference, and Constant Multiple 2.6 The four main derivative rules are:

  1. Constant Rule: The derivative of a constant is zero.
  2. Sum Rule: The derivative of the sum of two functions is equal to the sum of their individual derivatives.
  3. Difference Rule: The derivative of the difference between two functions is equal to the difference of their individual derivatives.
  4. Constant Multiple Rule: The derivative of a constant times a function is equal to the constant times the derivative of the function.

These rules are essential in finding the derivative of more complex functions and can be combined with other techniques such as the product rule, quotient rule, and chain rule to calculate derivatives of more complicated functions.

Derivatives of cos x, sin x, ex, and ln x 2.7 Understanding a limit as a definition of a derivative

Derivative of cos(x): ddx(cos(x)) = -sin(x)

Derivative of sin(x): ddx(sin(x)) = cos(x)

Derivative of ex: ddx(ex) = ex

Derivative of ln(x): ddx(ln(x)) = 1/x

The Product Rule 2.8 The product rule states that the derivative of the product of two functions, u(x) and v(x), is given by:

ddx(u(x) v(x)) = u(x) (ddx)v(x) + v(x) (ddx)u(x)

To take the derivative of a product of two functions, you multiply the first function by the derivative of the second function, and add to it the product of the second function and the derivative of the first function.

The Quotient Rule 2.9 The quotient rule states that the derivative of the quotient of two functions, u(x) and v(x), is given by:

(ddx)(u(x)v(x)) = [v(x) (ddx)u(x) – u(x) (ddx)v(x)] / [v(x)]2

To take the derivative of a quotient of two functions, you subtract the product of the first function and the derivative of the second function from the product of the second function and the derivative of the first function, and divide the result by the square of the second function.

Finding the Derivatives of Tangent, Cotangent, Secant, and/or Cosecant Functions 2.10
  • The derivative of the tangent function, tan(x), is sec2(x)
  • The derivative of the cotangent function, cot(x), is -csc2(x)
  • The derivative of the secant function, sec(x), is sec(x) tan(x)
  • The derivative of the cosecant function, csc(x), is -csc(x) cot(x)

To apply these rules, you’ll need to be familiar with the basic trigonometric identities, such as

  •  sin2(x)cos2(x) = 1

and the reciprocal identities, such as 

  • sec(x) = 1/cos(x).

Unit 3: Differentiation: Composite, Implicit, and Inverse Functions

Subtopic Subtopic Number AP Points to understand
The Chain Rule 3.1 The chain rule is a calculus formula that is used to differentiate composite functions.

If a function f(x) can be written as a composite function g(h(x)), where g is a function of one variable and h is a function of x, then the chain rule states that the derivative of f(x) is given by:

(f(g(h(x))))’ = f'(g(h(x))) g'(h(x)) h'(x)

In other words, to take the derivative of a composite function, you first find the derivative of the outermost function, evaluated at the inner function. Then you multiply that by the derivative of the inner function, and finally multiply that by the derivative of the input variable.

Implicit Differentiation 3.2 Implicit differentiation is a technique used in calculus to find the derivative of an implicitly defined function. 

An implicitly defined function is a function that is not given in the form y = f(x), but rather in a more complicated equation involving both x and y.

To find the derivative of an implicitly defined function, we differentiate both sides of the equation with respect to x, treating y as a function of x. This means that we use the chain rule to differentiate any terms that involve y.

Differentiating Inverse Functions 3.3 An inverse function is a function that “undoes” another function. More specifically, if f(x) is a function, and there exists a function g(x) such that g(f(x)) = x for all x in the domain of f, then g is called the inverse function of f, and is denoted byf-1(x).

To differentiate an inverse function f-1(x), we use the formula: (f-1)'(x) = 1 / f'(f-1(x))

Differentiating Inverse Trigonometric Functions 3.4 The inverse trigonometric functions are the inverse functions of the trigonometric functions. The inverse trigonometric functions and their derivatives are:

  • arcsin (or sin-1) = 1(1 – x2
  • arccos (or cos-1) = – 1(1 – x2
  • arctan (or tan-1) = 1(1 + x2)
Selecting Procedures for Calculating Derivatives 3.5 Having the ability and skill to understand the procedure of choosing differentiation rules to calculate derivatives. 
Calculating Higher-Order Derivatives 3.6 Higher order derivatives refer to the successive derivatives of a function beyond the first derivative. The nth derivative of a function f(x) is denoted as fⁿ(x) or dⁿy/dxⁿ, where n is a non-negative integer.

The second derivative of a function f(x) is the derivative of its first derivative. It can be represented using the following notations: f”(x) = d²y/dx² = (d/dx) (dy/dx)

The second derivative measures the rate of change of the slope of the function with respect to the independent variable x. 

Unit 4: Contextual Applications of Differentiation

Subtopic Subtopic Number AP Points to understand
Interpreting the Meaning of the Derivative in Context 4.1 The derivative of a function can be understood as the rate of change of the function at a particular point, or the slope of the tangent line at that point. 

This concept can be applied to various real-world situations to express information such as velocity, acceleration, and optimization. 

For example, the derivative of a position function can give us information about an object’s velocity, and the second derivative can tell us about its acceleration.

Straight-Line Motion: Connecting Position, Velocity and Acceleration 4.2 Derivatives are used extensively to solve rectilinear motion problems. Rectilinear motion refers to motion in a straight line, and is commonly used to model the motion of objects moving horizontally or vertically.

To understand how derivatives can be used to solve rectilinear motion problems, we can connect position, velocity, and acceleration through differentiation. 

  • The position of an object at any time t can be represented by a function s(t)
  • The derivative of s(t) with respect to time t gives us the velocity of the object, which we can represent as v(t) = s'(t)
  • The second derivative of s(t) with respect to time t gives us the acceleration of the object, which we can represent as a(t) = s”(t).
Rates of Change in Applied Contexts other than Motion 4.3 Rates of change can be applied to a wide range of real-life situations beyond motion. Here are a few examples:

  1. Population Growth: For instance, the rate of change of a population can be used to determine how quickly a disease will spread within a community.
  2. Finance: Rate of change of an investment’s value over time, or the rate of change of a company’s profits.
  3. Chemistry: To model chemical reactions and determine the rate at which reactants are consumed and products are formed.
  4. Biology: To model biological processes, such as the rate at which a tumor grows or the rate at which an enzyme catalyzes a reaction.
  5. Engineering: For modeling and optimizing processes, such as the rate at which heat is transferred in a system.
  6. Economics: To model economic scenarios, such as the rate of change of a product’s demand.
Introduction to Related Rates 4.4 Related rates is a concept in calculus that deals with finding the rate of change of one quantity with respect to another quantity, when the two quantities are related in some way. 

To solve related rates problems, there are several rules and techniques that can be used:

  1. Implicit differentiation: This technique is used when the relationship between the variables is not given explicitly, but rather implicitly in an equation. 
  2. Chain rule: The chain rule is used when we have a composite function.
  3. Product rule: The product rule is used when we have a product of two variables that are both changing with respect to time.
  4. Quotient rule: The quotient rule is used when we have a quotient of two variables that are both changing with respect to time. 
  5. Trigonometric identities: In some problems, trigonometric identities can be used to simplify expressions and make them easier to differentiate.
Solving Related Rates Problems 4.5 Related rates problems in calculus involve finding the rate of change of one variable with respect to another variable in a given situation, where the two variables are related by an equation or a formula.
Approximating Values of a Function using Local Linearity and Linearization 4.6 Local linearity refers to the property of a function where it can be well approximated by a linear function around a specific point. 

Linearisation is the process of finding this linear approximation of a function around a given point. The linearisation of a function f(x) at a point x=a is the linear function L(x) = f(a) + f'(a)(x-a), where f'(a) is the derivative of f(x) evaluated at x=a.

Using the equation of the tangent line, we can approximate the value of a function at a point nearby to where the tangent line intersects the curve. To do this, we evaluate the tangent line equation at the desired point, which gives an approximation of the function value at that point.

Using L’Hopital’s Rule for determining Limits of Indeterminate forms 4.7 Indeterminate forms are mathematical expressions that cannot be easily evaluated by simply plugging in a value for the variable. Indeterminate forms arise when we encounter expressions like 0/0, infinity/infinity, and 0 times infinity.

L’Hospital’s Rule is a technique used to evaluate limits of indeterminate forms. The rule states that if we have a limit of the form 0/0 or infinity/infinity, we can take the derivative of both the numerator and denominator with respect to the variable, and then evaluate the limit again. This process can be repeated as many times as necessary until we reach a limit that is no longer indeterminate.

The formal statement of L’Hospital’s Rule is:

If f(x) and g(x) are differentiable functions such that xaf(x)=0 and xag(x)=0 or xaf(x)=±∞  and xag(x)=±∞  then:

xaf(x)g(x) =xaf'(x)g'(x)  

provided that the limit on the right-hand side exists or is infinite.

Unit 5: Analytical Applications of Differentiation

Subtopic Subtopic Number AP Points to understand
Using the Mean Value Theorem 5.1 Mean Value Theorem: If f(x) is continuous on [a,b] and differentiable on (a,b), then there exists at least one point c in (a,b) such that: f'(c) = (f(b) – f(a)) / (b – a)

Application of MVT: One common application is to use the MVT to find the average rate of change of a function over an interval when only the values of the function at the endpoints of the interval are known. 

Extreme Value Theorem, Global Versus Local Extrema, and Critical Points 5.2 Extreme Value Theorem states that a continuous function on a closed and bounded interval must have both a global maximum and a global minimum value on that interval. 

Critical point of a function: A critical point of a function is a point where the derivative of the function is either zero or undefined. At a critical point, the slope of the tangent line to the graph of the function is either zero or does not exist.

Global versus Local Extrema: Global extrema refers to the highest or lowest values of a function over its entire domain or over a specific interval. Local extrema, on the other hand, refers to the highest or lowest values of a function in a particular region or interval, & can be found at critical points. 

Determining Intervals on which a function is Increasing or Decreasing 5.3 Increasing and Decreasing Functions: A function is considered increasing if its graph rises as you move from left to right, meaning that its derivative (slope) is positive. Conversely, a function is considered decreasing if its graph falls as you move from left to right, meaning that its derivative (slope) is negative.

The first derivative of a function provides information about the behavior of the function in terms of its increasing and decreasing intervals. Positive first derivatives mean the function is increasing at that point.

The first derivative helps to identify the intervals of the function where it is increasing, decreasing, or having a critical point. 

Using the First Derivative Test to determine Relative (Local) Extrema 5.4 The relative local extrema is defined as a point on a function where the function has a maximum or minimum value in a specific interval, and where the function is either increasing or decreasing on both sides of the extremum.

The first derivative test is a method used to find relative local extrema of a function. It involves taking the derivative of the function and analyzing its sign in intervals around critical points, which are points where the derivative is equal to zero or undefined.

Using the Candidates Test to determine Absolute (Global) Extrema 5.5 A function f(x) is said to have an absolute (global) extremum at a point c in its domain if f(c) is the highest (maximum) or lowest (minimum) value of f(x) over the entire domain of the function.
Determining Concavity of Functions over their Domains 5.6 The concavity of a function over its domain refers to the direction of the curvature of the function’s graph, whether it is upward or downward. 

A point of inflection on a graph is where the concavity of the graph changes from upward to downward, or vice versa. 

The second derivative of a function can be used to identify these points of inflection and determine the intervals of upward or downward concavity for the function’s graph.

Using the Second Derivative Test to determine Extrema 5.7 To determine whether a critical point is a local minimum or a local maximum, we can use the Second Derivative Test, which involves finding the second derivative of the function at the critical point.

The Second Derivative Test states that if the second derivative of a function evaluated at a critical point is:

  • Positive, the function has a local minimum at that point. 
  • Negative, the function has a local maximum at that point. 
  • Zero, the Second Derivative Test is inconclusive, and we need to use additional methods to determine the nature of the critical point.
Sketching Graphs of Functions and their Derivatives 5.8 It is important to understand the key features of both graphs to fully comprehend the concepts of calculus. Here are some of the key features of graphs of functions and their derivatives:

Graphs of Functions:

  • Domain and Range: the set of all possible inputs and outputs of the function, respectively
  • Intercepts: points where the graph intersects with the x- or y-axis
  • Symmetry: whether the graph is symmetric with respect to the x-axis, y-axis, or origin
  • Asymptotes: lines that the graph approaches but never touches
  • Extrema: maximum and minimum points on the graph
  • Inflection points: points where the concavity of the graph changes

Graphs of Derivatives:

  • Domain: the set of all possible inputs for the derivative function
  • Increasing and Decreasing Intervals: intervals where the function is increasing or decreasing, respectively
  • Local Extrema: points where the derivative is equal to zero and changes sign (i.e., from positive to negative or vice versa)
  • Concavity: whether the function is concave up or down
  • Inflection points: points where the concavity of the graph changes
Connecting a Function, its first derivative and its second derivative 5.9 The first derivative gives information about the direction of change of the function, while the second derivative provides information about the rate at which the function is changing. The second derivative gives the rate at which the slope of the tangent line is changing.

Together, these derivatives can help us understand the behavior of a function in great detail.

Introduction to Optimization Problems 5.10 Optimization problems typically involve finding the maximum or minimum value of a function subject to certain constraints or conditions. These problems often require the use of differentiation and critical point analysis to determine the optimum solution.
Solving Optimization Problems 5.11 Maximum values are often used to find the optimal solution for a real-world situation. This could involve determining the maximum area of a rectangle with a fixed perimeter or finding the maximum profit for a business given certain constraints.

For minimum values, example: to find the minimum cost of producing a certain number of items, we would need to use calculus to find the minimum point of the cost function, where the derivative is equal to zero. This minimum point would represent the lowest possible cost that can be achieved for producing that number of items.

Exploring Behaviors of Implicit Relations 5.12 An implicit relation is a relationship between two variables that is not explicitly defined as a function of one variable in terms of the other.

Some common behaviors of implicit relations that are studied in AP Calculus BC include:

  1. Finding the slope: The slope of an implicit relation can be determined by taking the derivative of both sides of the equation with respect to one of the variables.
  2. Finding the concavity: The concavity of an implicit relation can be determined by taking the second derivative of both sides of the equation with respect to one of the variables.
  3. Finding critical points: Critical points of an implicit relation can be found by solving the equation for when the first derivative is equal to zero.
  4. Finding inflection points: Inflection points of an implicit relation can be found by solving the equation for when the second derivative changes sign.
  5. Sketching the curve: By using the above techniques and the behavior of the curve at the critical points and inflection points, a sketch of the curve can be made.

The second derivative can be used to analyze the concavity and inflection points of the original function. Second derivatives involving implicit differentiation refer to the process of finding the second derivative of a function that is defined implicitly by an equation involving both x and y variables.

Unit 6: Integration and Accumulation of Change

Subtopic Subtopic Number AP Points to understand
Exploring Accumulations of Change 6.1 Accumulation of change refers to the process of finding the total change or net change in a function over a certain interval. This process is also known as “integration” and is denoted by the symbol ∫.

The positive accumulation of change refers to the total amount by which a function increases over a given interval. It is calculated by finding the definite integral of the function over the interval, where the function values are positive.

The negative accumulation of change refers to the total amount by which a function decreases over a given interval. It is calculated by finding the definite integral of the function over the interval, where the function values are negative.

Approximating Areas with Riemann Sums 6.2 Definite Integrals: It is a type of integral where the limits of integration (lower and upper bounds) are specified. It represents the area between the graph of a function and the x-axis over a given interval. 

Approximating definite integrals: The most common methods used are the trapezoidal rule, Simpson’s rule, and the midpoint rule. These methods involve dividing the interval of integration into smaller subintervals, and then using the values of the function at the endpoints of these subintervals to estimate the area.

Riemann’s sums are a way to approximate the value of a definite integral by dividing the interval of integration into subintervals and then approximating the area of each subinterval using a rectangle. The sum of these rectangle areas gives an estimate of the area under the curve of the function being integrated. 

Riemann Sums, Summation Notation, and Definite Integral Notation 6.3 Summation notation, also known as sigma notation, is a concise way of representing a sum of a sequence of terms. It is denoted by the Greek letter sigma, ∑, followed by the expression to be summed, and the range of the index variable.

Definite integral notation is used to represent the area under a curve between two points on the x-axis. It is denoted by the symbol ∫, and the expression to be integrated, with the integration limits (upper and lower bounds of integration) written as subscripts and superscripts.

The Fundamental Theorem of Calculus and Accumulation Functions 6.4 An accumulation function is a function that gives the total amount of some quantity accumulated over a certain period of time. It is usually defined using definite integrals.

Let f(x) be a continuous function on an interval [a,b]. The accumulation function F(x) is defined as:

F(x) = [a,x] f(t) dt 

Interpreting the Behaviour of Accumulation Functions Involving Area 6.5 Accumulation functions involving area refer to the integration of a function over a certain interval. The resulting value represents the net area between the function and the x-axis within that interval.
Applying Properties of Definite Integrals 6.6 Evaluating definite integrals using geometry involves using geometric shapes and properties to find the area under a curve. This includes using basic geometric shapes like rectangles, trapezoids, and triangles to estimate the area, as well as more complex shapes like circles and ellipses. 

In AP Calculus BC, definite integrals have several properties that are essential to understand.

  1. Constant multiple rule: The integral of a constant times a function is equal to the constant times the integral of the function. In other words, for any constant c, c f(x) dx =cf(x) dx
  2. Sum and difference rules: The integral of the sum (or difference) of two functions is equal to the sum (or difference) of the integrals of the two functions. In other words, [f(x) + g(x)] dx =f(x)dx + g(x)dx , and [f(x) – g(x)] dx =f(x)dx – g(x)dx
  3. Reversal of limits of integration: When the limits of integration are reversed, the sign of the integral changes. In other words, if a and b are the limits of integration, then abf(x)dx = – baf(x)dx
  4. Integration over adjacent intervals: The integral of a function over an interval can be split into the sum of integrals over adjacent subintervals. In other words, if c is a point in the interval [a,b], thenabf(x)dx =  acf(x)dx +cbf(x)dx
The Fundamental Theorem of Calculus and Definite Integrals 6.7 Antiderivatives are functions that when differentiated, give a given function. In other words, if f(x) is a function, an antiderivative of f(x) is a function F(x) whose derivative is f(x).

To evaluate a definite integral using antiderivatives and the Fundamental Theorem of Calculus, you first find an antiderivative of the integrand, and then evaluate that antiderivative at the limits of integration and subtract the results. This process is often called “evaluating the integral by antiderivatives.”

Finding Antiderivatives and Indefinite Integrals: Basic Rules and Notations 6.8 Determining antiderivatives in AP Calculus BC involves finding the function that, when differentiated, gives the original function. This is also known as integration.

Indefinite integral, also called an antiderivative, is the family of functions that differ by a constant that can be obtained by integrating a given function. It is written using the integral symbol and the function being integrated.

Differentiation rules are used to find the derivative of a function. Some common rules include the power rule, product rule, quotient rule, chain rule, and trigonometric rules.

Closed-form antiderivatives are functions that can be expressed using a finite number of elementary functions, such as polynomials, exponential functions, logarithmic functions, and trigonometric functions. However, not all functions have closed-form antiderivatives, and in these cases, numerical methods or approximation techniques may be used to approximate the value of the integral.

Integrating using Substitution  6.9 Integration by substitution is a technique used in calculus to simplify the integration of functions that are not easily integrable. The method involves using a substitution to replace a complicated expression in the integrand with a simpler one. The substitution is chosen such that it transforms the original integral into a new integral that can be easily evaluated.

Integration by substitution can also be used for definite integrals. In this case, the limits of integration must be adjusted to account for the change in variable. The new limits are found by substituting the original limits into the substitution equation.

The formula for integration by substitution is:

f(g(x)) g'(x) dx =  f(u)du  (where u = g(x)) 

This formula states that if we can write the integrand as f(g(x)) g'(x), we can make the substitution u = g(x) and rewrite the integral in terms of u.

Integration Functions using Long Division and Completing the Square 6.10 Integration Functions using Long Division and Completing the Square: This technique is used to integrate rational functions with polynomials of higher degree in the numerator or denominator. In long division, we divide the numerator by the denominator, and in completing the square, we manipulate the function by adding and subtracting a term to make it integrable.
Integrating Using Integration by Parts 6.11 Integrating Using Integration by Parts: This technique is used to integrate the product of two functions. Integration by parts involves choosing a u and dv term and using the product rule of differentiation to find the integral.
Integrating using Linear Partial Fractions 6.12 Integrating using Linear Partial Fractions: This technique is used to integrate rational functions with distinct linear factors in the denominator. We first decompose the function into partial fractions, where each fraction has a linear factor in the denominator, and then integrate each partial fraction separately.

A rational function is written in the form:

f(x) = P(x) / Q(x)

where P(x) and Q(x) are polynomials and Q(x) is not the zero polynomial.

Evaluating Improper Integrals 6.13 An improper integral is defined as the integral of a function over an infinite interval or over an interval that includes one or more points where the function is undefined or approaches infinity.

Specifically, an improper integral is a limit of a definite integral as one or both of the limits of integration become infinite or as one or more points within the interval of integration approach a singular point (such as a vertical asymptote).

An improper integral can be expressed as:

af(x) dx =tatf(x) dx   or

a f(x) dx =cbacf(x) dx+da+dbf(x) dx

where “a” and “b” are finite or infinite limits of integration and “f(x)” is the function being integrated. The limits “t”, “c”, and “d” approach infinity or singular points within the interval of integration.

Improper integrals can be evaluated using various techniques, such as comparison test, limit comparison test, integral test, or the convergence/divergence test.

Selecting Techniques for Antidifferentiation 6.14 Evaluating and understanding the techniques to antidifferentiate such as:

  1. Integration by substitution
  2. Integration by parts
  3. Partial fraction decomposition
  4. Trigonometric substitution
  5. Improper integration

Unit 7: Differential Equations

Subtopic Subtopic Number AP Points to understand
Modeling situations with Differential Equations 7.1 Differential equations are mathematical equations that involve derivatives of an unknown function. In AP Calculus BC, students study both ordinary differential equations (ODEs) and partial differential equations (PDEs).

They relate a function of an independent variable and the function’s derivatives.

Verifying Solutions for Differential Equations 7.2 Verifying solutions for differential equations involves substituting a given function into the differential equation and checking that it satisfies the equation. This can be done by taking derivatives of the function and plugging them into the equation.
Sketching Slope Fields 7.3 Slope fields are graphical representations of differential equations that show the direction of the solution curves at different points in the plane. They provide information about the behavior of solutions to the differential equation, including whether they are increasing or decreasing, and the location of critical points.
Reasoning using Slope Fields 7.4 Reasoning using slope fields involves using the information provided by the slope field to make predictions about the behavior of solutions to the differential equation.
Approximating Solutions using Euler’s Method 7.5 Euler’s method is a numerical method for approximating solutions to differential equations. It involves using the slope field to estimate the slope of the solution curve at a given point, and then using that slope to find the value of the function at a nearby point. This process is repeated iteratively to approximate the solution curve.
Finding General Solutions using Separation of Variables 7.6 Anti-differentiation is a technique used to find the general solution of a differential equation. Anti-differentiation involves reversing the process of differentiation by finding an indefinite integral of a given function.

To find the general solution of a differential equation, we use anti-differentiation to obtain an expression that contains an arbitrary constant (known as the constant of integration). This expression represents the family of all possible solutions to the differential equation.

Finding Particular Solutions using Initial Conditions and Separation of Variables 7.7 A general solution represents an infinite number of solutions to a differential equation, while a particular solution represents a single solution that passes through a specific point. 
Exponential Models with Differential Equations 7.8 Exponential models with differential equations are also studied, which involve equations that describe the rate of change of a quantity proportional to its current value. 

Specific applications of finding general and particular solutions to differential equations, such as exponential growth and decay models which are commonly used in various fields including biology, economics, and physics will be studied.

Logistic Models with Differential Equations 7.9 The logistic growth model is a mathematical equation that describes how a population grows over time. It takes into account the population’s current size, the maximum size that the population can reach (known as the carrying capacity), and the rate at which the population grows.

The purpose of the limiting value, or carrying capacity, in this logistic differential equation is to represent the maximum sustainable population size that can be supported by the available resources in a given environment. 

Unit 8: Applications of Integration

Subtopic Subtopic Number AP Points to understand
Finding the Average Value of a Function on an Interval 8.1 Finding the average value of a function on an interval involves using integration to calculate the mean of a function over a specified interval. This process is done by dividing the integral of the function over the given interval by the length of the interval.

The average value of a continuous function f on an interval [a, b] in AP Calculus BC is given by the formula:

avg (f) = 1(b-a)abf(x) dx

This formula calculates the mean value of f over the interval [a, b] and represents the height of a horizontal line that would have the same area as the region bounded by the graph of f, the x-axis, and the vertical lines x = a and x = b.

Connecting Position, Velocity and Acceleration of Functions using Integrals 8.2 Using integration to find position functions from velocity functions, and velocity functions from acceleration functions, by taking the integral of the corresponding derivative function.

The definite integral of velocity in rectilinear motion represents the displacement of an object over a given time interval. If v(t) represents the velocity of an object at time t, then the displacement of the object over the interval [a, b] is given by the definite integral:

abv(t) dt

Using Accumulation Functions and Definite Integrals in Applied Contexts 8.3 The definite integral is used to represent the accumulation of a quantity over a given interval. For example, the definite integral of the velocity function over a time interval represents the total displacement of an object during that interval.

The definite integral of the rate of change of a quantity represents the total change in that quantity over a given interval. For example, the definite integral of the rate of change of a population over a time interval represents the total change in the population during that interval.

Finding the Area between Curves expressed as Functions of x 8.4 Finding the area between two curves expressed as functions of x involves using definite integrals to calculate the difference in the areas bounded by the two curves. This is done by integrating the difference between the two functions with respect to x.
Finding the Area between Curves expressed as Functions of y 8.5 Same as the method described above but between the two functions with respect to y.
Finding the Area between Curves that Intersect at more than Two Points 8.6 In some cases, the two curves that bound the region of interest may intersect at more than two points. In this case, it is necessary to break the region up into smaller subregions, each of which can be integrated using the methods described above.
Volumes with Cross Sections: Squares and Rectangles 8.7 Volumes of solids with known cross sections can be calculated using definite integrals. This involves integrating the area of the cross section along the length of the solid, which yields the total volume. The formula of the area of these shapes can also be used to make the process simpler. 
Volumes with Cross Sections: Triangles and Semi circles 8.8 To calculate the volume of such solids, one integrates the area of the cross section along the length of the solid, just