Calculus BC course outline

CHA-1.A.1 · Calculus uses limits to understand and model dynamic change.

CHA-1.A.2 · Because an average rate of change divides the change in one variable by the change in another, the average rate of change is undefined at a point where the change in the independent variable would be zero.

CHA-1.A.3 · The limit concept allows us to define instantaneous rate of change in terms of average rates of change.

LIM-1.A.1 · Given a function f, the limit of f(x) as x approaches c is a real number R if f(x) can be made arbitrarily close to R by taking x sufficiently close to c (but not equal to c). If the limit exists and is a real number, then the common notation is lim(x→c) f(x)=R.

LIM-1.B.1 · A limit can be expressed in multiple ways, including graphically, numerically, and analytically.

LIM-1.C.1 · The concept of a limit includes one sided limits.

LIM-1.C.2 · Graphical information about a function can be used to estimate limits.

LIM-1.C.3 · Because of issues of scale, graphical representations of functions may miss important function behavior.

LIM-1.C.4 · A limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.

LIM-1.C.5 · Numerical information can be used to estimate limits.

LIM-1.C.1 · The concept of a limit includes one sided limits.

LIM-1.C.2 · Graphical information about a function can be used to estimate limits.

LIM-1.C.3 · Because of issues of scale, graphical representations of functions may miss important function behavior.

LIM-1.C.4 · A limit might not exist for some functions at particular values of x. Some ways that the limit might not exist are if the function is unbounded, if the function is oscillating near this value, or if the limit from the left does not equal the limit from the right.

LIM-1.C.5 · Numerical information can be used to estimate limits.

LIM-1.D.1 · One-sided limits can be determined analytically or graphically.

LIM-1.D.2 · Limits of sums, differences, products, quotients, and composite functions can be found using limit theorems.

LIM-1.E.1 · It may be necessary or helpful to rearrange expressions into equivalent forms before evaluating limits.

LIM-1.E.2 · The limit of a function may be found by using the squeeze theorem.

LIM-1.E.1 · It may be necessary or helpful to rearrange expressions into equivalent forms before evaluating limits.

LIM-1.E.2 · The limit of a function may be found by using the squeeze theorem.

LIM-2.A.1 · Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

LIM-2.A.2 · A function f is continuous at x=c provided that f(c) exists, lim(x→c) f(x) exists, and lim(x→c) f(x)=f(c).

LIM-2.A.1 · Types of discontinuities include removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.

LIM-2.A.2 · A function f is continuous at x=c provided that f(c) exists, lim(x→c) f(x) exists, and lim(x→c) f(x)=f(c).

LIM-2.B.1 · A function is continuous on an interval if the function is continuous at each point in the interval.

LIM-2.B.2 · Polynomial, rational, power, exponential, logarithmic, and trigonometric functions are continuous on all points in their domains.

LIM-2.C.1 · If the limit of a function exists at a discontinuity in its graph, then it is possible to remove the discontinuity by defining or redefining the value of the function at that point, so it equals the value of the limit of the function as x approaches that point.

LIM-2.C.2 · In order for a piecewise-defined function to be continuous at a boundary to the partition of its domain, the value of the expression defining the function on one side of the boundary must equal the value of the expression defining the other side of the boundary, as well as the value of the function at the boundary.

LIM-2.D.1 · The concept of a limit can be extended to include infinite limits.

LIM-2.D.2 · Asymptotic and unbounded behavior of functions can be described and explained using limits.

LIM-2.D.3 · The concept of a limit can be extended to include limits at infinity.

LIM-2.D.4 · Limits at infinity describe end behavior.

LIM-2.D.5 · Relative magnitudes of functions and their rates of change can be compared using limits.

LIM-2.D.1 · The concept of a limit can be extended to include infinite limits.

LIM-2.D.2 · Asymptotic and unbounded behavior of functions can be described and explained using limits.

LIM-2.D.3 · The concept of a limit can be extended to include limits at infinity.

LIM-2.D.4 · Limits at infinity describe end behavior.

LIM-2.D.5 · Relative magnitudes of functions and their rates of change can be compared using limits.

FUN-1.A.1 · If f is a continuous function on the closed interval [a, b] and d is a number between f (a) and f (b), then the Intermediate Value Theorem guarantees that there is at least one number c between a and b, such that f(c) = d.

CHA-2.A.1 · 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.

CHA-2.B.1 · The instantaneous rate of change of a function at x=a can be expressed by lim(h→0) [f(a+h)−f(a)]/h or lim(x→a) [f(x)−f(a)]/(x−a), provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted f′(a).

CHA-2.B.2 · The derivative of f is the function whose value at x is lim(h→0) [f(x+h)−f(x)]/h, provided this limit exists.

CHA-2.B.3 · For y=f(x), notations for the derivative include dy/dx, f′(x), and y′.

CHA-2.B.4 · The derivative can be represented graphically, numerically, analytically, and verbally.

CHA-2.B.1 · The instantaneous rate of change of a function at x=a can be expressed by lim(h→0) [f(a+h)−f(a)]/h or lim(x→a) [f(x)−f(a)]/(x−a), provided the limit exists. These are equivalent forms of the definition of the derivative and are denoted f′(a).

CHA-2.B.2 · The derivative of f is the function whose value at x is lim(h→0) [f(x+h)−f(x)]/h, provided this limit exists.

CHA-2.B.3 · For y=f(x), notations for the derivative include dy/dx, f′(x), and y′.

CHA-2.B.4 · The derivative can be represented graphically, numerically, analytically, and verbally.

CHA-2.C.1 · The derivative of a function at a point is the slope of the line tangent to a graph of the function at that point.

CHA-2.D.1 · The derivative at a point can be estimated from information given in tables or graphs.

CHA-2.D.2 · Technology can be used to calculate or estimate the value of a derivative of a function at a point.

FUN-2.A.1 · If a function is differentiable at a point, then it is continuous at that point. In particular, if a point is not in the domain of f, then it is not in the domain of f′.

FUN-2.A.2 · A continuous function may fail to be differentiable at a point in its domain.

FUN-3.A.1 · Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form f(x)=x^r.

FUN-3.A.2 · Sums, differences, and constant multiples of functions can be differentiated using derivative rules.

FUN-3.A.3 · The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.

FUN-3.A.4 · Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.

FUN-3.A.1 · Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form f(x)=x^r.

FUN-3.A.2 · Sums, differences, and constant multiples of functions can be differentiated using derivative rules.

FUN-3.A.3 · The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.

FUN-3.A.4 · Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.

FUN-3.A.1 · Direct application of the definition of the derivative and specific rules can be used to calculate the derivative for functions of the form f(x)=x^r.

FUN-3.A.2 · Sums, differences, and constant multiples of functions can be differentiated using derivative rules.

FUN-3.A.3 · The power rule combined with sum, difference, and constant multiple properties can be used to find the derivatives for polynomial functions.

FUN-3.A.4 · Specific rules can be used to find the derivatives for sine, cosine, exponential, and logarithmic functions.

LIM-3.A.1 · In some cases, recognizing an expression for the definition of the derivative of a function whose derivative is known offers a strategy for determining a limit.

FUN-3.B.1 · Derivatives of products of differentiable functions can be found using the product rule.

FUN-3.B.2 · Derivatives of quotients of differentiable functions can be found using the quotient rule.

FUN-3.B.3 · Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.

FUN-3.B.1 · Derivatives of products of differentiable functions can be found using the product rule.

FUN-3.B.2 · Derivatives of quotients of differentiable functions can be found using the quotient rule.

FUN-3.B.3 · Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.

FUN-3.B.1 · Derivatives of products of differentiable functions can be found using the product rule.

FUN-3.B.2 · Derivatives of quotients of differentiable functions can be found using the quotient rule.

FUN-3.B.3 · Rearranging tangent, cotangent, secant, and cosecant functions using identities allows differentiation using derivative rules.

FUN-3.C.1 · The chain rule provides a way to differentiate composite functions.

FUN-3.D.1 · The chain rule is the basis for implicit differentiation.

FUN-3.E.1 · The chain rule and definition of an inverse function can be used to find the derivative of an inverse function, provided the derivative exists.

FUN-3.E.2 · The chain rule applied with the definition of an inverse function, or the formula for the derivative of an inverse function, can be used to find the derivatives of inverse trigonometric functions.

FUN-3.E.1 · The chain rule and definition of an inverse function can be used to find the derivative of an inverse function, provided the derivative exists.

FUN-3.E.2 · The chain rule applied with the definition of an inverse function, or the formula for the derivative of an inverse function, can be used to find the derivatives of inverse trigonometric functions.

FUN-3.F.1 · Differentiating f ′ produces the second derivative f ″, provided the derivative of f ′ exists; repeating this process produces higher-order derivatives of f .

FUN-3.F.2 · Higher-order derivatives are represented with a variety of notations. For y=f(x), notations for the second derivative include d²y/dx², f″(x), and y″. Higher-order derivatives can be denoted dⁿy/dxⁿ or f⁽ⁿ⁾(x).

CHA-3.A.1 · The derivative of a function can be interpreted as the instantaneous rate of change with respect to its independent variable.

CHA-3.A.2 · The derivative can be used to express information about rates of change in applied contexts.

CHA-3.A.3 · The unit for f'(x) is the unit for f divided by the unit for x.

CHA-3.B.1 · The derivative can be used to solve rectilinear motion problems involving position, speed, velocity, and acceleration.

CHA-3.C.1 · The derivative can be used to solve problems involving rates of change in applied contexts.

CHA-3.D.1 · The chain rule is the basis for differentiating variables in a related rates problem with respect to the same independent variable.

CHA-3.D.2 · Other differentiation rules, such as the product rule and the quotient rule, may also be necessary to differentiate all variables with respect to the same independent variable.

CHA-3.E.1 · The derivative can be used to solve related rates problems; that is, finding a rate at which one quantity is changing by relating it to other quantities whose rates of change are known.

CHA-3.F.1 · The tangent line is the graph of a locally linear approximation of the function near the point of tangency.

CHA-3.F.2 · For a tangent line approximation, the function’s behavior near the point of tangency may determine whether a tangent line value is an underestimate or an overestimate of the corresponding function value.

LIM-4.A.1 · When the ratio of two functions tends to 0/0 or ∞/∞ in the limit, such forms are said to be indeterminate.

LIM-4.A.2 · Limits of the indeterminate forms 0/0 or ∞/∞ may be evaluated using L’Hospital’s Rule.

FUN-1.B.1 · If a function f is continuous over the interval [a, b] and differentiable over the interval (a, b), then the Mean Value Theorem guarantees a point within that open interval where the instantaneous rate of change equals the average rate of change over the interval.

FUN-1.C.1 · If a function f is continuous over the interval [a, b], then the Extreme Value Theorem guarantees that f has at least one minimum value and at least one maximum value on [a, b].

FUN-1.C.2 · A point on a function where the first derivative equals zero or fails to exist is a critical point of the function.

FUN-1.C.3 · All local (relative) extrema occur at critical points of a function, though not all critical points are local extrema.

FUN-4.A.1 · The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

FUN-4.A.2 · The first derivative of a function can determine the location of relative (local) extrema of the function.

FUN-4.A.3 · Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

FUN-4.A.4 · The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.

FUN-4.A.5 · The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.

FUN-4.A.6 · The second derivative of a function may be used to locate points of inflection for the graph of the original function.

FUN-4.A.7 · The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.

FUN-4.A.8 · When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

FUN-4.A.9 · Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

FUN-4.A.10 · Graphical, numerical, and analytical information from f' and f" can be used to predict and explain the behavior of f.

FUN-4.A.11 · Key features of the graphs of f, f', and f" are related to one another.

FUN-4.A.1 · The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

FUN-4.A.2 · The first derivative of a function can determine the location of relative (local) extrema of the function.

FUN-4.A.3 · Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

FUN-4.A.4 · The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.

FUN-4.A.5 · The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.

FUN-4.A.6 · The second derivative of a function may be used to locate points of inflection for the graph of the original function.

FUN-4.A.7 · The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.

FUN-4.A.8 · When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

FUN-4.A.9 · Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

FUN-4.A.10 · Graphical, numerical, and analytical information from f' and f" can be used to predict and explain the behavior of f.

FUN-4.A.11 · Key features of the graphs of f, f', and f" are related to one another.

FUN-4.A.1 · The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

FUN-4.A.2 · The first derivative of a function can determine the location of relative (local) extrema of the function.

FUN-4.A.3 · Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

FUN-4.A.4 · The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.

FUN-4.A.5 · The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.

FUN-4.A.6 · The second derivative of a function may be used to locate points of inflection for the graph of the original function.

FUN-4.A.7 · The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.

FUN-4.A.8 · When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

FUN-4.A.9 · Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

FUN-4.A.10 · Graphical, numerical, and analytical information from f' and f" can be used to predict and explain the behavior of f.

FUN-4.A.11 · Key features of the graphs of f, f', and f" are related to one another.

FUN-4.A.1 · The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

FUN-4.A.2 · The first derivative of a function can determine the location of relative (local) extrema of the function.

FUN-4.A.3 · Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

FUN-4.A.4 · The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.

FUN-4.A.5 · The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.

FUN-4.A.6 · The second derivative of a function may be used to locate points of inflection for the graph of the original function.

FUN-4.A.7 · The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.

FUN-4.A.8 · When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

FUN-4.A.9 · Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

FUN-4.A.10 · Graphical, numerical, and analytical information from f' and f" can be used to predict and explain the behavior of f.

FUN-4.A.11 · Key features of the graphs of f, f', and f" are related to one another.

FUN-4.A.1 · The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

FUN-4.A.2 · The first derivative of a function can determine the location of relative (local) extrema of the function.

FUN-4.A.3 · Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

FUN-4.A.4 · The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.

FUN-4.A.5 · The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.

FUN-4.A.6 · The second derivative of a function may be used to locate points of inflection for the graph of the original function.

FUN-4.A.7 · The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.

FUN-4.A.8 · When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

FUN-4.A.9 · Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

FUN-4.A.10 · Graphical, numerical, and analytical information from f' and f" can be used to predict and explain the behavior of f.

FUN-4.A.11 · Key features of the graphs of f, f', and f" are related to one another.

FUN-4.A.1 · The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

FUN-4.A.2 · The first derivative of a function can determine the location of relative (local) extrema of the function.

FUN-4.A.3 · Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

FUN-4.A.4 · The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.

FUN-4.A.5 · The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.

FUN-4.A.6 · The second derivative of a function may be used to locate points of inflection for the graph of the original function.

FUN-4.A.7 · The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.

FUN-4.A.8 · When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

FUN-4.A.9 · Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

FUN-4.A.10 · Graphical, numerical, and analytical information from f' and f" can be used to predict and explain the behavior of f.

FUN-4.A.11 · Key features of the graphs of f, f', and f" are related to one another.

FUN-4.A.1 · The first derivative of a function can provide information about the function and its graph, including intervals where the function is increasing or decreasing.

FUN-4.A.2 · The first derivative of a function can determine the location of relative (local) extrema of the function.

FUN-4.A.3 · Absolute (global) extrema of a function on a closed interval can only occur at critical points or at endpoints.

FUN-4.A.4 · The graph of a function is concave up (down) on an open interval if the function’s derivative is increasing (decreasing) on that interval.

FUN-4.A.5 · The second derivative of a function provides information about the function and its graph, including intervals of upward or downward concavity.

FUN-4.A.6 · The second derivative of a function may be used to locate points of inflection for the graph of the original function.

FUN-4.A.7 · The second derivative of a function may determine whether a critical point is the location of a relative (local) maximum or minimum.

FUN-4.A.8 · When a continuous function has only one critical point on an interval on its domain and the critical point corresponds to a relative (local) extremum of the function on the interval, then that critical point also corresponds to the absolute (global) extremum of the function on the interval.

FUN-4.A.9 · Key features of functions and their derivatives can be identified and related to their graphical, numerical, and analytical representations.

FUN-4.A.10 · Graphical, numerical, and analytical information from f' and f" can be used to predict and explain the behavior of f.

FUN-4.A.11 · Key features of the graphs of f, f', and f" are related to one another.

FUN-4.B.1 · The derivative can be used to solve optimization problems; that is, finding a minimum or maximum value of a function on a given interval.

FUN-4.C.1 · Minimum and maximum values of a function take on specific meanings in applied contexts.

FUN-4.D.1 · A point on an implicit relation where the first derivative equals zero or does not exist is a critical point of the function.

FUN-4.E.1 · Applications of derivatives can be extended to implicitly defined functions.

FUN-4.E.2 · Second derivatives involving implicit differentiation may be relations of x, y, and dy/dx.

CHA-4.A.1 · The area of the region between the graph of a rate of change function and the x axis gives the accumulation of change.

CHA-4.A.2 · In some cases, accumulation of change can be evaluated by using geometry.

CHA-4.A.3 · If a rate of change is positive (negative) over an interval, then the accumulated change is positive (negative).

CHA-4.A.4 · The unit for the area of a region defined by rate of change is the unit for the rate of change multiplied by the unit for the independent variable.

LIM-5.A.1 · Definite integrals can be approximated for functions that are represented graphically, numerically, analytically, and verbally.

LIM-5.A.2 · Definite integrals can be approximated using a left Riemann sum, a right Riemann sum, a midpoint Riemann sum, or a trapezoidal sum; approximations can be computed using either uniform or nonuniform partitions.

LIM-5.A.3 · Definite integrals can be approximated using numerical methods, with or without technology.

LIM-5.A.4 · Depending on the behavior of a function, it may be possible to determine whether an approximation for a definite integral is an underestimate or overestimate for the value of the definite integral.

LIM-5.B.1 · The limit of an approximating Riemann sum can be interpreted as a definite integral.

LIM-5.B.2 · A Riemann sum, which requires a partition of an interval I, is the sum of products, each of which is the value of the function at a point in a subinterval multiplied by the length of that subinterval of the partition.

LIM-5.C.1 · The definite integral of a continuous function f over the interval [a,b], denoted by ∫ₐᵇ f(x) dx, is the limit of Riemann sums as the widths of the subintervals approach 0. That is, ∫ₐᵇ f(x) dx = lim(max Δxᵢ→0) Σᵢ₌₁ⁿ f(xᵢ*)Δxᵢ, where n is the number of subintervals, Δxᵢ is the width of the ith subinterval, and xᵢ* is a value in the ith subinterval.

LIM-5.C.2 · A definite integral can be translated into the limit of a related Riemann sum, and the limit of a Riemann sum can be written as a definite integral.

FUN-5.A.1 · The definite integral can be used to define new functions.

FUN-5.A.2 · If f is a continuous function on an interval containing a, then d/dx {∫ₐˣ f(t) dt}=f(x), where x is in the interval.

FUN-5.A.3 · Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as g(x)=∫ₐˣ f(t) dt.

FUN-5.A.1 · The definite integral can be used to define new functions.

FUN-5.A.2 · If f is a continuous function on an interval containing a, then d/dx {∫ₐˣ f(t) dt}=f(x), where x is in the interval.

FUN-5.A.3 · Graphical, numerical, analytical, and verbal representations of a function f provide information about the function g defined as g(x)=∫ₐˣ f(t) dt.

FUN-6.A.1 · In some cases, a definite integral can be evaluated by using geometry and the connection between the definite integral and area.

FUN-6.A.2 · Properties of definite integrals include the integral of a constant times a function, the integral of the sum of two functions, reversal of limits of integration, and the integral of a function over adjacent intervals.

FUN-6.A.3 · The definition of the definite integral may be extended to functions with removable or jump discontinuities.

FUN-6.B.1 · An antiderivative of a function f is a function g whose derivative is f.

FUN-6.B.2 · If a function f is continuous on an interval containing a, the function defined by F(x)=∫ₐˣ f(t) dt is an antiderivative of f for x in the interval.

FUN-6.B.3 · If f is continuous on the interval [a,b] and F is an antiderivative of f, then ∫ₐᵇ f(x) dx=F(b)−F(a).

FUN-6.C.1 · ∫ f(x) dx is an indefinite integral of the function f and can be expressed as ∫ f(x) dx=F(x)+C, where F′(x)=f(x) and C is any constant.

FUN-6.C.2 · Differentiation rules provide the foundation for finding antiderivatives.

FUN-6.C.3 · Many functions do not have closed-form antiderivatives.

FUN-6.D.1 · Substitution of variables is a technique for finding antiderivatives.

FUN-6.D.2 · For a definite integral, substitution of variables requires corresponding changes to the limits of integration.

FUN-6.D.3 · Techniques for finding antiderivatives include rearrangements into equivalent forms, such as long division and completing the square.

FUN-6.D.1 · Substitution of variables is a technique for finding antiderivatives.

FUN-6.D.2 · For a definite integral, substitution of variables requires corresponding changes to the limits of integration.

FUN-6.D.3 · Techniques for finding antiderivatives include rearrangements into equivalent forms, such as long division and completing the square.

FUN-6.E.1 · Integration by parts is a technique for finding antiderivatives.

FUN-6.F.1 · Some rational functions can be decomposed into sums of ratios of linear, nonrepeating factors to which basic integration techniques can be applied.

LIM-6.A.1 · An improper integral is an integral that has one or both limits infinite or has an integrand that is unbounded in the interval of integration.

LIM-6.A.2 · Improper integrals can be determined using limits of definite integrals.

FUN-7.A.1 · Differential equations relate a function of an independent variable and the function’s derivatives.

FUN-7.B.1 · Derivatives can be used to verify that a function is a solution to a given differential equation.

FUN-7.B.2 · There may be infinitely many solutions to a differential equation.

FUN-7.C.1 · A slope field is a graphical representation of a differential equation on a finite set of points in the plane.

FUN-7.C.2 · Slope fields provide information about the behavior of solutions to first-order differential equations.

FUN-7.C.3 · Solutions to differential equations are functions or families of functions.

FUN-7.C.4 · Euler’s method provides a procedure for approximating a solution to a differential equation or a point on a solution curve.

FUN-7.C.1 · A slope field is a graphical representation of a differential equation on a finite set of points in the plane.

FUN-7.C.2 · Slope fields provide information about the behavior of solutions to first-order differential equations.

FUN-7.C.3 · Solutions to differential equations are functions or families of functions.

FUN-7.C.4 · Euler’s method provides a procedure for approximating a solution to a differential equation or a point on a solution curve.

FUN-7.C.1 · A slope field is a graphical representation of a differential equation on a finite set of points in the plane.

FUN-7.C.2 · Slope fields provide information about the behavior of solutions to first-order differential equations.

FUN-7.C.3 · Solutions to differential equations are functions or families of functions.

FUN-7.C.4 · Euler’s method provides a procedure for approximating a solution to a differential equation or a point on a solution curve.

FUN-7.D.1 · Some differential equations can be solved by separation of variables.

FUN-7.D.2 · Antidifferentiation can be used to find general solutions to differential equations.

FUN-7.E.1 · A general solution may describe infinitely many solutions to a differential equation. There is only one particular solution passing through a given point.

FUN-7.E.2 · The function F defined by F(x)=y₀+∫ₐˣ f(t) dt is a particular solution to the differential equation dy/dx=f(x), satisfying F(a)=y₀.

FUN-7.E.3 · Solutions to differential equations may be subject to domain restrictions.

FUN-7.F.1 · Specific applications of finding general and particular solutions to differential equations include motion along a line and exponential growth and decay.

FUN-7.F.2 · The model for exponential growth and decay that arises from the statement “The rate of change of a quantity is proportional to the size of the quantity” is dy/dt=ky.

FUN-7.G.1 · The exponential growth and decay model, dy/dt=ky, with initial condition y=y₀ when t=0, has solutions of the form y=y₀e^(kt).

FUN-7.H.1 · The model for logistic growth that arises from the statement “The rate of change of a quantity is jointly proportional to the size of the quantity and the difference between the quantity and the carrying capacity” is dy/dt=ky(a−y).

FUN-7.H.2 · The logistic differential equation and initial conditions can be interpreted without solving the differential equation.

FUN-7.H.3 · The limiting value (carrying capacity) of a logistic differential equation as the independent variable approaches infinity can be determined using the logistic growth model and initial conditions.

FUN-7.H.4 · The value of the dependent variable in a logistic differential equation at the point when it is changing fastest can be determined using the logistic growth model and initial conditions.

CHA-4.B.1 · The average value of a continuous function f over an interval [a,b] is [1/(b−a)]∫ₐᵇ f(x) dx.

CHA-4.C.1 · For a particle in rectilinear motion over an interval of time, the definite integral of velocity represents the particle’s displacement over the interval of time, and the definite integral of speed represents the particle’s total distance traveled over the interval of time.

CHA-4.D.1 · A function defined as an integral represents an accumulation of a rate of change.

CHA-4.D.2 · The definite integral of the rate of change of a quantity over an interval gives the net change of that quantity over that interval.

CHA-4.E.1 · The definite integral can be used to express information about accumulation and net change in many applied contexts.

CHA-5.A.1 · Areas of regions in the plane can be calculated with definite integrals.

CHA-5.A.2 · Areas of regions in the plane can be calculated using functions of either x or y.

CHA-5.A.3 · Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.

CHA-5.A.1 · Areas of regions in the plane can be calculated with definite integrals.

CHA-5.A.2 · Areas of regions in the plane can be calculated using functions of either x or y.

CHA-5.A.3 · Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.

CHA-5.A.1 · Areas of regions in the plane can be calculated with definite integrals.

CHA-5.A.2 · Areas of regions in the plane can be calculated using functions of either x or y.

CHA-5.A.3 · Areas of certain regions in the plane may be calculated using a sum of two or more definite integrals or by evaluating a definite integral of the absolute value of the difference of two functions.

CHA-5.B.1 · Volumes of solids with square and rectangular cross sections can be found using definite integrals and the area formulas for these shapes.

CHA-5.B.2 · Volumes of solids with triangular cross sections can be found using definite integrals and the area formulas for these shapes.

CHA-5.B.3 · Volumes of solids with semicircular and other geometrically defined cross sections can be found using definite integrals and the area formulas for these shapes.

CHA-5.B.1 · Volumes of solids with square and rectangular cross sections can be found using definite integrals and the area formulas for these shapes.

CHA-5.B.2 · Volumes of solids with triangular cross sections can be found using definite integrals and the area formulas for these shapes.

CHA-5.B.3 · Volumes of solids with semicircular and other geometrically defined cross sections can be found using definite integrals and the area formulas for these shapes.

CHA-5.C.1 · Volumes of solids of revolution around the x-or y-axis may be found by using definite integrals with the disc method.

CHA-5.C.2 · Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.

CHA-5.C.3 · Volumes of solids of revolution around the x-or y-axis whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-5.C.4 · Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-5.C.1 · Volumes of solids of revolution around the x-or y-axis may be found by using definite integrals with the disc method.

CHA-5.C.2 · Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.

CHA-5.C.3 · Volumes of solids of revolution around the x-or y-axis whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-5.C.4 · Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-5.C.1 · Volumes of solids of revolution around the x-or y-axis may be found by using definite integrals with the disc method.

CHA-5.C.2 · Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.

CHA-5.C.3 · Volumes of solids of revolution around the x-or y-axis whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-5.C.4 · Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-5.C.1 · Volumes of solids of revolution around the x-or y-axis may be found by using definite integrals with the disc method.

CHA-5.C.2 · Volumes of solids of revolution around any horizontal or vertical line in the plane may be found by using definite integrals with the disc method.

CHA-5.C.3 · Volumes of solids of revolution around the x-or y-axis whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-5.C.4 · Volumes of solids of revolution around any horizontal or vertical line whose cross sections are ring shaped may be found using definite integrals with the washer method.

CHA-6.A.1 · The length of a planar curve defined by a function can be calculated using a definite integral.

CHA-3.G.1 · Methods for calculating derivatives of real-valued functions can be extended to parametric functions.

CHA-3.G.2 · For a curve defined parametrically, the value of dy/dx at a point on the curve is the slope of the line tangent to the curve at that point. dy/dx, the slope of the line tangent to a curve defined using parametric equations, can be determined by dividing dy/dt by dx/dt, provided dx/dt does not equal zero.

CHA-3.G.3 · d²y/dx² can be calculated by dividing d/dt {dy/dx} by dx/dt.

CHA-3.G.1 · Methods for calculating derivatives of real-valued functions can be extended to parametric functions.

CHA-3.G.2 · For a curve defined parametrically, the value of dy/dx at a point on the curve is the slope of the line tangent to the curve at that point. dy/dx, the slope of the line tangent to a curve defined using parametric equations, can be determined by dividing dy/dt by dx/dt, provided dx/dt does not equal zero.

CHA-3.G.3 · d²y/dx² can be calculated by dividing d/dt {dy/dx} by dx/dt.

CHA-6.B.1 · The length of a parametrically defined curve can be calculated using a definite integral.

CHA-3.H.1 · Methods for calculating derivatives of real-valued functions can be extended to vector-valued functions.

FUN-8.A.1 · Methods for calculating integrals of real-valued functions can be extended to parametric or vector-valued functions.

FUN-8.B.1 · Derivatives can be used to determine velocity, speed, and acceleration for a particle moving along a curve in the plane defined using parametric or vector-valued functions.

FUN-8.B.2 · For a particle in planar motion over an interval of time, the definite integral of the velocity vector represents the particle’s displacement (net change in position) over the interval of time, from which we might determine its position. The definite integral of speed represents the particle’s total distance traveled over the interval of time.

FUN-3.G.1 · Methods for calculating derivatives of real-valued functions can be extended to functions in polar coordinates.

FUN-3.G.2 · For a curve given by a polar equation r = f(θ), derivatives of r, x, and y with respect to θ, and first and second derivatives of y with respect to x can provide information about the curve.