AP Math Calculus AB #320 Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code: Derivatives The students will be -Determine the derivative -written -Derivative presented graphically, numerically, able to differentiate of a composite function documentation of and analytically composite functions using the Chain Rule problem solving -Derivative interpreted as an instantaneous using the Chain Rule, -Utilize the Chain Rule -applications to rate of change find derivatives using two or more times to real life problems -Derivative defined as the limit of the the Power Rule for determine the derivative difference quotient Rational Powers, and -Understand the use and -AP Calculus -Relationship between differentiability and find derivatives using notation for an implicitly exam practice continuity implicit differentiation. defined function questions -Slope of a curve at a point -Find the tangent and -technology -Tangent line to a curve at a point and local normal lines to a conic applications linear approximation -Find a higher order -classroom -Instantaneous rate of change as the limit of derivative implicitly discussion average rate of change -Apply the Power Rule -Approximate rate of change from graphs and for Rational Powers of tables of values the exponent -Corresponding characteristics of graphs of f and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f"
Transcript
AP Math Calculus AB #320
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Determine the derivative -written -Derivative presented graphically, numerically, able to differentiate of a composite function documentation of and analytically composite functions using the Chain Rule problem solving -Derivative interpreted as an instantaneous using the Chain Rule, -Utilize the Chain Rule -applications to rate of change find derivatives using two or more times to real life problems -Derivative defined as the limit of the the Power Rule for determine the derivative difference quotient Rational Powers, and -Understand the use and -AP Calculus -Relationship between differentiability and find derivatives using notation for an implicitly exam practice continuity implicit differentiation. defined function questions -Slope of a curve at a point -Find the tangent and -technology -Tangent line to a curve at a point and local normal lines to a conic applications linear approximation -Find a higher order -classroom -Instantaneous rate of change as the limit of derivative implicitly discussion average rate of change -Apply the Power Rule -Approximate rate of change from graphs and for Rational Powers of tables of values the exponent -Corresponding characteristics of graphs of f and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f"
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity -Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Define symbolically a -written 2.11.11.A -Derivative presented graphically, numerically, able to calculate slopes derivative documentation of and analytically and derivatives using -Determine the derivative problem solving -Derivative interpreted as an instantaneous the definition of the of a function using the -applications to rate of change derivative, and graph f definition of a derivative real life problems -Derivative defined as the limit of the from the graph of f', -Graph f' given f difference quotient graph f' from the graph -Graph f given f' -AP Calculus -Relationship between differentiability and of f, and graph the -Graph the derivative for exam practice continuity derivative of a function a table of data questions -Slope of a curve at a point given numerically with -Determine one-sided -technology -Tangent line to a curve at a point and local data. derivatives applications linear approximation -classroom -Instantaneous rate of change as the limit of discussion average rate of change -Approximate rate of change from graphs and tables of values -Corresponding characteristics of graphs of f and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Understand how a -written 2.11.11.C -Derivative presented graphically, numerically, able to find where a derivative fails to exist documentation of and analytically function is not -Understand the problem solving -Derivative interpreted as an instantaneous differentiable and relationship between -applications to rate of change approximate a differentiability and local real life problems -Derivative defined as the limit of the derivative numerically linearity difference quotient and graphically. -Compute a numerical -AP Calculus -Relationship between differentiability and derivative exam practice continuity -Understand the questions -Slope of a curve at a point relationship between -technology -Tangent line to a curve at a point and local differentiability and applications linear approximation continuity -classroom -Instantaneous rate of change as the limit of -Apply the Intermediate discussion average rate of change Value Theorem for -Approximate rate of change from graphs and Derivatives tables of values -Corresponding characteristics of graphs of f and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Apply differentiation -written -Derivative presented graphically, numerically, able to use the rules of rules for integer powers, documentation of and analytically differentiation to multiples, sums and problem solving -Derivative interpreted as an instantaneous calculate derivatives, differences -applications to rate of change including second and -Determine the derivative real life problems -Derivative defined as the limit of the higher order of a product or quotient difference quotient derivatives of basic of functions -AP Calculus -Relationship between differentiability and functions, including -Determine second and exam practice continuity trigonometric functions, higher order derivatives questions -Slope of a curve at a point inverse trigonometric -Determine the -technology -Tangent line to a curve at a point and local functions, and derivatives of the six applications linear approximation exponential and basic trigonometric -classroom -Instantaneous rate of change as the limit of logarithmic functions. functions discussion average rate of change -Determine the -Approximate rate of change from graphs and derivatives of the six tables of values basic inverse -Corresponding characteristics of graphs of f trigonometric functions and f' -Determine the derivative -Relationship between the increasing and of an exponential decreasing behavior of f and the sign of f' function -The Mean Value Theorem and its geometric -Determine the derivative consequences of a power function -Equations involving derivatives. Verbal -Determine the derivative descriptions are translated into equations of both a natural and involving derivatives and vice versa. common logarithmic -Corresponding characteristics of the graphs function of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Use the definition of a -written 2.5.11.A, -Derivative presented graphically, numerically, able to use derivatives derivative to determine documentation of 2.8.11.D, and analytically to analyze straight line an object's instantaneous problem solving 2.8.11.P, -Derivative interpreted as an instantaneous motion and solve other rate of change -applications to 2.11.11.A, rate of change problems involving -Understand the real life problems 2.11.11.B, -Derivative defined as the limit of the rates of change. difference between 2.11.11.C difference quotient speed and velocity -AP Calculus -Relationship between differentiability and -Utilize the concept of a exam practice continuity derivative to determine questions -Slope of a curve at a point an object's velocity and -technology -Tangent line to a curve at a point and local acceleration given its applications linear approximation position -classroom -Instantaneous rate of change as the limit of -Use the derivative to discussion average rate of change model other rates of -Approximate rate of change from graphs and change such as free fall, tables of values sensitivity to change, -Corresponding characteristics of graphs of f and economics and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Determine absolute -written 2.11.11.A, -Derivative presented graphically, numerically, able to determine local extreme values documentation of 2.11.11.B and analytically or global extreme -Apply the Extreme Value problem solving -Derivative interpreted as an instantaneous values of a function, Theorem -applications to rate of change apply the Mean Value -Determine all critical real life problems -Derivative defined as the limit of the Theorem, and find the points of a function difference quotient intervals on which a -Determine all local -AP Calculus -Relationship between differentiability and function is increasing extreme values exam practice continuity or decreasing. -Apply the Mean Value questions -Slope of a curve at a point Theorem for Derivatives -technology -Tangent line to a curve at a point and local applications linear approximation -Determine the intervals -classroom -Instantaneous rate of change as the limit of over which a function is discussion-writte average rate of change increasing and n documentation -Approximate rate of change from graphs and decreasing of problem solving tables of values -Understand the concept -Corresponding characteristics of graphs of f of an antiderivative and -applications to and f' the process of real life problems -Relationship between the increasing and antidifferentiation decreasing behavior of f and the sign of f' -AP Calculus -The Mean Value Theorem and its geometric exam practice consequences questions -Equations involving derivatives. Verbal -technology descriptions are translated into equations applications involving derivatives and vice versa. -classroom -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative discussion extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Determine a linear -written 2.5.11.A, -Derivative presented graphically, numerically, able to find approximation using a documentation of 2.5.11.B, and analytically linearizations, estimate linearization process problem solving 2.5.11.C, -Derivative interpreted as an instantaneous the change in a -Estimate the change in a -applications to 2.5.11.D rate of change function using function using real life problems -Derivative defined as the limit of the differentials, and solve differentials difference quotient related rate problems. -Understand the -AP Calculus -Relationship between differentiability and difference between exam practice continuity absolute, relative, and questions -Slope of a curve at a point percentage change -technology -Tangent line to a curve at a point and local -Determine the related applications linear approximation rate equation of a -classroom -Instantaneous rate of change as the limit of defined quantity discussion average rate of change -Apply the concept of -Approximate rate of change from graphs and related rates to real life tables of values situations -Corresponding characteristics of graphs of f and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Apply the first derivative -written 2.5.11.A, -Derivative presented graphically, numerically, able to apply the first test for local extrema documentation of 2.5.11.D, and analytically and second derivative -Determine the concavity problem solving 2.11.11.A, -Derivative interpreted as an instantaneous tests, determine the of a function using the -applications to 2.11.11.B rate of change concavity of a second derivative test real life problems -Derivative defined as the limit of the function, locate points -Determine the points of difference quotient of inflection, and solve inflection of a function -AP Calculus -Relationship between differentiability and problems involving -Utilize the second exam practice continuity minimum or maximum derivative test for local questions -Slope of a curve at a point values of a function. extrema -technology -Tangent line to a curve at a point and local -Develop a mathematical applications linear approximation model to optimize a given -classroom -Instantaneous rate of change as the limit of situation discussion average rate of change -Approximate rate of change from graphs and tables of values -Corresponding characteristics of graphs of f and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Derivatives The students will be -Solve initial value -written 2.8.11.Q -Derivative presented graphically, numerically, able to construct slope differential problems documentation of and analytically fields using -Construct a slope field problem solving -Derivative interpreted as an instantaneous technology, interpret using graphical -applications to rate of change slope fields as techniques and real life problems -Derivative defined as the limit of the visualizations of technology difference quotient differential equations, -Predict a functions -AP Calculus -Relationship between differentiability and and use Euler's Method equation given the slope exam practice continuity to find approximate field of the function questions -Slope of a curve at a point solutions to differential -Use Euler's Method to -technology -Tangent line to a curve at a point and local equations with initial find approximations for applications linear approximation values. initial value problems -classroom -Instantaneous rate of change as the limit of -Visualize Euler's Method discussion average rate of change using graphical solutions -Approximate rate of change from graphs and to initial value problem tables of values approximations -Corresponding characteristics of graphs of f and f' -Relationship between the increasing and decreasing behavior of f and the sign of f' -The Mean Value Theorem and its geometric consequences -Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa. -Corresponding characteristics of the graphs of f, f', and f" -Relationship between the concavity of f and the sign of f" -Points of inflections as places where concavity changes -Analysis of curves, including the notions of monotonicity and concavity
Instructional Unit Derivatives AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
-Optimization, both absolute and relative extrema -Modeling rates of change, including related rates problems -Use of implicit differentiation to find the derivative of an inverse function -Interpretation of the derivative as a rate of change in varies applied contexts, including velocity, speed, and acceleration -Numerical solution of differential equations using Euler's method -Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse trigonometric functions -Basic rules for the derivative of sums, products, and quotients of functions -Chain rule and implicit differentiation -Derivatives of parametric, polar, and vector functions
Instructional Unit Functions, Graphs, and Limits AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Functions, Graphs, and Limits The students will be -Define both in words -written 2.8.11.Q, -An intuitive understanding of the limiting able to find and verify and symbolically a documentation of 2.11.11.A process end behavior models horizontal asymptote problem solving -Calculating limits (both one- and two-sided) for various functions, -Apply properties of the -applications to using algebra calculate limits as x limit as x approaches real life problems -Estimating limits from graphs and tables of approaches both both positive and data positive and negative negative infinity -AP Calculus -Understanding asymptotes in terms of infinity, and identify -Define a vertical exam practice graphical behavior vertical and horizontal asymptote using the questions -Describing asymptotic behavior in terms of asymptotes. concept of a limit -technology limits involving infinity -Determine both a right applications -Comparing relative magnitudes of functions and left end behavior -classroom and their rates of change model of a function discussion -An intuitive understanding of continuity -Understanding continuity in terms of limits -Geometric understanding of graphs of continuous functions
Instructional Unit Functions, Graphs, and Limits AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Functions, Graphs, and Limits The students will be -Understand the -written 2.8.11.S -An intuitive understanding of the limiting able to evaluate the definition of a limit documentation of process limits of functions. -Apply properties of limits problem solving -Calculating limits (both one- and two-sided) to applications -applications to using algebra -Evaluate both one-sided real life problems -Estimating limits from graphs and tables of and two-sided limits data -Apply the Sandwich -AP Calculus -Understanding asymptotes in terms of Theorem exam practice graphical behavior questions -Describing asymptotic behavior in terms of -technology limits involving infinity applications -Comparing relative magnitudes of functions -classroom and their rates of change discussion -An intuitive understanding of continuity -Understanding continuity in terms of limits -Geometric understanding of graphs of continuous functions
Instructional Unit Functions, Graphs, and Limits AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Functions, Graphs, and Limits The students will be -Determine the continuity -written 2.8.11.O -An intuitive understanding of the limiting able to identify the at both an endpoint and documentation of process intervals upon which a an interior point on the problem solving -Calculating limits (both one- and two-sided) given function is domain of a function -applications to using algebra continuous, -Define continuity of a real life problems -Estimating limits from graphs and tables of understand the function data meaning of a -Understand and apply -AP Calculus -Understanding asymptotes in terms of continuous function, properties of continuous exam practice graphical behavior and be able to remove functions questions -Describing asymptotic behavior in terms of removable -Analyze the continuity -technology limits involving infinity discontinuities by of a function through the applications -Comparing relative magnitudes of functions extending or modifying use of composite -classroom and their rates of change a function. functions discussion -An intuitive understanding of continuity -Apply the Intermediate -Understanding continuity in terms of limits Value Theorem for -Geometric understanding of graphs of Continuous Functions continuous functions
Instructional Unit Integrals AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Integrals The students will be -Develop a strategy for -written 2.5.11.A, -Computation of Riemann sums using left, able to adapt their modeling with integrals documentation of 2.5.11.B, right, and midpoint evaluation points knowledge of integral -Model the effects on the problem solving 2.5.11.C, -Use of Riemann and trapezoidal sums to calculus to model interrelationships -applications to 2.5.11.D approximate definite integrals of functions problems involving between position, real life problems represented algebraically, graphically, and by rates of change in a velocity, and acceleration tables of values variety of applications. -AP Calculus -Definite integral as a limit of Riemann sums -Use integral calculus to exam practice over equal subdivisions model net change in data questions -Definite integral of the rate of change of a -technology quantity over an interval -Review applications of applications -Basic properties of definite integrals integral calculus to the -classroom -Use of the Fundamental Theorem of Calculus area of physics discussion to evaluate definite integrals -Review applications of -Use of the Fundamental Theorem of Calculus integral calculus to the to represent a particular antiderivative, and the area of statistics analytical and graphical analysis of functions so defined -Antiderivatives following directly from derivatives of basic functions -Antiderivatives by substitution of variables and parts -Improper integrals -Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations -Finding specific antiderivatives using initial conditions, including applications to motion along a line -Solving separable differential equations and using them in modeling. In particular, studying the equation y'=ky and exponential growth -Solving logistic differential equations and using them in modeling
Instructional Unit Integrals AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Integrals The students will be -Understand and apply -written 2.11.11.E -Computation of Riemann sums using left, able to apply rules for rules for definite integrals documentation of right, and midpoint evaluation points definite integrals, find problem solving -Use of Riemann and trapezoidal sums to the average value of a -Apply the Mean Value -applications to approximate definite integrals of functions function, and apply the Theorem for definite real life problems represented algebraically, graphically, and by Fundamental Theorem integrals tables of values of Calculus. -Examine the -AP Calculus -Definite integral as a limit of Riemann sums connections between exam practice over equal subdivisions differential and integral questions -Definite integral of the rate of change of a calculus -technology quantity over an interval -Understand the proof of applications -Basic properties of definite integrals the Fundamental -classroom -Use of the Fundamental Theorem of Calculus Theorem of Calculus discussion to evaluate definite integrals -Apply the Fundamental -Use of the Fundamental Theorem of Calculus Theorem of Calculus to represent a particular antiderivative, and the -Use technology to graph analytical and graphical analysis of functions an integral function so defined -Review applications of -Antiderivatives following directly from the Fundamental derivatives of basic functions Theorem of Calculus -Antiderivatives by substitution of variables and parts -Improper integrals -Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations -Finding specific antiderivatives using initial conditions, including applications to motion along a line -Solving separable differential equations and using them in modeling. In particular, studying the equation y'=ky and exponential growth -Solving logistic differential equations and using them in modeling
Instructional Unit Integrals AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Integrals The students will be -Evaluate integrals using -written -Computation of Riemann sums using left, able to compute a substitution method documentation of right, and midpoint evaluation points indefinite and definite -Understand and apply problem solving -Use of Riemann and trapezoidal sums to integrals using the the integration by parts -applications to approximate definite integrals of functions methods of formula real life problems represented algebraically, graphically, and by substitution, parts, -Investigate the repeated tables of values tabular, and integral use of integration by -AP Calculus -Definite integral as a limit of Riemann sums tables. parts to evaluate certain exam practice over equal subdivisions integrals questions -Definite integral of the rate of change of a -Use tabular integration -technology quantity over an interval as a shortcut to problems applications -Basic properties of definite integrals fitting a specific pattern -classroom -Use of the Fundamental Theorem of Calculus discussion to evaluate definite integrals -Use of the Fundamental Theorem of Calculus -Use the procedure of to represent a particular antiderivative, and the partial fractions to analytical and graphical analysis of functions evaluate the integral of a so defined rational expression -Antiderivatives following directly from -Utilize integral tables to derivatives of basic functions evaluate nonroutine -Antiderivatives by substitution of variables integration procedures and parts -Utilize trigonometric -Improper integrals substitution to evaluate -Geometric interpretation of differential integrals fitting a specific equations via slope fields and the relationship pattern between slope fields and solution curves for differential equations -Finding specific antiderivatives using initial conditions, including applications to motion along a line -Solving separable differential equations and using them in modeling. In particular, studying the equation y'=ky and exponential growth
-Solving logistic differential equations and using them in modeling
Instructional Unit Integrals AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Integrals The students will be -Understand and explain -written 2.11.11.C -Computation of Riemann sums using left, able to solve problems the law of exponential documentation of right, and midpoint evaluation points involving exponential change problem solving -Use of Riemann and trapezoidal sums to growth and decay, and -Apply the law of -applications to approximate definite integrals of functions exponential or logistic exponential change to real life problems represented algebraically, graphically, and by population growth. such areas as tables of values compounded interest, -AP Calculus -Definite integral as a limit of Riemann sums radioactivity, Newton's exam practice over equal subdivisions law of cooling, and questions -Definite integral of the rate of change of a resistance proportional to -technology quantity over an interval an object's velocity applications -Basic properties of definite integrals -Understand and apply -classroom -Use of the Fundamental Theorem of Calculus the exponential model discussion to evaluate definite integrals -Understand and apply -Use of the Fundamental Theorem of Calculus the logistic growth model to represent a particular antiderivative, and the analytical and graphical analysis of functions -Use technology to so defined explore a logistic growth -Antiderivatives following directly from model derivatives of basic functions -Antiderivatives by substitution of variables and parts -Improper integrals -Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations -Finding specific antiderivatives using initial conditions, including applications to motion along a line -Solving separable differential equations and using them in modeling. In particular, studying the equation y'=ky and exponential growth
-Solving logistic differential equations and using them in modeling
Instructional Unit Integrals AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Integrals The students will be -Determine the area -written 2.11.11.E -Computation of Riemann sums using left, able to use integration between two documentation of right, and midpoint evaluation points to calculate areas of nonintersecting curves in problem solving -Use of Riemann and trapezoidal sums to regions in a plane and a plane -applications to approximate definite integrals of functions volumes of solids of -Determine the area real life problems represented algebraically, graphically, and by revolution and known enclosed by the tables of values cross sections. intersection of curves in -AP Calculus -Definite integral as a limit of Riemann sums a plane exam practice over equal subdivisions -Determine the area questions -Definite integral of the rate of change of a enclosed by intersecting -technology quantity over an interval curves by partitioning the applications -Basic properties of definite integrals region into several -classroom -Use of the Fundamental Theorem of Calculus subregions discussion to evaluate definite integrals -Integrate an enclosed -Use of the Fundamental Theorem of Calculus area with respect to both to represent a particular antiderivative, and the x and y analytical and graphical analysis of functions -Determine the volume of so defined a solid of known -Antiderivatives following directly from integrable cross section derivatives of basic functions areas -Antiderivatives by substitution of variables -Apply the disc and and parts washer methods to -Improper integrals determine volumes of -Geometric interpretation of differential solids equations via slope fields and the relationship between slope fields and solution curves for differential equations -Finding specific antiderivatives using initial conditions, including applications to motion along a line -Solving separable differential equations and using them in modeling. In particular, studying the equation y'=ky and exponential growth -Solving logistic differential equations and using them in modeling
Instructional Unit Integrals AP Math Calculus AB #320 Unit Content Objective Performance Indicator Performance Task State Standards Code:
Integrals The students will be -Utilize and apply the -written 2.11.11.E -Computation of Riemann sums using left, able to approximate the three basic rectangular documentation of right, and midpoint evaluation points area under the graph approximation methods problem solving -Use of Riemann and trapezoidal sums to of a function by using (LRAM, RRAM, MRAM) -applications to approximate definite integrals of functions rectangle -Apply rectangular real life problems represented algebraically, graphically, and by approximation approximation methods to tables of values methods, interpret the three dimensional -AP Calculus -Definite integral as a limit of Riemann sums area under a graph, volume estimations exam practice over equal subdivisions and express the area -Express the definite questions -Definite integral of the rate of change of a under a curve as a integral as a limit of -technology quantity over an interval definite integral and as Riemann sums applications -Basic properties of definite integrals a limit of Riemann sums -Understand and utilize -classroom -Use of the Fundamental Theorem of Calculus or Trapezoidal Rule. the terminology and discussion to evaluate definite integrals notation of integration -Use of the Fundamental Theorem of Calculus -Apply the concept of a to represent a particular antiderivative, and the definite integral to analytical and graphical analysis of functions compute the area under so defined a curve -Antiderivatives following directly from -Use the Trapezoidal derivatives of basic functions Rule to approximate the -Antiderivatives by substitution of variables area under a curve and parts -Evaluate the integral of a -Improper integrals constant -Geometric interpretation of differential -Use technology to equations via slope fields and the relationship evaluate an integral between slope fields and solution curves for differential equations -Finding specific antiderivatives using initial conditions, including applications to motion along a line -Solving separable differential equations and using them in modeling. In particular, studying the equation y'=ky and exponential growth -Solving logistic differential equations and using them in modeling
