West Windsor-Plainsboro Regional School District AP Calculus BC Grades 9-12

Transcription

1 West Windsor-Plainsboro Regional School District AP Calculus BC Grades 9-12

2 Unit 1: Limits and Continuity What is a limit? Definition of limit, continuous, Sandwich Theorem, Intermediate Value Theorem Apply definition of limit to evaluate the average rate of change Apply definition of limit to approximate the instantaneous rate of change Evaluate limits at infinity Apply the sandwich theorem Show, by definition, if a function is continuous Determine the equation of a line, tangent to a curve at a given point Suggested :

3 Unit 2: Derivatives What is a derivative? How do we use derivatives? Terms: Derivative, differentiable, chain rule, implicit differentiation, instantaneous rate of change Apply the definition of derivative Determine whether a function is differentiable Apply the power rule, product rule, and quotient rule to evaluate the derivative of function Evaluate the instantaneous rate of change of a function Evaluate the velocity of a function Differentiate trigonometric functions Differentiate functions by applying the chain rule Apply the method of implicit differentiation Differentiate inverse trigonometric functions Differentiate exponential and logarithmic functions

4 Apply the method of logarithmic differentiation Suggested :

5 Unit 3: Applications of Derivative How can we use derivatives solve problems? Terms: Absolute (global) extreme values, local (relative) extreme values, linearization, Newton s Method, differentials, and the Mean Value Theorem Apply calculus to find the maximum/minimum values of a function Apply the Mean Value Theorem Analyze and make the connections between and Solve real world optimization problems Find the linearization of a function and use it to approximate values Compute the differential of a function Approximate roots by using Newton s method Solve related rate applications

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7 Unit 4: The Definite Integral How can we use integrals to find areas under a curve? How can we use integrals to solve problems? The Fundamental Theorem of Calculus Terms, the trapezoidal rule, Simpson s method and the term definite integral, Riemann Sum and antiderivative Estimate the region under a curve by using finite sums Approximate the definite integral Apply properties of the definite integral Find the average value of a function Connect differential and integral calculus by applying the Fundamental Theorem of Calculus Approximate definite integrals by applying the trapezoidal rule Approximate definite integrals by applying Simpson s rule

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9 Unit 5: Differential Equations and Math Modeling How can we use integration to solve problems? Terms: Slope fields, separable differential equations, Euler s Method Construct a slope field Sketch an antiderivative using a slope field Apply properties of indefinite integrals Solve initial value problems Integrate functions by using the substitution method Apply integration by parts to integrate functions Solve exponential growth and decay applications including Newton s Law of Cooling Model population with a logistic differential equation Approximate values using numerical methods including Euler s Method

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11 Unit 6: Applications of Definite Integrals How can we use integrals to find areas and volumes? How can we use integrals to solve science and statistical problems? Terms: Displacement, net change, fluid pressure Approximate Riemann Sums and evaluate from a given set of data Find the area of a region bounded by two curves Find the volume of a solid obtained by a rotation, using the disk and washer methods Find the volume of a solid obtained by a rotation, using the shell method Find the volume of a solid with a known cross section Find the length of an arc Evaluate the surface area of a function Solve science and statistical applications

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13 Unit 7: L Hopitals Rule, Improper Integrals & Partial Fractions How can we evaluate limits? How can we compare relative rates of growth? Terms: Indeterminate Form, L Hopitals Rule, improper integrals, convergence, divergence Evaluate limits in indeterminate form and apply L Hopitals Rule Compare relative rates of growth Evaluate improper integrals with infinite integration limits Evaluate improper integrals with infinite discontinuities Apply the Direct Comparison Test to determine if a function converges/diverges Apply the Limit Comparison Test to determine if a function converges/diverges Apply improper integrals to finding the volume of an infinite solid Use the method of partial fractions to evaluate antiderivatives Use an integral table Integrate functions using trigonometric substitutions

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15 Unit 8: Infinite Series How do we apply tests to determine convergence of a series? Terms: Power Series, Taylor Series, Maclaurin Series, n th Term Test, Ratio Test, absolute and conditional convergence Find a power series for other functions Find a power series by differentiation Find a power series by integration Construct a Taylor Series or a Maclaurin Series Approximate error of a Taylor Polynomial Apply the n th term test to determine convergence of a series Apply the Direct Comparison Test to determine convergence of a series Apply the Ratio Test to determine convergence of a series Apply the Integral Test to determine convergence of a series including a p series Apply the Limit Comparison Test to determine convergence of a series

16 Apply the Alternating Series Test to determine convergence of a series Determine absolute and conditional convergence including intervals of convergence for a power series Suggested :

17 Unit 9: Parametric and Vector Functions How do we evaluate vector functions? Terms: Vectors, vector operations, component form of a vector, dot product Evaluate derivatives in parametric form Evaluate the arc Length of a smooth parametrized curve Evaluate the surface area from a smooth parametrized curve Find vectors tangent to a curve Evaluate velocity, speed, acceleration and direction of motion of a position vector Apply differentiation rules for vector functions Evaluate the antiderivative of a differentiable vector function Model projectile motion Suggested :

18 Unit 10: Polar Functions What do we know about polar functions? Terms: Polar Coordinates, polar graphs Graph polar equations Find the slope of the tangent line to a polar curve Evaluate the area between the origin and the curve of the polar function Evaluate the area between polar curves Evaluate the length of a polar curve Evaluate the area of a surface generated by revolving a polar graph Suggested :

19 Unit 11: Differential Equations How do we solve differential equations? Terms: Second order homogeneous linear equations, nonhomogeneous differential equations Solve a first order linear differential equation Solve a second order homogeneous linear equation Solve a nonhomogeneous differential equation using variation of parameter Solve a nonhomogeneous differential equations using the method of undetermined coefficients Suggested :

20 Unit 12: Moments, Center of Mass and Centroids Standard What is meant by the center of mass? Terms: Moments, centroids, planar lamina Find the center of mass of a linear system Find the center of mass of a two dimensional system Find the center of mass of the lamina bounded by a function Find the centroid of a region bounded by two functions

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