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1 Unit 1 Days 1-9 Evaluate one-sided two-sided limits, given the graph of a function. Limits, Evaluate limits using tables calculators. Continuity Evaluate limits using direct substitution. Differentiability Apply limit properties to evaluate limits. Evaluate limits analytically/algebraically when direct substitution yields 0/0 form. Methods include multiplication/simplification, factoring/reducing, finding LCD simplifying complex fraction, multiplying numerator denominator by conjugate. EK 1.1 A1 EK 1.1 A2 EK 1.1 A3 EK 1.1 B1 EK 1.1 C1 EK 1.1 C2 EK 1.1 C3 EK 1.1 D1 EK 1.2 A1 EK 1.2 A2 EK 1.2 A3 EK 1.2 B1 EK 1.1 D1 Evaluate limits involving difference quotients. Evaluate limits of piecewise-defined functions or absolute value functions. Evaluate the limit of a rational function involving algebraic expressions (including radicals) as x approaches positive or negative infinity. Find equations of vertical horizontal asymptotes of a rational function. Evaluate limits involving one of the seven indeterminate forms by rewriting the problem into a form for which L'Hopital's Rule can be applied. Identify values of x for which a function is discontinuous, given a graph or a functional expression. Name intervals on which a function is continuous, given a graph or a functional expression. Determine whether a piecewise-defined function is continuous everywhere or where it has discontinuities. Classify discontinuities as removable or non-removable. Classify discontinuities as point, jump, infinite or oscillating. Determine the value of a constant which makes a piecewise-defined function continuous everywhere. Identify values of x for which a function is non-differentiable, given a graph or a functional expression. Name intervals on which a function is differentiable, given a graph or a functional expression. Recognize that if a function is differentiable, then it is also continuous, but that the converse of this statement is not necessarily true. Solve problems by applying the Intermediate Value Theorem. Page 1
2 Unit 2 Days Write the limit definition of the derivative using h or delta-x. EK 2.1 A1 Find the derivative of a function using basic differentiation rules. EK 2.1 A2 Evaluate the derivative of a function at a point. EK 2.1 A3 Write the equation of a tangent line to a function at a point, expressing answers in EK 2.1 A4 point-slope, slope-intercept general forms. EK 2.1 A5 Use features of the graphics calculator to evaluate the derivative of a function at a EK 2.1 C1 point. EK 2.2 B2 Find points where a function has a horizontal tangent line. EK 2.3 B1 EK 2.1 A4 Determine the value of constants which makes a piecewise-defined function continuous everywhere. EK 2.1 C2 EK 2.1 C3 Find the average rate of change or average velocity of a function. EK 2.1 D1 Find the instantaneous rate of change or instantaneous velocity of a function. EK 2.1 D2 Solve problems involving position, velocity acceleration using derivatives. EK 2.3 A1 Find the derivative of a function using the product or quotient rule. EK 2.3 A2 Solve problems involving equation of a tangent line, horizontal tangent line or EK 2.3 C1 value of the derivative at a point using the product or quotient rule. EK 2.3 D1 Determine the most efficient method of calculating the derivative: rewriting EK 1.1 C3 simplifying or using the product or quotient rule. EK 2.1 B1 In AP Exam-type questions, recognize that the product or quotient rule is the concept needed to solve a problem involving a graph or table of information then correctly apply the rule to solve the problem. EK 2.1 A2 EK 2.1 C3 EK 2.1 D1 Use the quotient rule to derive formulas for tan, cot, sec csc. EK 2.1 D2 Use L'Hopital's Rule to evaluate limits, applying the product rule when needed. EK 1.1 C3 Evaluate higher order derivatives. EK 2.1 C3 Find the derivative of a function using the Chain Rule. EK 2.1 C4 Find the derivative of a function using the General (Extended) Power Rule. Find the derivative of a function by applying more than one derivative rule. Find the derivative of expressions involving the natural logarithm function. Find the derivative of expressions involving the natural exponential function. Find the derivative of expressions involving log or exponential functions with a base other than "e". Page 2
3 Unit 2 Days Find the derivative of expressions involving inverse trigonometric functions. EK 2.1 C2 Given a set of mixed problems, decide which differentiation rule(s) applies then find the derivative of the expression. EK 2.1 C3 EK 2.1 C4 Find the derivative of an expression using implicit differentiation. EK 2.1 C6 Solve problems involving related rates. These include problems with circles, EK 2.1 C5 spheres, rectangles, cubes, triangles, cones, cylinders, similar triangles, trig EK 2.1 D2 functions graphs. EK 2.3 C2 Identify absolute local (relative) extrema (maxima/minima) on a graph. EK 1.2 B1 Identify critical numbers on a graph. EK 2.2 A1 EK 2.4 A1 Determine critical numbers of a function using the derivative. EK 2.1 A5 Determine extrema on a closed interval by checking endpoints using the EK 2.1 B1 derivative of the function. EK 2.3 D1 Apply the Extreme Value Theorem to solve problems. EK 2.2 A1 Apply Rolle's Theorem to solve problems. EK 2.2 A2 Apply the Mean-Value Theorem to solve problems. EK 2.2 A3 Interpret the meaning of the derivative in the context of an application problem. Estimate the value of the derivative at a point using information in tables or graphs. Sketch the graph of a basic function, using algebra rules for translation, transformation reflection. Use the First Derivative Test to locate intervals where a function increases or decreases to locate relative extrema, then sketch a graph of the function. Use the graph of the derivative of a function to determine where the original function increases or decreases has relative extrema. Use the Second Derivative its number line test to determine intervals where a function is concave up or concave down where it has inflection points, then sketch a graph of the function. Sketch a graph of a polynomial or rational function using skills from calculus algebra. Evaluate infinite limits involving radical or exponential functions. Solve AP Exam-type questions involving analysis of graphs of the function or of the graph of its first or second derivative. Page 3
4 Unit 2 Days Solve optimization problems, including cylinder or cone inscribed in a sphere, Norman windows, distance from a point to a curve, volume girth of a box, distance/rate/time. EK 2.3 C3 EK 2.3 C3 EK 2.3 B2 Find the differential of a function. Compare the values of delta-y "dy" for a function. Estimate the value of a function using the equation of a tangent line (local linearization). Use differentials to solve problems involving propagated error, relative error percent error. Solve basic business economics applications using derivatives differentials. These include problems with marginal cost, profit, or revenue maximizing profit or revenue, minimizing total or average cost per unit. Unit 3 Days Find the antiderivative evaluate indefinite integrals using the power rule, constant multiple rule, sum difference rule rules for trig functions. EK 3.1 A1 EK 3.1 A2 Find the general solution of basic differential equations using antiderivatives. EK 3.2 A1 Find the particular solution, given initial condition(s) of basic differential EK 3.2 A2 equations using antiderivatives. EK 2.3 A3 Solve physics problems involving an object in free-fall using basic differential EK 3.2 B1 equations. EK 3.2 B2 Approximate area under a curve on a closed interval using numerical integration EK 3.3 B3 techniques Riemann sums (left, right, midpoint) trapezoidal sums. EK 3.3 B4 Evaluate a definite integral using the limit of a Riemann sum. EK 3.4 A3 EK 3.4 C1 Find the antiderivative evaluate indefinite integrals using the power rule, constant multiple rule, sum difference rule rules for trig functions. EK 3.5 A1 Find the general solution or particular solution, given initial condition(s), of basic differential equations using antiderivatives. Solve physics problems involving an object in free-fall using basic differential equations. Approximate area under a curve on a closed interval using numerical integration techniques Riemann sums (left, right, midpoint) trapezoidal sums. Evaluate a definite integral using the limit of a Riemann sum. Page 4
5 Unit 3 Days Write a definite integral to represent the area under a curve above the x-axis on a closed interval. EK 3.2 C1 EK 3.2 C2 Find the area under a curve above the x-axis on a closed interval using an appropriate geometry formula /or by evaluating a definite integral. EK 3.2 C3 EK 3.3 B2 Use properties of the definite integral to solve problems. EK 3.4 B1 State the First Fundamental Theorem of Calculus apply it to evaluate a EK 4.3 D1 definite integral. EK 3.3 B5 Make a sketch of a region bounded by 2 or more graphs, set up a definite integral EK 3.4 D2 to find the area of the region find the area by evaluating the definite integral. Evaluate integrals involving absolute values. Find the average value of a function on a closed interval the value(s) of x where the function equals its average value. Find the area between 2 curves using integration, without with use of graphics calculator capabilities. Evaluate indefinite integrals using u-substitution. Evaluate definite integrals by changing limits of integration while using u- substitution. Evaluate integrals (indefinite definite) involving natural log forms. Evaluate integrals (indefinite definite) involving natural exponential functions. Evaluate integrals (indefinite definite) involving inverse trigonometric forms. Evaluate integrals (indefinite definite) using algebraic manipulation techniques such as long division completing the square. Evaluate integrals (indefinite definite) involving natural exponential functions with bases other than "e". Find the volume generated when a curve or the region between 2 or more curves is revolved around the x-axis, the y-axis, or other horizontal vertical lines using the disk, washer or shell methods. Set up integral(s) to find volume of revolution. Evaluate these integrals manually or using graphics calculator capabilities. Set up integral(s) evaluate integral(s) to find volume of a solid with known cross-section perpendicular to the x-axis or y-axis. Page 5
6 Unit 3 Days Solve problems by applying the 2 nd Fundamental Theorem of Calculus. EK 1.2 A2 Interpret the meaning of an integral as an accumulator function within the context of an application problem. EK 2.1 A3 EK 2.1 C6 Solve problems involving the Net Change Theorem. EK 3.3 A1 Solve problems involving displacement or total distance traveled using EK 3.3 A2 integration. EK 3.3 A3 Solve AP Exam-type problems involving the definition of the derivative. EK 3.3 B1 Solve AP Exam-type problems involving the derivative of an inverse function. EK 3.3 B2 EK 3.4 A1 Find general particular solutions to differential equations using separation of variables. EK 3.4 A2 EK 3.4 C1 Solve differential equation problems involving growth decay. EK 3.4 E1 Solve AP Exam-type problems involving slope fields differential equations. EK 2.3 E1 Verify that a particular function is a solution of a differential equation. EK 2.3 E2 Solve differential equation problems using Euler's Method. EK 2.3 F1 EK 3.5 A1 EK 3.5 A2 EK 3.5 A3 EK 3.5 A4 EK 3.5 B1 EK 2.3 F2 Unit 4 Arc Length Surface Area of Revolution Days Find the length of a curve (arc length) over a closed interval of a function given in rectangular form. Find the surface area of revolution of a curve (arc length) about the x-axis or y- axis over a closed interval of a function given in rectangular form. EK 3.4 D3 Page 6
7 Unit 5 Advanced Days Decide which integration technique is required to evaluate a given integral. Methods include simplifying using basic integration rules, u-substitution, use of trigonometric identities to rewrite first, long division, completing the square, using an inverse trig form, using a Pythagorean conjugate. EK 3.3 B5 Evaluate integrals using integration by parts. Days Evaluate trigonometric integrals by using u-substitution, integration by parts, rewriting using trig or power-reducing identities. Evaluate integrals using trigonometric substitution. Days Evaluate integrals using partial fractions. Denominators may include linear factors, repeated linear factors, irreducible quadratic factors /or repeated irreducible quadratic factors. Synthetic division long division may need to be applied to simplify factor. Evaluate integrals using a table of integrals. Days Evaluate improper integrals. Integrals may contain endpoint discontinuities, interior discontinuities, /or infinite lower or upper limit of integration. Determine whether an improper integral converges or diverges. Days Review Test on Unit 5 EK 3.3 B5 EK 3.3 B5 EK 3.2 D1 EK 3.2 D2 EK 3.3 B5 EK 3.2 D1 EK 3.2 D2 EK 4.1 A1 EK 4.1 A2 EK 4.1 A3 EK 4.1 A6 EK 4.1 B1 Unit 6 Days Write an expression for the nth term of a sequence. Sequences Simplify expressions involving factorial notation. Series Determine whether an infinite sequence converges or diverges. Apply special sequence limit theorems. Derive a formula for continuously compounded interest. Determine whether a sequence is monotonic /or bounded. Explain the difference between a sequence a series. Use a sequence of partial sums to determine if a series converges or diverges. Solve problems involving finite geometric series, including annuity problems. Determine the sum of a telescoping series. Identify an infinite geometric series determine whether it converges or diverges. If it converges, find its sum. Use the nth-term Test for Divergence to show that a series diverges. Page 7
8 Unit 6 Sequences Series Days Determine whether a series converges or diverges using the Integral Test, the p- series Test, the Direct Comparison Test, the Limit Comparison Test, the Ratio Test or the Root Test. Days Determine whether an alternating series converges or diverges using the Alternating Series Test. Determine whether an alternating series converges conditionally or absolutely or if it diverges. Select apply one of the 10 tests for series convergence/divergence. Use Taylor's Formula to write an nth-degree Taylor or Maclaurin polynomial for a given function at a specified center. Use an nth-degree Taylor or Maclaurin polynomial to approximate a functional value. Write a power series for a function using Taylor's formula. Determine the radius of convergence of a power series using the Ratio Test. Determine the interval of convergence of a power series, checking convergence at interval endpoints separately, using the Ratio Test. Find the derivative of a power series. Find the integral of a power series. Find a geometric power series for a rational function. Use long division to determine a power series. Determine a power series for a function using derivatives integrals of known functions. Determine a Taylor or Maclaurin series for a function using substitution, multiplication or division with known power series for sine, cosine, natural log natural exponential functions. Use the Alternating Series Remainder Theorem to determine the maximum error bound when using an nth-degree Taylor or Maclaurin polynomial to approximate a value. Use the Lagrange Remainder Theorem to determine the maximum error bound when using an nth-degree Taylor or Maclaurin polynomial to approximate a value. EK 4.1 A3 EK 4.1 A4 EK 4.1 A5 EK 4.1 A6 EK 4.1 B2 EK 4.1 B3 EK 4.2 A1 EK 4.2 A2 EK 4.2 A3 EK 4.2 A4 EK 4.2 A5 EK 4.2 B1 EK 4.2 B2 EK 4.2 B3 EK 4.2 B4 EK 4.2 B5 EK 4.2 C1 EK 4.2 C2 EK 4.2 C3 EK 4.2 C4 Approximate the value of a definite integral using power series. Page 8
9 Unit 7 Calculus with Days Sketch the graph of a function (manually by using graphics calculators) defined by parametric equations, identifying domain range. EK 2.1 C7 EK 3.4 D3 Parametric Convert parametric equations into rectangular form by eliminating the parameter. Equations, Find the parametric form of the first derivative use to find the slope of the Polar curve at a specific value of the parameter or specific point. Coordinates, Find the equation of tangent line(s) at a point on a curve defined by parametric Vectors, equations. Logistics Find the 2 nd derivative in parametric form use to determine the concavity of Equations the curve. Find points where a curve defined by parametric equations has horizontal or vertical tangent lines. Use information obtained by 1 st 2 nd derivatives to sketch the graph of a curve defined by parametric equations. Find the arc length of a curve defined by parametric equations over a given interval of parameter values. Days 125- Convert coordinates equations between rectangular polar forms. EK 2.1 C7 134 Sketch the graph of a function (manually by using graphics calculators) defined by a polar equation, identifying domain range. EK 2.2 A4 EK 3.4 D1 Find the polar form of the first derivative use to find the slope of the curve at a specific value of the angle or specific point. EK 3.4 D3 Find the equation of tangent line(s) at a point on a curve defined by a polar equation. Find points where a curve defined by a polar equation has horizontal or vertical tangent lines or tangent lines at the pole. Find relative extrema of "r". Use information obtained by 1 st derivative to sketch the graph of a curve defined by a polar equation. Find the arc length of a curve defined by a polar equation over a given interval of angle values. Find points of intersection of polar graphs. Find the area of a polar region bounded by a single curve. Find the area of a polar region between two curves. Page 9
10 Unit 7 Calculus with Days Find the component form linear combination form of a vector, given its magnitude direction angle. EK 2.1 C7 EK 2.3 C4 Parametric Equations, Find the magnitude direction angle of a vector, given its components or linear combination form. EK 3.4 C2 EK 3.4 D3 Polar Solve problems, including physics application problems, using basic vector EK 3.5 A1 Coordinates, operations (addition, scalar multiplication). EK 3.5 B2 Vectors, Find the limit of a vector-valued function. Logistics Find the derivative of a vector-valued functions. Equations Find the indefinite integral of a vector-valued function. Find the definite integral of a vector valued function, given initial condition(s). Find the velocity acceleration vectors, given a position vector. Find a specific velocity acceleration vector, given a position vector, at a specific point or value of the parameter. Find the acceleration vector, given the velocity vector. Find the velocity or speed of a particle moving along a curve defined by parametric or vector-valued functions. Use integration the Net Change Theorem to determine the position (at a specific time) of a particle moving along a curve defined by parametric or vectorvalued functions. Days Use integration to find the displacement or total distance traveled over a given interval for a particle moving along a curve defined by parametric or vectorvalued functions. Identify a differential equation that models logistic growth. Given a logistic differential equation, identify the carrying capacity, the limit of the function as time approaches infinity, the point at which the maximum rate of change of the function occurs. Identify characteristics of the graph of a logistics equation. Review for AP Calculus BC Exam. Exam is given on Day 165 (approx.) All stards from the course thus far. Page 10
11 Unit 8 Days 166- Evaluate a function of two or more variables for a given set of values. Beyond AP Introduction to Calculus Find the first partial derivative with respect to any variable for a function of two or more variables. Calculus Stards Find higher order partial derivatives with respect to any variable for a function of two or more variables. Use right-to-left left-to-right notation when working with partial derivatives. Find the total differential of a function of two or more variables. Use the total differential to find propagated, relative percent error in experiment calculations. Evaluate double integrals. Evaluate triple integrals. Unit 9 Days 170- Find the order of a matrix. Beyond AP Introduction to Linear Algebra 175 Perform basic matrix operations (addition, subtraction, scalar multiplication, matrix multiplication). Calculus Stards Evaluate a 2 x 2 determinant. Find the minor of any entry in a matrix. Evaluate a 3 x 3 determinant using the diagonals method or expansion by minors along any row or column. Evaluate a 4 x 4 determinant using expansion by minors. Solve a 2 x 2 or 3 x 3 system of equations using Cramer's Rule. Solve a 2 x 2 or 3 x 3 system of equations using Gaussian elimination or Gauss- Jordan elimination. Find the inverse of a matrix by simultaneously using elementary row operations on a given matrix the identity matrix of the same order. Solve systems of equations using inverse matrices. Unit 10 More If time Find the distance between the midpoint of two points in space. Beyond AP on Vectors permits Solve problems involving general stard equations of a sphere. Calculus Find the dot product of two vectors. Stards Find the cross product of two vectors. Use the dot product to find the angle between two vectors. Use the dot product to determine if two vectors are orthogonal. Solve application problems using the dot product or cross product. Page 11
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