Sept

Oct

Nov

Dec

Jan

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March

April

May

June

 Themes

Prerequisites for Calculus

Limits and Continuity

Derivatives

Derivatives/Applications of Derivatives

Applications of Derivatives

The Definite Integral

The Definite Integral/Differential Equations and Mathematical Modeling.

Differential Equations and Mathematical Modeling/Applications of Definite Integrals

Applications of Definite Integrals/ Sequences, L’Hopital’s Rule, and Improper Integrals.

Infinite Series

Major Skills

1.1 Lines: Students will be able to: find increments, the slope of a line, tell if lines are parallel and perpendicular, find the equation of a line, and use lines to make predictions.

1.2. Functions and Graphs: Students will be able to: Find the domain and range of a function, tell if it is even or odd, determine symmetry, graph a piecewise function, absolute value function, and composite function.

1.3. Exponential Functions: Students will be able to model exponential growth and decay, and use the models to predict.

1.5. Students will be able to tell whether a function is one to one, find inverses of functions, and find logarithmic functions and use them for prediction.

1.6: Trigonometric Functions: Students will be able to use Radian measure, find graphs of trig. Functions, tell which are even and odd, and find inverse trig. functions.

2.1: Rates of Change and Limits. Students will be able to find average and instantaneous speed, one sided and two sided limits, and apply the sandwich theorem.

2.2: Limits involving infinity. Students will be able to find finite limits as x approaches positive or negative infinity, find infinite limits as x approaches a, and use end behavior models.

2.3: Continuity: Students will be able to determine continuity at a point, determine continuous functions, and use the intermediate value theorem for continuous functions.

2.4: Rates of Change and Tangent Lines: Students will be able to find an average rate of change, find a line tangent to a curve, and the slope of a curve.

3.1: Derivative of a Function: Students will: know the definition of a derivative, be able to tell the relationship between the graphs of f and f’, use data to graph the derivative, and find one sided derivatives.

3.2: Differentiability: Students will be able to tell how f’(a) might fail to exist, know that differentiability implies local linearity and continuity, find derivatives on the calculator, and use the Intermediate Value Theorem for Derivatives.

3.3: Rules for Differentiation: Students will be able to find derivatives of positive integer powers, multiples, sums and differences, products and quotients, negative integer powers of x, and second and higher order derivatives.

3.4: Velocity and other rates of change: Students will be able to find instantaneous rates of change and model motion along a line.

3.5 Derivatives of Trigonometric Functions: Students will be able to find the derivative of the sine, cosine and other basic trigonometric functions. They will also understand the concept of jerk.

3.6: Chain Rule

3.7: Implicit Differentiation

3.8: Derivatives of Inverse Trigonometric Functions

3.9: Derivatives of Exponential and Logarithmic Functions.

4.1: Students will be able to determine the local or global extreme values of a function.

4.2: Students will be able to apply the Mean Value Theorem and find the intervals on which a function is increasing or decreasing.

4.3 Students will be able to use the First and Second Derivative Tests to determine the local extreme values of a function. SWBAT determine the concavity of a function and locate the points of inflection by analyzing the second derivative. SWBAT graph f using information about f’.

4.4: SWBAT solve application problems involving finding minimum or maximum values of functions.

4.5: SWBAT find linearizations and use Newton’s method to approximate the zeros of a function.

SWBAT estimate the change in a function using differentials.

4.6: SWBAT solve related rate problems.

5.1: SWBAT approximate the area under the graph of a nonnegative continuous function by using rectangle approximation methods.

SWBAT interpret the area under a graph as a net accumulation of a rate of change.

5.2: SWBAT express the area under a curve as a definite integral and as a limit of Riemann Sums. SWBAT compute the area under a curve using a numerical integration procedure.

5.3: SWBAT apply the rules for definite integrals and find the average value over a closed interval.

5.4: SWBAT apply the Fundamental Theorem of Calculus. Students will understand the relationship between the derivative and the definite integral.

5.5: SWBAT approximate the definite integral by using the Trapezoidal rule and Simpson’s rule, and estimate the error in using each.

6.1: SWBAT construct antiderivatives using the Fundamental Theorem of Calculus, solve initial value problems of the form dy/dx = f(x), y0 = f(x0), construct slope fields using technology and interpret slope fields as visualizations of different equations, and use Euler’s method for graphing a solution to an initial value problem.

6.2: SWBAT compute indefinite and definite integrals using the method of substitution.

6.3: SWBAT use integration by parts to evaluate indefinite and definite integrals, use tabular integration or the method of solving for the unknown integral in order to evaluate integrals that require repeated use of integration by parts. SWBAT use integration by parts to integrate inverse trigonometric and logarithmic functions.

6.4: SWBAT solve problems involving exponential growth and decay in a variety of applications.

6.5: SWBAT solve problems involving exponential or logistic population growth.

7.1: SWBAT solve problems in which a rate is integrated to find the net change over time in a variety of applications.

7.2: SWBAT use integration to calculate areas of regions in a plane.

7.3: SWBAT use integration (by slices or shells) to calculate volumes of solids. SWBAT use integration to calculate surface areas of solids of a revolution.

Wednesday, May 7 th: AP exam

7.4: SWBAT use integration to calculate lengths of curves in a plane.

8.1: SWBAT define limits explicitly or recursively, define explicit and recursive rules for arithmetic and geometric sequences, graph sequences using parametric or graphing mode, use properties of limits to find the limit of a sequence.

8.2: SWBAT find the limits of indeterminate forms using l’Hopital’s rule.

8.3: SWBAT use l’Hopital’s rule to compare growth rates of functions.

8.4: SWBAT use limits to evaluate improper integrals. SWBAT use the direct comparison test and the limit comparison test to determine the convergence or divergence of improper integrals.

9.1: SWBAT apply the properties of geometric series. SWBAT differentiate, integrate, or substitute into a known power series in order to find additional power series representations.

9.2: SWBAT use derivatives to find the McLaurin series or Taylor series generated by a different function.

9.3: SWBAT approximate a function with a Taylor Polynomial.

SWBAT analyze the truncation error of a series using graphical methods or the Remainder Estimation Theorem.

SWBAT use Euler’s formula to relate the functions sin(x), cos(x) and e^x.

Textbook Chapters

CH 1: Prerequisites for Calculus

Ch 2 / Ch 3

Ch 3

Ch 3/4

Ch. 4

Ch. 4/5

Chapter 5/6

Chapter 6/7

Chapter 7,8

Chapter 9