COURSE GOALS
- Students should be able to work with functions represented in a variety of ways: graphical, numerical, analytical, or verbal. They should understand the connections among these representations.
- Students should understand the meaning of the derivative in terms of a rate of change and local linear approximation, and should be able to use derivatives to solve a variety of problems.
- Students should understand the meaning of the definite integral both as a limit of Riemann sums and as the net accumulation of change, and should be able to use integrals to solve a variety of problems.
- Students should understand the relationship between the derivative and the definite integral as expressed in both pads of the Fundamental Theorem of Calculus.
- Students should be able to communicate mathematics and explain solutions to problems both verbally and in written sentences.
- Students should be able to model a written description of a physical situation with a function, a differential equation, or an integral.
- Students should be able to use technology to help solve problems, experiment, interpret results, and support conclusions.
- Students should be able to determine the reasonableness of solutions, including sign, size, relative accuracy, and units of measurement.
- Students should develop an appreciation of calculus as a coherent body of knowledge and as a human accomplishment.
OBJECTIVES
These objectives are taken from the 2012 College Board Calculus Course Description.
I. Functions, Graphs, and Limits
Analysis of graphs. With the aid of technology, graphs of functions are often easy to produce. The emphasis is on the interplay between the geometric and analytic information and on the use of calculus both to predict and to explain the observed local and global behavior of a function.
Limits of functions (including one-sided limits)
- An intuitive understanding of the limiting process.
- Calculating limits using algebra.
- Estimating limits from graphs or tables of data.
Asymptotic and unbounded behavior
- Understanding asymptotes in terms of graphical behavior.
- Describing asymptotic behavior in terms of limits involving infinity.
- Comparing relative magnitudes of functions and their rates of change (for example, contrasting exponential growth,
polynomial growth, and logarithmic growth).
Continuity as a property of functions
- An intuitive understanding of continuity. (The function values can be made as close as desired by taking sufficiently close values of the domain.)
- Understanding continuity in terms of limits.
- Geometric understanding of graphs of continuous functions (Intermediate Value Theorem and Extreme Value Theorem).
- Parametric, polar, and vector functions. The analysis of planar curves includes those given in parametric form, polar form,
and vector form.
II. Derivatives
Concept of the derivative
- Derivative presented graphically, numerically, and analytically.
- Derivative interpreted as an instantaneous rate of change.
- Derivative defined as the limit of the difference quotient.
- Relationship between differentiability and continuity.
Derivative at a point
- Slope of a curve at a point. Examples are emphasized, including points at which there are vertical tangents and points at which there are no tangents.
- Tangent line to a curve at a point and local linear approximation.
- Instantaneous rate of change as the limit of average rate of change.
- Approximate rate of change from graphs and tables of values.
Derivative as a function
- Corresponding characteristics of graphs of ƒ and ƒ•.
- Relationship between the increasing and decreasing behavior of ƒ and the sign ofƒ•.
- The Mean Value Theorem and its geometric interpretation.
- Equations involving derivatives. Verbal descriptions are translated into equations involving derivatives and vice versa.
Second derivatives
- Corresponding characteristics of the graphs of ƒ, ƒ•, and ƒ •.
- Relationship between the concavity of ƒ and the sign of ƒ •.
- Points of inflection as places where concavity changes.
Assignment:
Through exploration students will investigate critical points both numerically and graphically. Students will then verbalize their findings, describing when a point will be a maximum, minimum or point of inflection.
Applications of derivatives
- Analysis of curves, including the notions of monotonicity and concavity.
- Analysis of planar curves given in parametric form, polar form, and vector form, including velocity and acceleration.
- Optimization, both absolute (global) and relative (local) 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 varied applied contexts, including velocity, speed, and acceleration.
- Geometric interpretation of differential equations via slope fields and the relationship between slope fields and solution curves for differential equations.
- Numerical solution of differential equations using Euler’s method.
- L’Hospital’s Rule, including its use in determining limits and convergence of improper
integrals and series.
Assignment:
This assignment uses a graphical, numerical and analytic approach to help students to develop an understanding of the relationship between position, velocity and acceleration. Students will learn to reason from tables and graphs and to communicate the relationships using correct language and notation.
Computation of derivatives
- Knowledge of derivatives of basic functions, including power, exponential, logarithmic, trigonometric, and inverse
trigonometric functions. - Derivative rules for sums, products, and quotients of functions.
- Chain rule and implicit differentiation.
- Derivatives of parametric, polar, and vector functions.
Assignment:
Parametric Equations:
- Sketch the curve defined by the parametric equations and eliminate the parameter.
- Find and evaluate them for a given value of t.
- Write an equation for the tangent line to the curve for a given value of t.
- Find the points of horizontal and vertical tangency.
- Find the length of an arc of a curve given by parametric equations.
Vector Equations:
- Find the velocity and acceleration vectors when given the position vector.
- Given the components of the velocity vector and the position of the particle at
a particular value oft, find the position at another value of t. - Given the components of the acceleration vector and the velocity of the particle at a particular value of t, find the velocity at another value of t.
- Find the slope of the path of the particle for a given value of t.
- Write an equation for the tangent line to the curve for a given value of t.
- Find the values oft at which the line tangent to the path of the particle is horizontal or vertical.
Assignment:
Reasoning from Tabular Data
Students analyze data from a table. Students will use data to estimate rates or change, net changeand average value from values of functions given in the table. They will need to use a variety of methods to find derivatives and integrals as they relate to real world applications.
Assignment:
Students Discover the First and Second Fundamental Theorems of Calculus through graphical exploration.
Assignment:
Students discover and explore Second Fundamental Theorem of Calculus. Students then use theorem toanalyze the F(x) from the graph of the derivative of the function f.
III. Integrals
Interpretations and properties of definite integrals
- Definite integral as a limit of Riemann sums.
- Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval:
- Basic properties of definite integrals (examples include additivity and linearity).
- Applications of integrals. Appropriate integrals are used in a variety of applications to model physical, biological, or economic situations. Although only a sampling of applications can be included in any specific course, students should be able to adapt their knowledge and techniques to solve other similar application problems.
Whatever applications are chosen, the emphasis is on using the method of setting up an approximating Riemann sum and representing its limit as a definite integral. To provide a common foundation, specific applications should include finding the area of a region (including a region bounded by polar curves), the volume of a solid with known cross sections, the average value of a function, the distance traveled by a particle along a line, the length of a curve (including a curve given in parametric form), and accumulated change from a rate of change.
Fundamental Theorem of Calculus
- Use of the Fundamental Theorem to evaluate definite integrals.
- Use of the Fundamental Theorem to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined.
Techniques of antidifferentiation
- Antiderivatives following directly from derivatives of basic functions.
- Antiderivatives by substitution of variables (including change of limits for
definite integrals), parts, and simple partial fractions (nonrepeating linear factors only). - Improper integrals (as limits of definite integrals).
Applications of antidifferentiation
- Finding specific antiderivatives using initial conditions, including applications to motion along a line.
- Solving separable differential equations and using them in modeling (including the study of the equation y• = ky and exponential growth). + Solving logistic differential equations and using them in modeling.
Numerical approximations to definite integrals. Use of Riemann sums (using left, right, and midpoint evaluation points) and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically, and by tables of values
ASSESSMENTS
Students will be expected to complete sample problems in each unit as practice for unit tests. Unit tests include multiple-choice and free-response questions from AP Released Exams. Students will be asked to periodically meet with the instructor in the instructor’s chat room to review unit tests and discuss the scoring guidelines. In addition to practice problems and unit tests, students will be asked to complete activities requiring use of graphing calculators and written justifications of their solutions. Sample activities include estimating the limit from a graph and comparing to the numerical computation of the limit, performing differentiation and integration, calculate definite integrals and verify properties of functions. Grades will be determined by student participation in completing practice problems and graded activities as well as by unit test scores.
GRADING SYSTEM
A 100% – 90%
B 89% – 80%
C 79% – 70%
D 69% – 60%
F 59% and below
MATERIALS
Technology Requirement
You will want to have a graphing calculator. I recommend the TI-84
We will use the calculator in a variety of ways including:
- Conduct explorations.
- Graph functions within arbitrary windows.
- Solve equations numerically.
- Analyze and interpret results.
- Justify and explain results of graphs and equations.
