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HomeSubjectsAdvanced MathematicsCalculus

Calculus

Teachers
Momeena Garcia
Category:
Advanced Mathematics/
Calculus
Duration: 45- or 90-hour course
Lectures: Daily classwork and homework in DarwinPrep Diagnostic and summative exams Accreditation in California and New York

Course Description:

Calculus is the mathematical study of change and is divided into two different branches, differential Calculus and integral Calculus. This introductory course in Calculus serves both as a gateway to other, more advanced courses of mathematics, and as platform which will allow students to begin to engage in analytical mathematical thought. The course covers many skills which require intricate analysis and abstract thinking, and balances those concepts with activities that allow students to make the connection between the abstract and the world around them.

Curriculum:

Introduction and Course overview

Introduction to Calculus
Welcome to your lesson review.

Unit 1: Introduction to Limits

Introduction to Limits
Introduction to Limits

Students will demonstrate knowledge of both the formal definition and the graphical interpretation of the limit of values of functions.
Students will prove and use theorems evaluating the limits of sums, products, quotients, and composition of functions.
Understanding the concept of limit and identifying real life applications

Unit 2: Continuity

Continuity
Continuity

Students demonstrate knowledge of both the formal definition and the graphical interpretation of continuity of a function.
Students demonstrate an understanding and the application of the intermediate value theorem and the extreme value theorem.

Unit 3: Derivatives and rates of Change

Derivatives and rates of Change
Derivatives and rates of Change

Students demonstrate an understanding of the formal definition of the derivative of a function at a point and the notion of differentiability.
Students demonstrate an understanding of the derivative of a function as the slope of the tangent line to the graph of the function.
Students demonstrate an understanding of the interpretation of the derivative as an instantaneous rate of change. Students can use derivatives to solve a variety of problems from physics, chemistry, economics, and so forth that involve the rate of change of a function.
Students understand the relation between differentiability and continuity.
Students derive derivative formulas and use them to find the derivatives of algebraic, trigonometric, inverse trigonometric, exponential, and logarithmic functions.
Students know the chain rule and its proof and applications to the calculation of the derivative of a variety of composite functions.
Students compute derivatives of higher orders.
Students know and can apply Rolle’s Theorem, the mean value theorem, and L’Hôpital’s rule.

Unit 4: Other Applications of Derivatives

Other Applications of Derivatives
Other Applications of Derivatives

Students find the derivatives of parametrically defined functions and use implicit differentiation in a wide variety of problems in physics, chemistry, economics, and so forth.
Students use differentiation to sketch, by hand, graphs of functions. They can identify maxima, minima, inflection points, and intervals in which the function is increasing and decreasing.
Students know Newton’s method for approximating the zeros of a function.
Students use differentiation to solve optimization (maximum-minimum problems) in a variety of pure and applied contexts.
Students use differentiation to solve related rate problems in a variety of pure and applied contexts.

Unit 5: Introduction to Integrals

Introduction to Integrals
Introduction to Integrals

Students know the definition of the definite integral by using Riemann sums. They use this definition to approximate integrals.
Students apply the definition of the integral to model problems in physics, economics, and so forth, obtaining results in terms of integrals.
Students demonstrate knowledge and proof of the fundamental theorem of calculus and use it to interpret integrals as antiderivatives.
Students use definite integrals in problems involving area, velocity, acceleration, volume of a solid, area of a surface of revolution, length of a curve, and work.

Unit 6: Techniques of Integration

Techniques of Integration
Techniques of Integration

Students compute, by hand, the integrals of a wide variety of functions by using techniques of integration, such as substitution, integration by parts, and trigonometric substitution. They can also combine these techniques when appropriate.
Students know the definitions and properties of inverse trigonometric functions and the expression of these functions as indefinite integrals.
Students compute, by hand, the integrals of rational functions by combining the techniques in standard 17.0 with the algebraic techniques of partial fractions and
completing the square.
Students compute the integrals of trigonometric functions by using the techniques noted above.
Students understand the algorithms involved in Simpson’s rule and Newton’s method. They use calculators or computers or both to approximate integrals numerically.
Students understand improper integrals as limits of definite integrals.

Unit 7: Series and Sequences

Series and Sequences
Series and Sequences

Students demonstrate an understanding of the definitions of convergence and divergence of sequences and series of real numbers. By using such tests as the comparison test, ratio test, and alternate series test, they can determine whether a series converges.
Students understand and can compute the radius (interval) of the convergence of power series.
Students differentiate and integrate the terms of a power series in order to form new series from known ones.
Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.
Students understand and can compute the radius (interval) of the convergence of power series.
Students differentiate and integrate the terms of a power series in order to form new series from known ones.
Students calculate Taylor polynomials and Taylor series of basic functions, including the remainder term.
Students know the techniques of solution of selected elementary differential equations and their applications to a wide variety of situations, including growth-and-decay problems.

About Instructors

Momeena Garcia
Momeena Garcia teaches math (K-12 and college), English language arts (elementary and middle school), special education (K-12), and Spanish, as well as preparation for exams such as the SAT and ACT.
Duration: 45- or 90-hour course
Lectures: Daily classwork and homework in DarwinPrep Diagnostic and summative exams Accreditation in California and New York

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