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Year course.

Prerequisite: Honors Precalculus and teacher’s recommendation. 

TEXT: THOMAS´CALCULUS

AUTHOR: R.L. Finney, M.D. Weir and F.R. Giordano

PUBLISHER:  Addison Wesley Longman  Copyright 2001.

COURSE OVERVIEW:

This is a college level course in differential and integral calculus.

ESSENTIAL LEARNINGS:

Students will be able to:

Introduction PRELIMINARIES. FUNCTIONS (10 days)

·        Use Equations of lines.

·        Graph Functions.

·        Understand Inverse Functions and Logarithms.

·        Use trigonometric Functions.

·        Use exponential and Logarithmic Functions

·        Use Parametric Equations.

CHAPTER 1.- LIMITS AND CONTINUITY. (20 days)

·        Define and apply rate of change and limits.

·        Find velocity and rate.

·        Use limit theorems.

·        Use the Squeezing Theorem

·        Evaluate limits.

·        Find one-sided limits.

·        Use the definition of continuity.

·        To find limits at infinity.

·        Find the slope of tangents to a curve

CHAPTER 2.- DERIVATIVES (20  days)

·        Define derivative.

·        To understand the derivative as rate of change.

·        Find derivatives of polynomial and rational functions.

·        Use the chain rule and parametric equations.

·        Find the derivative by implicit differentiation.

·        Find derivatives of composite functions.

·        Find derivatives of trigonometric functions.

·        Use Related Rates in Problem Solving.   

CHAPTER 3.- APPLICATIONS OF DERIVATIVES (20 days)

·        Find the extreme values of functions.

·        Use Rolle’s Theorem.

·        Use the Mean Value Theorem.

·        Use the First and Second Derivative Tests.

·        Find Concavity and Inflection Points.

·        Use Derivatives to Sketch Graphs.

·        Solve graphically Autonomous Differential Equations.

·        Use Modeling and learn some techniques of optimization.

·        To use Newton´s method to solve equations.

·        Linearize functions.

CHAPTER 4.-THE INTEGRAL (15 days)

·        Indefinite integrals, Differential Equations and Modeling.

·        Find the area under a curve.

·        Integrate Numerically and to use Riemann Sums.

·        Understand and apply the Fundamental Theorem of Calculus.

·        Use integration rules.

·        Integrate by substitution.(END OF FIRST SEMESTER)

CHAPTER 5.- APPLICATION OF INTEGRALS (20 days)

·        Find the Volume using the Cross-Sectional Method

·        Find the Volume using the Shell and Washer Method

·        Find the Length of a Curve

·        To solve problems related with  Springs, Pumping and Lifting.

·        To apply integration to Fluid Forces.

CHAPTER 6.-TRANSCENDENTAL FUNCTIONS AND DIFFERENTIAL EQUATIONS  (20 days)

·        Find the derivative of a natural logarithm.

·        Find the derivative and the integral of e^x.

·        Find the derivative of a y = a^x, y = a^mx, and y = logn[u(x)].

·        Derivatives of Inverse Trigonometric Functions. Integrals

·        To Solve simple Differential Equations.

·        To use L'Hopital's Rule

·        Hyperbolic Functions.

CHAPTER 7 .- TECHNIQUES OF INTEGRATION (30 days)

·        To Integrate by Parts

·        To Integrate using Partial Fractions

·        To integrate using Trigonometric Substitutions.

·        To solve Improper Integrals.

CHAPTER 8.- INFINITE SERIES (15 days)

·        To find Limits of Sequences of Numbers.

·        To understand Infinite Series.

·        To use Power Series.

·        To apply the Taylor and Maclaurin Series. (END OF SECOND SEMESTER)