Calculus Course Description and Philosophy

Transcription

1 Calculus Course Description and Philosophy The Math 5 Course presented at the high school level is intended to serve those students who have successfully completed Math 4 (Pre-calculus). The general aim of the course is to develop the basic topics of differential and integral calculus that are generally found in any college level Calculus 1 course. Course content includes the study of functions, limits, continuity, slope of the tangent line, differentiation and integration of algebraic and trigonometric functions, differentials, applications of derivatives, Mean Value Theorem, Fundamental Theorem of Calculus, the Riemann integral, and areas under or between curves. Each new skill will primarily be taught as a solution to a problem. The Calculator (TI-84) and the overhead will be used extensively, as well as lecture and cooperative problem solving groups. Students will be evaluated through the use of quizzes, tests, homework, and projects. Text Reference: C. H. Edwards, Jr. and David E Penney Calculus with Analytic Geometry, 4 th edition, 1994 by Prentice Hall, Englewood Cliffs, NJ, USA Written 2007 Unit I: Introduction and Review of Pre-calculus Essential Question: What are the two basic questions answered by Calculus and how does knowledge of the graphs of functions enable us to answer them? Objectives: Students will be able to: 1

2 find the average rate of change of an object in motion evaluate a function for a given value of x find the domain and range of a given function graph linear functions write the equation of a line in slope intercept form given sufficient information use the distance formula and Pythagorean theorem graph parabolas find the equation of circles and sketch them draw and recognize sketches of the identity function, cubic functions, square root functions, absolute value functions, and sine, cosine, and tangent functions use vertical line test to determine if a graph is a function 2

3 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit I Topic 1: Finding Average Rates Analyze the difference between Test/Quizzes Text Lecture Overhead average and instantaneous rate Class activities Teacher-made Discussion Calculator find the average rate of an object over a period of time SmartBoard Calculator Labs Worksheets Small Group Work 4.2.C, 4.3.B.1, 4.3.B.2 Homework Topic 2: Using Functions Determining whether a relation is a function To evaluate a function for a particular value To write the domain and range of a function in interval form Topic 3: Linear Functions To write equations of lines given points, slope, etc To use the distance formula to find the distance between points To find slopes of parallel and perpendicular lines Topic 4: Parabolas To graph a parabola To write the standard form of a parabola To find the x intercepts of a parabola Topic 5: Circles To write the equation of a circle To graph a circle Topic 6: Special Functions To graph cubic functions To graph the square root functions To graph the absolute value function To graph the sine, cosine and tangent functions To graph the greatest integer function Unit II: LIMITS Essential Question: How does one find the instantaneous velocity of an object in motion and why do we care? Objectives: Students will be able to: 3

4 explain the concept of a limit and its use evaluate limits evaluate limits involving infinity evaluate one sided limits use limits to determine continuity of a function use the Intermediate Value Theorem to find intervals of continuity 4

5 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit II Topic 1: The Concept of a Limit Instantaneous rate of change Test/Quizzes Text Lecture 4.1.B.1, 4.1.C.1 Finding limits graphically Teacher created 4.2.D.2, 4.3.A.2 The formal definition of limit worksheets 4.3.A.3, 4.3.C.1 The limit laws 4.5.A.1, 4.5.A.2 One sided limits and continuity 4.5.B.ALL Intermediate Value Theorem Discussion Small Group Work PowerPoint Report Homework ibooks graphing calculator Media Center VCR/SmartBoard Topic 2: Evaluating Limits Evaluating using the properties of a limit Using substitution and factoring to evaluate limits Evaluating limits using more difficult techniques Evaluating limits at infinity Evaluating limits using Trigonometric functions Differentiated Learning Activities: Students may use graphs to visually interpret the concepts of a limit Projects on page 91 of text may be done in groups. Students will use diagrams, Written explanations, and numerical proofs Unit III: Derivatives and the Evaluation of Derivatives Essential Question: Is there a reliable accurate method for finding instantaneous rates of change instead of the approximation given to us by the limit? Objectives: Students will be able to: 5

6 Find the formal definition of a derivative Find the slope of a tangent line Find the equation of the tangent line to a curve Determine if a function is differentiable Find a derivative using techniques such as the chain rule, power rule, product rule, quotient rule, and implicit differentiation. 6

7 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit III Topic 1: Understanding the Derivative Rates of change, secants, and tangents Test/Quizzes Text Lecture 4.3.B.2, 4.3.B.3 Finding instantaneous velocity Teacher Created 4.3.C.1, 4.5.A.1 The formal definition of derivative worksheets 4.5.A.2, 4.5.A.3 Differentiability and Continuity Discussion Digital Projector 4.5.B, 4.5.C.4 Lab Activity Small Group Work PowerPoint Report 4.5.F Homework graphing calculator Media Center Lab Work SmartBoard Topic 2: Using the derivative The slope of the tangent line Instantaneous rate The equation of the tangent line The short cut for finding derivatives Topic 3: Evaluation of the Derivative Evaluate derivatives using the power rule Evaluate the derivative to find velocity and acceleration of an object in motion Evaluate the derivative using the product rule Evaluate the derivative using the quotient rule Evaluate the derivative using the chain rule The derivative of trigonometric functions Evaluate the derivatives of trigonometric functions using the derivative rules Topic 4: Implicit Differentiation Introduction to implicit differentiation and why we use it How to solve problems using implicit differentiation Using implicit differentiation to solve related rates problems Differentiated Learning: Students may present the applications through the use of pictures, paragraph explanations, or algebraic proofs. Projects involve applications of investigations and may include text projects such as those on page 183 in text Unit IV: Applications of the Derivative Essential Question: What are some of the ways in which we can use the derivative to solve real life problems that cannot be solved any other way? 7

8 Objectives: Students will be able to: Evaluate the maximum or minimum values of a function to find such quantities as maximum profit, maximum area, etc. Solve classic Box and Fence type problems Solve Box-Volume Type problems Solve Maximum Volume problems involving cylinders Solve Wire Problems involving cutting wire to maximize or minimize areas Use derivatives and the tangent line to approximate difficult numbers Use differentials to find linear approximations Use Newton s Method of successive iterations to approximate the roots of a function Find intervals on which a function is increasing or decreasing and use this to identify maxima or minima Use the first derivative test to determine where a function has a maxima or minima, and its possible critical points Use the second derivative test to determine the possible inflection points of a function and the intervals on which the function is concave up or concave down Determine the locations of any asymptotes of a function including vertical, horizontal, and oblique asymptotes Sketch curves using first and second derivative tests including asymptotes 8

9 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit IV Topic 1: Finding Maxima and Minima Finding extreme values Test/Quizzes Text Lecture Internet 4.3.B.2, 4.3.B.4 Relative vs. absolute extreme graphing calculator 4.3.C.1, 4.5.A.all Finding critical points algebraically 4.5.B.all, 4.5.F.1 Finding critical points graphically Teacher-created- SmartBoard The Mean Value Theorem worksheets Rolle s Theorem Discussion Lab Activity Small Group Work Homework Media Center Topic 2: Applications of Optimization The Box and Fence Problem The Box Problem The Can Problem The Wire Problem Topic 3: Linearization Using the tangent line to approximate difficult numbers ( or how your calculator does it!) Use differentials to find linear approximations Newton s Method of Successive Iterations Topic 4: Curve Sketching Find intervals on which a curve is increasing or decreasing using critical numbers and sketch Sketch a complete curve using zeros, critical numbers, and behavior at infinity Using the second derivative test to determine concavity and inflection points and apply them to curve sketching Finding the asymptotes of a function including vertical, horizontal, and oblique Sketching a curve with asymptotes Differentiated Learning Activities: Students will be graphing visually and interpreting their graphs during discussions. Ethical Decision Making/Character Education: What might be some ethical problems in the optimization process? How might maximizing profit affect quality of work, employee morale, etc? Unit V: The Integral Essential Question: How do we find the equation to predict the path of an object in motion from some piece of information about velocity or acceleration? Objectives: Students will be able to: 9

10 Understand the purpose of integral calculus and find the antiderivative of powers of x Evaluate indefinite integrals including some with trigonometry Solve Initial Value Problems and differential equations Find the integral of a function using U-Substitution Find the integral of a function using u-substitution with trigonometry Write a series using summation notation Use summation formulas to evaluate summations Evaluate summations with a telescoping series Approximate the area of an exotic region using a Riemann Sum Approximate the area using the Riemann Sum by letting the number of subdivisions approach infinity Use the Fundamental Theorem of Calculus to find the definite integral Evaluate the definite integral 10

11 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit V Topic 1:Antiderivatives Antidifferentiation and the indefinite Integral Test/Quizzes Text Lecture 4.3.A.1, 4.3.B Antiderivatives and powers of x Discussion Overhead Projector 4.5.A.1.2 Antiderivatives of trigonometric functions Lab Activity Small Group Work Graphing Calculator 4.5.B.1.2 Initial Value Problems Homework SmartBoard 4.5.C.all Topic 2: Integration by Substitution Undoing the Chain Rule Integrating Polynomials using U-substitution U-substitution involving trigonometric functions Topic 3: Summation Summation Notation The Summation Formulas for evaluation The Telescoping Series Summations as a method of approximating areas of regions The Riemann Sum Topic 4: The Definite Integral The Riemann Sum as it applies to the Definite Integral Evaluating the Definite Integral The Mean Value Theorem Lab Work 4.5.E.all 4.5.F.3.4 Unit VI: Applications of the Integral Essential Question: How do we find the area or volume of an exotic region? Objectives: Students will be able to: Find the definite integral as it relates to the area under a curve Set up a definite integral to find the area under a curve and evaluate 11

12 Sketch the area to be evaluated under a curve Set up a definite integral to find the area of a region bounded by several curves and evaluate Set up a definite integral to find the area between two or more curves and evaluate Find the area of a region by integrating with respect to y Find the volume using cross sections Finding Volume using Cylindrical Shells 12

13 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit VI 4.2.A.1.2 Topic 1:The Fundamental Theorem of Calculus The Definite Integral Test/Quizzes Text Lecture Graphing Calculator 4.2.E.2 Find the Area under a curve 4.3.C.1 The area between two curves 4.5.A.1 The area between curves that intersect 4.5.E.1 Regions bounded by several curves Teacher-created Discussion Overhead Projector Lab Activity worksheets Small Group Work SmartBoard 4.5.F Homework Lab Work Topic 2: Integration with respect to y Finding areas with respect to y Finding areas with respect to y with several curves and intersecting curves Topic 3: Volumes Finding Volumes using cross sectional slices Solids of Revolution The disk method along the y axis The Washer Method across the x-axis The Washer Method across the y-axis The Shell Method Differentiated Learning: Students may present their projects in any format including pictorial representations, paragraph proofs, or algebraic proofs Unit VII: Exponential and Logarithmic Functions Essential Question: How do we use Calculus to study Problems of Growth and Decay? ex Objectives: Students will be able to: graph an exponential function and understand the tremendous increase of exponential growth use the discovery process to determine the derivative of the exponential function 13

14 solve applications that include derivatives of ex find the integral involving exponential functions evaluate logarithmic equations using properties of logarithms find the derivatives of functions including natural logarithms find the integrals of functions involving natural logarithms find derivatives and integrals of general exponential and logarithmic functions solve applications involving natural growth and decay such as population growth, radioactive decay, continuously compounded interest, drug elimination in the blood stream, elimination of pollutants, and sales decline 14

15 Topic/Content Skills Assessment Resources Instructional Method Tech Infusion NJCCCS: Unit VII 4.2.A.1 Topic 1: The Exponential Function 4.3.B.1.2 Test/Quizzes Text Lecture Graphing Calculator 4.4.A.1 Graphs of the exponential function 4.5.B.1 Derivatives of the exponential function 4.5.A.1 Digital projector 4.5.E, 4.5.F Integrals with the exponential function Discussion Overhead projector Lab Activity Small Group Work SmartBoard Homework Individual Research Derivatives of functions that include ex Topic 2: Logarithmic Functions The concept of the logarithm Properties of Logarithms Derivatives of logarithms Integrals with logarithms Topic 3: General exponential and logarithmic functions Derivatives with general exponential and logarithmic functions Integrals with general exponential and logarithmic functions Topic 4: Growth and Decay The natural growth equation Population growth Radioactive decay Continuously compounded interest Drug elimination Sales decline Elimination of pollutants Differentiated Learning: Students will select a problem to solve such as a population growth problem and write a short paper on how they might be able to influence the outcome of the problem through non- mathematical solutions, such as politics, media, science, conservation, etc. Ethical Decision Making/Character Education How do the laws of natural growth and decay affect how we should make our long-term decisions in business, the environment, population growth and politics? 15