AP Calculus lessons University High School Normal,IL http://www.uhigh.ilstu.edu/math/thompson/ipod/apcalc.xml 2008 http://blogs.law.harvard.edu/tech/rss en Mon, 15 Mar 2010 13:29:53 -0500 math_man77@yahoo.com Fri, 26 Feb 2010 12:42:05 -0600 FeedForAll v2.0 (2.0.2.9) http://www.feedforall.com Dr Thompson's AP Calculus University High School AP calculus lessons University High School Normal,IL Dr Kevin Thompson Dr Kevin Thompson math_man77@yahoo.com AP Calculus University High School Normal, IL Dr Thompson no no http://www.uhigh.ilstu.edu/math/thompson/images/seal100.jpg http://www.uhigh.ilstu.edu/math/thompson/ipod/apcalc.xml 100 98 Some example problems involving force and work. Lifting a leaky bucket and pumping to fill a tank. http://www.uhigh.ilstu.edu/math/thompson/ipod/7.5.mp4 1847C794-DCA2-43A2-A478-02846DD5D522 Mon, 15 Mar 2010 13:29:50 -0500 7.5 day 1 and day 2 Applications involving force and work Some example problems involving force and work. Lifting a leaky bucket and pumping to fill a tank. 1:04:50 Dr Thompson work, force, leaky bucket, pump, integral no no Methods of disks and washers are explored in calculating the volume of a solid. http://www.uhigh.ilstu.edu/math/thompson/ipod/7.3day2.m4v 73833AD0-4FD7-456B-B1E0-E6687DFB26F7 Fri, 26 Feb 2010 12:42:05 -0600 7.3 day 2 volumes of revolution Methods of disks and washers are explored in calculating the volume of a solid. 39:47 Dr Thompson volume, revolution, volume of revolution, washer, disk, washers, revolve, calculus no no Methods of disks and washers are explored in calculating the volume of a solid. http://www.uhigh.ilstu.edu/math/thompson/ipod/7.3day1.m4v A11BAD37-3499-4024-9C34-F7538D28E969 Thu, 25 Feb 2010 12:50:41 -0600 7.3 day 1 volumes of revolution Methods of disks and washers are explored in calculating the volume of a solid. 39:16 Dr Thompson volume, revolution, volume of revolution, washer, disk, washers, revolve, calculus no no Several problems are explored finding the area between two curves. Integrating with respect to y is explored as well. http://www.uhigh.ilstu.edu/math/thompson/ipod/7.2.m4v 7E54A83E-0328-4251-B110-E80F8750CC42 Mon, 22 Feb 2010 16:49:51 -0600 7.2 Several problems are explored finding the area between two curves. Integrating with respect to y is explored as well. 31:45 Dr Thompson area, integration, dy, with respect to y, between curves no no Focus upon real life problems and the integral as the net change. Force, Work, Acceleration, Velocity are tied together. Units are discussed. A problem involving electricity is explored. http://www.uhigh.ilstu.edu/math/thompson/ipod/7.1day2.m4v 394676CF-726B-4B89-AB56-F98289A3AC39 Wed, 17 Feb 2010 15:20:22 -0600 7.1 integral as net change Focus upon real life problems and the integral as the net change. Force, Work, Acceleration, Velocity are tied together. Units are discussed. A problem with an initial velocity is examined. 30:51 Dr Thompson integral, net change, acceleration no no Focus upon real life problems and the integral as the net change. Force, Work, Acceleration, Velocity are tied together. Units are discussed. A problem with an initial velocity is examined. http://www.uhigh.ilstu.edu/math/thompson/ipod/7.1.m4v 4174D6A0-089A-4EE8-B64F-DC89099ECB17 Wed, 17 Feb 2010 15:20:04 -0600 7.1 integral as net change Focus upon real life problems and the integral as the net change. Force, Work, Acceleration, Velocity are tied together. Units are discussed. A problem with an initial velocity is examined. 41:51 Dr Thompson integral, net change, acceleration no no This lesson focuses upon developing the exponential function A=Pe^(rt) from a separable differential equation. Next Newton's Law of cooling is explored. http://www.uhigh.ilstu.edu/math/thompson/ipod/6.4.m4v C4818DD7-A68A-4690-B95D-9EE75FDE6C36 Tue, 9 Feb 2010 15:20:15 -0600 6.4 Newton's law of cooling and solving a separable differential equation. This lesson focuses upon developing the exponential function A=Pe^(rt) from a separable differential equation. Next Newton's Law of cooling is explored. 43:20 Dr Thompson Newton, Law of cooling, cooling, Newton's law of cooling, separable differential equation no no Integration by parts. Tabular integration and a problem involving an unknown integral is investigated. http://www.uhigh.ilstu.edu/math/thompson/ipod/6.3day2.m4v F0A12D1B-6A19-4AAB-8032-17DFB7B6E371 Tue, 9 Feb 2010 15:12:24 -0600 6.3 day 2 integration by parts Integration by parts. Tabular integration and a problem involving an unknown integral is investigated. 48:42 Dr Thompson integration by parts, integration, tabular integration, unknown integral no no Using integration by parts. Investigate problems using this specific method. http://www.uhigh.ilstu.edu/math/thompson/ipod/6.3day1.m4v 63B83986-1258-400B-84E2-1A8C66B59031 Tue, 9 Feb 2010 15:12:23 -0600 6.3 Integration by parts second day Using integration by parts. Investigate problems using this specific method. 29:53 Dr Thompson integration, parts, integration by parts no no 2 examples of using a method of u substitution for indefinite integrals. http://www.uhigh.ilstu.edu/math/thompson/ipod/6.2_day1.m4v 12F96D97-2824-41C7-8025-FDEF60E24610 Thu, 28 Jan 2010 15:03:36 -0600 6.2 Using u substitution for indefinite integrals 2 examples of using a method of u substitution for indefinite integrals. 39:10 Dr Thompson u substitution, integral, indefinite, antiderivative no no An introduction to first order differential equations and slope fields. A slope field is graphed using a TI 89. Connections to the fundamental theorem of calculus are made. http://www.uhigh.ilstu.edu/math/thompson/ipod/6.1_day1.m4v 757A60EB-B51B-45D1-8C84-D41F0232AF51 Mon, 25 Jan 2010 12:46:07 -0600 6.1 differential equations and slope fields An introduction to first order differential equations and slope fields. A slope field is graphed using a TI 89. Connections to the fundamental theorem of calculus are made. 48:51 Dr Thompson slope fields, differential equation, first order, fundamental theorem of calculus, calculus, rates of change, slope no no Investigates the second half of the Thm. Time is spent looking at examples as well as finding anti derivatives with specific conditions http://www.uhigh.ilstu.edu/math/thompson/ipod/5.4day2.m4v 653033E6-3BBE-4FEF-BFE2-37DDD8153523 Tue, 19 Jan 2010 10:52:32 -0600 5.4 day2 Fundamental Thm of calculus Investigates the second half of the Thm. Time is spent looking at examples as well as finding anti derivatives with specific conditions 40:05 Dr Thompson antiderivative, fundamental theorem, calculus, integration no no The lesson begins with a connection being illustrated between position and velocity. This is a review. However, the area under the velocity graph is calculated and the connection back to position is explored. This lesson also examined a graph and a function defined as the integral of another function. The derivative of the integral is examined. http://www.uhigh.ilstu.edu/math/thompson/ipod/5.4day1.m4v A3A6F476-6A65-42C8-9F51-51071ADDD207 Fri, 15 Jan 2010 20:06:05 -0600 5.4 Fundamental Thm of Calc_ an intorduction The lesson begins with a connection being illustrated between position and velocity. This is a review. However, the area under the velocity graph is calculated and the connection back to position is explored. This lesson also examined a graph and a function defined as the integral of another function. The derivative of the integral is examined. 23:51 Dr Thompson fundamental theorem, calculus no no Exploring through Sketchpad, the connection between a function and a function representing the signed area from a to x is explored. The function [F(x)] that is the signed area from a to x for the function f(x) is revealed to be the anti-derivative. http://www.uhigh.ilstu.edu/math/thompson/ipod/5.3day2.m4v 74F90512-D792-4655-A47A-32BDD6A50161 Fri, 15 Jan 2010 19:54:00 -0600 5.3 day 2 Exploring through Sketchpad, the connection between a function and a function representing the signed area from a to x is explored. The function [F(x)] that is the signed area from a to x for the function f(x) is revealed to be the anti-derivative. 47:18 Dr Thompson definite integral, anti-derivative no no This section looks at several properties of definite integrals including how to combine or separate them over a given interval, combining functions, and reversing the limits. Average value of a function is mentioned briefly at the end. http://www.uhigh.ilstu.edu/math/thompson/ipod/5.3.m4v 349DF238-6B76-47B1-8196-A7016A99FDE7 Fri, 15 Jan 2010 19:47:43 -0600 5.3 definite integral Rules for the definite integral are explored. Average value of a function. 33:24 Dr Thompson integral, antiderivative, definite integral no no A reimann sum is constructed and then developed into the definite integral. A connection to area is explored. Definite integrals are evaluated using knowledge of geometry. http://www.uhigh.ilstu.edu/math/thompson/ipod/5.2.m4v B55DA8D0-E90F-41D4-9DA5-35B5A62A5D56 Fri, 8 Jan 2010 12:45:07 -0600 5.2 the definite integral and reimann sums A reimann sum is constructed and then developed into the definite integral. A connection to area is explored. Definite integrals are evaluated using knowledge of geometry. 29:14 Dr Thompson integral, integrand, definite integral, integration, reimann sum, Reimann sum, area no no Explore how the area under a curve can connect to a real life situation involving velocity and time. Rectangle methods are explored to approximate the area. http://www.uhigh.ilstu.edu/math/thompson/ipod/5.1_day1.m4v AEB5A4A2-D91B-4D2C-ADF0-1FBDB4B89AA0 Tue, 5 Jan 2010 12:45:54 -0600 5.1 Rectangle approximation methods Explore how the area under a curve can connect to a real life situation involving velocity and time. Rectangle methods are explored to approximate the area. 27:48 Dr Thompson rectangle approximation method, rram, lram, trapezoidal method, area under a curve, velocity, approximation, ram no no Examine two related rate questions http://www.uhigh.ilstu.edu/math/thompson/ipod/4.6.m4v F75E3430-F8DB-4C2E-B70D-DD74DA975BA8 Mon, 7 Dec 2009 12:13:27 -0600 4.6 Related Rates Examine two related rate questions 28:16 Dr Thompson related rates, derivative, rate of change no no Examines how the slope of the tangent line at a point could aid in the estimation of a function value or in estimation of the x intercepts. http://www.uhigh.ilstu.edu/math/thompson/ipod/4.5.m4v 351A02CE-8213-42BF-9DCF-DDFAB4554564 Fri, 4 Dec 2009 14:52:41 -0600 4.5 Linearization and Newton's Method Examines how the slope of the tangent line at a point could aid in the estimation of a function value or in estimation of the x intercepts. 28:02 Dr Thompson newton, newton's method, newtons method, linearization, derivative no no Examples of optimization of profit and average cost http://www.uhigh.ilstu.edu/math/thompson/ipod/4.4_day2.m4v A2B6D259-9C28-4498-8A7C-8EF2BBF2E2B4 Wed, 2 Dec 2009 10:00:16 -0600 4.4 day 2 maximize profit and minimize average cost. Examples of optimization of profit and average cost 15:54 Dr Thompson profit, maximize, minimize, average cost, marginal cost, revenue, marginal prfit, derivative no no Using the derivative to maximize or minimize. 2 geometric examples are explored. http://www.uhigh.ilstu.edu/math/thompson/ipod/4.4.m4v DBB4B385-AAB3-449E-87E2-41893B4CAD2D Mon, 30 Nov 2009 12:27:56 -0600 4.4 Optimization Using the derivative to maximize or minimize. 2 geometric examples are explored. 24:41 Dr Thompson optimization, derivative, maximum, max, area, volume no no A single question is investigated where the derivative is given. From this derivative. We determine where the original function is inc/dec/max/min/cc up/cc down/point of infl. http://www.uhigh.ilstu.edu/math/thompson/ipod/4.3day3.m4v 056CE487-AD3D-4C0C-916E-4F3636F4B303 Thu, 19 Nov 2009 12:43:40 -0600 4.3 starting with f' and determining the attributes of f A single question is investigated where the derivative is given. From this derivative. We determine where the original function is inc/dec/max/min/cc up/cc down/point of infl. 12:30 Dr Thompson concave up, concave down, point of inflection, critical number no no Examines the graphical connections between concavity and the first and second derivative http://www.uhigh.ilstu.edu/math/thompson/ipod/4.3day2.m4v 00FC581F-E645-4433-9B9C-AA2567AA0CCF Wed, 18 Nov 2009 12:57:52 -0600 4.3 day 2 connecting the graphs to concavity and points of inflection Examines the graphical connections between concavity and the first and second derivative 29:24 Dr Thompson point of inflection, second derivative, concave up, concave down, concavity no no analyzing the connection between pos/neg derivatives and if f' exists or is zero at a specific value http://www.uhigh.ilstu.edu/math/thompson/ipod/4.3day1.m4v A24A9895-8D31-4FB3-9DA7-8BC716793CCC Mon, 16 Nov 2009 10:30:25 -0600 4.3 intro-derivatives and the connection to graphs analyzing the connection between pos/neg derivatives and if f' exists or is zero at a specific value 17:14 Dr Thompson extrema, increasing, decreasing, derivative no no Investigation of the antiderivative of a function. We look at how there are multiple functions that can satisfy a specific derivative. http://www.uhigh.ilstu.edu/math/thompson/ipod/4.2.m4v 76D5720C-456A-4BF3-B159-5A4D0D5950F1 Fri, 13 Nov 2009 13:11:05 -0600 4.2 The Antiderivative Investigation of the antiderivative of a function. We look at how there are multiple functions that can satisfy a specific derivative. 18:03 Dr Thompson derivative, constant, C, antiderivative, anti-derivative no no Investigate the definition of a critical point. Examine various functions and their extrema. http://www.uhigh.ilstu.edu/math/thompson/ipod/4.1_day1.m4v CEF41660-1AEA-4E00-9B6F-8E95A64EF88D Mon, 9 Nov 2009 11:00:35 -0600 4.1 day 1 Critical points and extrema Investigate the definition of a critical point. Examine various functions and their extrema. 40:53 Dr Thompson extrema, critical points, local maximum, local minimum no no Milk has been set out on the counter. How does it warm up? This question is explored. http://www.uhigh.ilstu.edu/math/thompson/ipod/3.9_day2.m4v 846FA3A6-1E08-41E9-8B26-D8FD76643F90 Tue, 3 Nov 2009 12:30:43 -0600 3.9 day 2. Application involving the derivative of an exponential Milk has been set out on the counter. How does it warm up? This question is explored. 19:53 Dr Thompson derivativce, exponential, milk, warming, room temperature no no Determines the general formula for exponential and natural logarithms. the derivative of e^x is explored. Sketchpad is used to show the connection of slope for e^x. http://www.uhigh.ilstu.edu/math/thompson/ipod/3.9_day1.m4v C2AC7B80-1CF6-420D-B43C-8D72E4E925A1 Mon, 2 Nov 2009 16:04:46 -0600 3.9 day 1 Derivatives of exponentials and logarithms Determines the general formula for exponential and natural logarithms. the derivative of e^x is explored. Sketchpad is used to show the connection of slope for e^x. 35:40 Dr Thompson natural base, natural log, derivative, exponentials no no Explores the derivatives of the inverse trigonometric functions http://www.uhigh.ilstu.edu/math/thompson/ipod/3.8_day2.mp4 CD0D3A6E-B463-4910-BA4A-47A9EE672879 Fri, 30 Oct 2009 09:52:42 -0500 3.8 day 2 Inverse trigonometric functions Explores the derivatives of the inverse trigonometric functions 28:57 Dr Thompson derivative, trigonometric functions, inverse functions, trig, inverse no no Explores the connection between the derivatives of a function and its inverse. Graphical connections are made. http://www.uhigh.ilstu.edu/math/thompson/ipod/3.8_day1.m4v FB36C4C5-C5EB-4314-89DC-62C748B07DB8 Wed, 28 Oct 2009 12:48:43 -0500 3.8 day 1, intro to derivatives of inverse functions Explores the connection between the derivatives of a function and its inverse. Graphical connections are made. 30:52 Dr Thompson derivative, inverse, functions, no no Taking derivatives implicitly http://www.uhigh.ilstu.edu/math/thompson/ipod/3.7.m4v 8A50974E-EC20-4CF0-90AF-7961F486AD3F Mon, 26 Oct 2009 11:44:46 -0500 3.7 Implicit Differentiation Taking derivatives implicitly 27:53 Dr Thompson implicit, differentiation, derivatives, implicit differentiation no no A connection between the rates of change of a gear combination is investigated. This is then connected to the concept of function composition leading to the chain rule. http://www.uhigh.ilstu.edu/math/thompson/ipod/3.6.m4v EC626063-D2C4-4E33-99BA-884F09F52FDE Tue, 20 Oct 2009 13:48:09 -0500 3.6 Chain Rule A connection between the rates of change of a gear combination is investigated. This is then connected to the concept of function composition leading to the chain rule. 34:20 Dr Thompson chain rule, composition of functions, gears, derivative, rate of change no no Explore the use of product and quotient rules for derivatives http://www.uhigh.ilstu.edu/math/thompson/ipod/3.3_day2.m4v 6EFB123B-599F-448E-A604-958245A625F9 Fri, 2 Oct 2009 10:42:13 -0500 3.3 day 2 the product and quotient rules Explore the use of product and quotient rules for derivatives 38:20 Dr Thompson product rule, quotient rule no no Time is spent to develop patterns so that we no longer need to take the limit of the difference quotient. The proof is provided as to why if f(x)=x^n the f'(x)=n*x^(n-1) http://www.uhigh.ilstu.edu/math/thompson/ipod/3.3_day1.m4v A71BB8DD-9B61-4839-9DA2-47FE03B6A71B Thu, 1 Oct 2009 11:39:40 -0500 3.3 day 1: Rules for differentiation Time is spent to develop patterns so that we no longer need to take the limit of the difference quotient. The proof is provided as to why if f(x)=x^n the f'(x)=n*x^(n-1) 32:33 Dr Thompson rules for differentiation, derivative no no Examines the idea of locally linear. Verifies that at a specific value a function has a vertical tangent. Examines how the calculator can give a numerical derivative answer--also its limitations regarding the use of a symmetric difference quotient. http://www.uhigh.ilstu.edu/math/thompson/ipod/3.2_day2.m4v 652A25A4-3F7E-4AC0-A341-930B537DC6B9 Tue, 29 Sep 2009 10:27:47 -0500 3.2 day 2 why f' may fail to exist Examines the idea of locally linear. Verifies that at a specific value a function has a vertical tangent. Examines how the calculator can give a numerical derivative answer--also its limitations regarding the use of a symmetric difference quotient. 39:07 Dr Thompson locally linear, vertical tangent, derivative, numerical derivative, NDERIVE no no Investigates four cases as to why the derivative may fail to exist. Sharp point, cusp, vertical tangent, and discontinuity. A discussion of the limit values at the points motivates why the derivate will fail to exist. A review of the alternate form of a derivate is also presented. http://www.uhigh.ilstu.edu/math/thompson/ipod/3.2_day1.m4v 4748F8D6-9C1D-47D1-91C5-3085845420B3 Mon, 28 Sep 2009 10:54:10 -0500 3.2 day 1_why f' may fail to exist Investigates four cases as to why the derivative may fail to exist. Sharp point, cusp, vertical tangent, and discontinuity. A discussion of the limit values at the points motivates why the derivate will fail to exist. A review of the alternate form of a derivate is also presented. 27:15 Dr Thompson cusp, sharp point, vertical tangent, reasons why the derivative does not exist, no no Explore the relationship between the graph of a function and how the graph of a derivative looks. Attention is paid to the the behavior of the slope of the tangent in helping understand this relationship. Images of a GSP sketch are provided to help clarify and explore the graphs. http://www.uhigh.ilstu.edu/math/thompson/ipod/3.1.m4v E5685EED-A1C9-4272-BF26-E97BEA9D1DA1 Fri, 25 Sep 2009 13:05:51 -0500 3.1 Graphs of f and f' Explore the relationship between the graph of a function and how the graph of a derivative looks. Attention is paid to the the behavior of the slope of the tangent in helping understand this relationship. Images of a GSP sketch are provided to help clarify and explore the graphs. 41:23 Dr Thompson graphs of f and f', tangent line, derivative no no Types of discontinuity is discussed. The intermediate value theorem is also mentioned. http://www.uhigh.ilstu.edu/math/thompson/ipod/2.3.m4v A8C61FF4-3E9D-43E3-8A75-AFF3257A17FA Thu, 10 Sep 2009 12:27:07 -0500 2.3 Continuity Types of discontinuity is discussed. The intermediate value theorem is also mentioned. 37:14 Dr Thompson continuous, continuity, intermediate value theorem no no Address limits as x->infinity or as x->-infinity The connection to horizontal asymptotes is explored as well as end behavior. Graphing f(1/x) is discussed as a means of better "seeing" limits as x->infinity http://www.uhigh.ilstu.edu/math/thompson/ipod/2..2.m4v B9A30AE6-F4B3-4DA2-A555-40441E1C039E Tue, 8 Sep 2009 12:49:07 -0500 2.2 Limits at infinity Address limits as x->infinity or as x->-infinity The connection to horizontal asymptotes is explored as well as end behavior. Graphing f(1/x) is discussed as a means of better "seeing" limits as x->infinity 38:06 Dr Thompson limits, infinity, horizontal asyptotes, end behavior no no Lesson finishes the intro to limits. A specific case of the limit of a polynomial is investigated leading to an easy way to evaluate many limits. The sandwich theorem is explored for a situation where substitution will not work. http://www.uhigh.ilstu.edu/math/thompson/ipod/apcalc.xml 56ED2F0C-F3CE-473B-A640-F351AC2614AC Fri, 4 Sep 2009 10:27:53 -0500 2.1 day 3: Limits of polynomials, using the sandwich theorem Lesson finishes the intro to limits. A specific case of the limit of a polynomial is investigated leading to an easy way to evaluate many limits. The sandwich theorem is explored for a situation where substitution will not work. 33:06 Dr Thompson limits, sandwhich theorem no no The idea of what a limit is is explored. Also the limit concept is explored graphically. Properties of limits are also discussed with graphical justification provided. http://www.uhigh.ilstu.edu/math/thompson/ipod/apcalc.xml 9EA8BB80-2BF5-49D8-B3BB-213187DB779B Thu, 3 Sep 2009 11:11:14 -0500 2.1 day 2: What is a limit? An informal definition of a limit. The idea of what a limit is is explored. Also the limit concept is explored graphically. Properties of limits are also discussed with graphical justification provided. 26:16 Dr Thompson limit properties, limit no no We develop from the equation that would provide the distance an object falls in t seconds the concept of average to instantaneous speed. This lesson culminates with an idea of a limit though it has yet to be defined. http://www.uhigh.ilstu.edu/math/thompson/ipod/apcalc.xml A694EDA2-CF33-407F-BE8B-0739B970A9C3 Wed, 2 Sep 2009 12:47:07 -0500 2.1 Moving from average speed to instantaneous We develop from the equation that would provide the distance an object falls in t seconds the concept of average to instantaneous speed. This lesson culminates with an idea of a limit though it has yet to be defined. 45:55 Dr Thompson limit, speed, instanteous speed, average speed no no intro to podcast and sites http://www.uhigh.ilstu.edu/math/thompson/ipod/apcalc.xml 1340CEB7-1F1E-4B28-9A26-4CD6B403F6CC Thu, 20 Aug 2009 12:48:21 -0500 Intro to podcast intro to podcast and sites 1:09 Dr Thompson no no