Ideal lesson plan
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This document provides a lesson plan for teaching students how to solve linear systems of equations by graphing. The teacher will present a PowerPoint to introduce the topic and have students complete practice problems in study groups. Students will graph lines and find their intersections to solve systems of equations. At the end, the teacher will check for understanding and assign additional practice problems to reinforce the material.
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7.2 volumes by slicing disks and washersdicosmo178 This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.
7.1 area between curves
7.1 area between curvesdicosmo178 The document discusses calculating the area between two curves. It explains that this area is defined as the limit of sums of the areas of rectangles between the curves as the number of rectangles approaches infinity, which is represented by a definite integral. It provides examples of finding the area between curves defined by various functions through setting up and evaluating the appropriate definite integrals.
6.3 integration by substitution
6.3 integration by substitutiondicosmo178 This document discusses integration by substitution. It provides an example of recognizing a composite function and rewriting the integral in terms of the inside and outside functions. Specifically, it shows rewriting the integral of (x2 +1)2x dx as the integral of the outside function (x2 + 1) with the inside function (x) plugged in, plus a constant. It then provides additional practice problems applying the technique of substitution to rewrite integrals in terms of u-substitutions.
6.2 the indefinite integral 
6.2 the indefinite integral dicosmo178 This document discusses the indefinite integral and antiderivatives. It defines an antiderivative as a function whose derivative is the original function, and notes that there are infinitely many antiderivatives that differ by a constant. The process of finding antiderivatives is called indefinite integration or antidifferentiation. Initial conditions can be used to determine a unique particular solution by solving for the constant of integration.
6.1 & 6.4 an overview of the area problem area
6.1 & 6.4 an overview of the area problem areadicosmo178 The document discusses different methods for approximating the area under a curve:
- Lower estimate (LAM) uses the left endpoints of intervals
- Upper estimate (RAM) uses the right endpoints
- Average estimate (MAM) uses the midpoints
Formulas are provided for calculating the area using each method by summing the areas of rectangles. Examples are shown for finding the area under y=x^2 from 0 to 2 using each method. Finally, the document introduces using the antiderivative method to find the exact area under a curve by calculating the antiderivative and evaluating it over the bounds.
5.8 rectilinear motion
5.8 rectilinear motiondicosmo178 This document discusses rectilinear motion and concepts related to position, velocity, speed, and acceleration for objects moving along a straight line. It defines velocity as the rate of change of position with respect to time and speed as the magnitude of velocity. Acceleration is defined as the rate of change of velocity with respect to time. Examples are given to show how to calculate position, velocity, speed, and acceleration functions from a given position function. The document also analyzes position versus time graphs to determine characteristics of the particle's motion at different points in time.
5.7 rolle's thrm & mv theorem
5.7 rolle's thrm & mv theoremdicosmo178 This document discusses Rolle's theorem and the mean value theorem. It provides the definitions and formulas for each theorem. It then gives examples of applying each theorem to find values of c where a derivative is equal to zero or a tangent line is parallel to a secant line. Rolle's theorem examples find values of c where the derivative of a function over an interval is zero. The mean value theorem examples find values of c where the slope of a tangent line equals the slope of a secant line over an interval.
5.5 optimization
5.5 optimizationdicosmo178 1. Optimization problems involve finding the maximum or minimum value of a function subject to certain constraints.
2. To solve maximum/minimum problems: draw a figure, write the primary equation relating quantities, reduce to one variable if needed, take the derivative(s) to find critical points, and check solutions in the domain.
3. Examples show applying this process to find the dimensions that maximize volume of an open box, minimize cost of laying pipe between points, maximize area of two corrals with a fixed fence length, and find the largest volume cylinder that can fit in a cone.
5.4 absolute maxima and minima
5.4 absolute maxima and minimadicosmo178 The document provides information on finding absolute maximum and minimum values (absolute extrema) of functions on different interval types. It discusses determining absolute extrema on closed, infinite, and open intervals. Examples are provided finding the absolute extrema of specific functions on given intervals, including finding any critical points and limits to determine if absolute extrema exist. Practice problems are also provided at the end to find the absolute extrema of additional functions on specified intervals.
5.3 curve sketching
5.3 curve sketchingdicosmo178 This document provides guidance on sketching graphs of functions by considering key features such as symmetries, intercepts, extrema, asymptotes, concavity, and inflection points. It then works through an example of sketching the graph of the function f(x) = (2x^2 - 8)/(x^2 - 16). Key steps include finding vertical and horizontal asymptotes, critical points and inflection points, intervals of increase/decrease, and finally sketching the graph.
5.2 first and second derivative test
5.2 first and second derivative testdicosmo178 This document discusses methods for finding relative extrema of functions:
1. The First Derivative Test (FDT) states that a critical point is a relative maximum if the derivative changes from positive to negative, and a relative minimum if the derivative changes from negative to positive.
2. The Second Derivative Test (SDT) states that a critical point is a relative maximum if the second derivative is negative, and a relative minimum if the second derivative is positive.
3. Examples are provided to demonstrate using the FDT and SDT to find relative extrema of functions.
5.1 analysis of function i
5.1 analysis of function idicosmo178 This document discusses increasing and decreasing functions, concavity of functions, and finding intervals where functions are increasing, decreasing, concave up, or concave down. It provides examples of finding the intervals for the functions f(x)=x-4x^2+3 and f(x)=x-5x^4+9x showing the steps to determine where the functions are increasing or decreasing and where they are concave up or concave down. It also discusses inflection points and provides an example of finding intervals of increase, decrease, concavity and the inflection point for the function f(x)=x^3-3x^2+1.
4.3 derivatives of inv erse trig. functions
4.3 derivatives of inv erse trig. functionsdicosmo178 This document discusses derivatives of inverse trigonometric functions and differentiability of inverse functions. It provides examples of finding the derivative of inverse trig functions like sin^-1(x^3) and sec^-1(e^x). It also explains that if a function f(x) is differentiable on an interval I, its inverse f^-1(x) will also be differentiable if f'(x) is not equal to 0. It gives the formula for the derivative of the inverse function and an example confirming this formula. It also discusses monotonic functions and how if f'(x) is always greater than 0 or less than 0, f(x) is one-to-one and its inverse will be different
8.2 integration by parts
8.2 integration by partsdicosmo178 Integration by parts is a technique for evaluating integrals of the form ∫udv, where u and v are differentiable functions. It works by expressing the integral as uv - ∫vdu. Some examples of integrals solved using integration by parts include ∫xe^xdx, ∫lnxdx, and ∫xe^-xdx. Repeated integration by parts may be necessary when the integral ∫vdu generated cannot be directly evaluated. Integration by parts also applies to definite integrals between limits a and b using the formula ∫_a^budv = uv|_a^b - ∫_a^bvdu.
8.7 numerical integration
8.7 numerical integrationdicosmo178 The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.
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Ideal lesson plan
- 1. LINEAR SYSTEMS Subject CP Algebra 2 Prepared by IlonaDiCosmo Grade Level 10/11 Materials Required Highlighters and graph paper. Teacher Guide Objectives Student Guide Teacher will enable students to solve linear systems of equations by graphing. Students will be able to solve linear systems of equations by graphing. Additional Resources Information Teacher will prepare PPT presentation for students to view at home. Students will watch PPT presentation at home and record notes and possible questions. Verification Teacher will ask questions about slopes of lines to recall previous information needed for this lesson and check for understanding of the lesson objectives. Students will review notes from previous lesson about slopes and ask questions if needed to clarify the objectives. Activity Teacher will assign group activity to practice graphing lines and looking for interceptions. The PPT will be projected on the white screen and the IWB will be available to access quick interactive tutorials if needed. Teacher will provide help to any student in need and encourage group work to promote peer tutoring. Students will form study groups to help each other with practicing graphing lines. Students will take advantage of the resources provided via IWB and the PPT. Some students may take a seat in on the couches, if that is more comfortable, to interact and hold a discussion. Summary At the end of the period, teacher will once again state the objective and check for the understanding to see whether the task was accomplished. Teacher will assign a few additional practice problems to reinforce the topic. At the end of the period, students will make sure they understand the topic clearly and they are able to complete any assignment independently. Internet access to PPT presentation. Additional Notes Hard copy of the PPT presentation and additional set of practice problems.
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