Draft Material. Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then


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1 CHAPTER : DERIVATIVES Specific Expectations Addressed in the Chapter Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, f(x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f (x) dy dy or dx, and make connections between the graphs of f(x) and f (x) or y and dx [e.g., when f(x) is linear, f (x) is constant; when f(x) is quadratic, f (x) is linear; when f(x) is cubic, f (x) is quadratic] [.] f(x h) f(x) Determine the derivatives of polynomial functions by simplifying the algebraic expression lim h and then h 0 f(x h) f(x) taking the limit of the simplified expression as h approaches zero [i.e., determining lim h ] [.] h 0 Verify the power rule for functions of the form f(x) x n, where n is a natural number [e.g., by determining the equations of f(x h) f(x) the derivatives of the functions f(x) x, f(x) x, f(x) x 3, and f(x) x 4 algebraically using lim h and h 0 graphically using slopes of tangents] [.] Verify the constant, constant multiple, sum, and difference rules graphically and numerically [e.g., by using the function g(x) kf(x) and comparing the graphs of g (x) and kf (x); by using a table of values to verify that f(x h) f(x) f (x) g (x) (f g) (x), given f(x) x and g(x) 3x], and read and interpret proofs lim h of the constant, h 0 constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required) [.] Determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs [.] Verify that the power rule applies to functions of the form f(x) x n, where n is a rational number [e.g., by comparing values of the slopes of tangents to the function f(x) x with values of the derivative function determined using the power rule], and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for f(x) (5x 3 3 ) by using the chain rule and by differentiating the simplified form, f(x) 5 3 x ] and the product rule using polynomial functions [e.g., by determining the same derivative for f(x) (3x )(x ) by using the product rule and by differentiating the expanded form f(x) 6x 3 4x 3x ] [.3] Solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions [e.g., by expressing f(x) x x as the product f(x) (x )(x ), radical functions [e.g., by expressing as the power f(x) (x 5) f(x) x 5 and other simple combinations of functions c e.g., f(x) x sin x, f(x) sin x cos x d [.3,.4,.5] Solve problems, using the derivative, that involve instantaneous rates of change, including problems arising from realworld applications (e.g., population growth, radioactive decay, temperature changes, hours of daylight, heights of tides), given the equation of a function. Sample problem: The size of a population of butterflies is given by the function P(t) 6000 where t is the time in days. Determine the rate of growth in the population after five days using the 49(0.6) t derivative, and verify graphically using technology. [.,.3,.4,.5] Prerequisite Skills Needed for the Chapter An understanding of the properties of exponents The ability to substitute real numbers and expressions into equations Facility expanding and simplifying rational expressions involving polynomials and radicals Evaluating limits An understanding of the difference between average rate of change and instantaneous rate of change The ability to determine the equation of a line if the slope and a point on the line are known. Chapter Introduction
2 What big ideas should students develop in this chapter? Students who have successfully completed the work of this chapter and who understand the essential concepts and procedures will know the following: f (a h) f (a) The derivative of a function f at a point (a, f (a)) is f (a) lim h, or f (x) f (a) h 0 f (a) lim x a, if the limit exists. x a A function is said to be differentiable at a if f (a) exists. A function is differentiable on an interval if it is differentiable at every number in the interval. f(x h) f(x) The derivative function for any function f (x) is given by f (x) lim h, if the h 0 limit exists. The value of the derivative at a point can be interpreted as the slope of the line tangent to the point or the instantaneous rate of change at the point. The derivative function can be determined using various rules, including: the constant function rule, the power rule, the constant multiple rule, the sum rule, and the difference rule. The rules for calculating derivatives of functions make the process easier than determining the derivative from first principles. The derivative of a product of differentiable functions is not the product of their derivatives. There are rules for determining the derivative of a function that is a product or a power of another function. The derivative of a quotient of two differentiable functions is not the quotient of their derivatives. The quotient rule for differentiation simplifies the process for determining the derivative of a function that is written as a quotient. The chain rule is used to determine the derivative of a composite function. If h(x) f (g(x)), then h (x) f (g(x)) g (x). Chapter : Planning Chart Section Title Section Goal Pacing days Materials/ Masters Needed Review of Prerequisite Skills Use concepts and skills developed day Diagnostic Test pp. xx xx prior to this chapter. Section.: The Derivative Introduce the definition of the derivative day graphing calculator Function pp. xx xx function. Section. Extra Practice Section.: The Derivatives of Begin to develop some rules for day graphing calculator Polynomial Functions pp. xx xx differentiation. Section. Extra Practice Section.3: The Product Rule Introduction of the product rule for day graphing calculator pp. xx xx differentiation. Section.3 Extra Practice Section.4: The Quotient Rule Introduction of the quotient rule for day Section.4 Extra Practice pp. xx xx differentiation. Section.5: The Derivatives of Introduce the chain rule to deal with day graphing calculator Composite Functions pp. xx xx composite functions. Section.5 Extra Practice MidChapter Review: pp. xx xx 5 days MidChapter Review Chapter Review: pp. xx xx Extra Practice; Chapter Test: p. xx Chapter Review Career Link: p. xx Extra Practice Calculus and Vectors: Chapter : Derivatives
3 REVIEW OF PREREQUISITE SKILLS Using the Review Discuss with students the review concepts listed on page XX of the student book. Be sure to emphasize the following points. When taking the product of exponential expressions with the same base, add the exponents, i.e., a x a y a x y. When taking the quotient of exponential expressions with the same base, subtract the exponents, i.e., a x a y a x y. When taking an exponential expression to a power, multiply the powers, i.e., (a x ) y a xy. When simplifying radicals, the square root distributes over multiplication, i.e., ab a b. Students should be able to then factor out perfect squares from radical expressions, i.e., 50 (5)() 5 5. Students should also be able to rationalize denominators of numerical expressions by multiplying the numerator and denominator by the conjugate. Parallel lines have the same slope. Perpendicular lines have slopes that are opposite reciprocals; equivalently, two lines are perpendicular if the product of their slopes is. When two rational expressions are being multiplied, multiply the numerators and multiply denominators. When two rational expressions are being divided, rewrite the problem as multiplication by the reciprocal of the denominator. When adding or subtracting rational expressions, a common denominator must first be obtained. After the first operation, combine like terms, factor, and cancel out common factors to simplify. Students should be able to expand the product of two polynomials. Students should also be able to factor quadratic trinomials, expressions that involve finding a greatest common factor first, and expressions that involve special forms, such as perfect square trinomials, the difference of squares, and the sum and difference of cubes. Students should also be able to factor polynomial expressions of degree three or greater using various techniques, including the Factor Theorem. Students should know the difference quotient, how to evaluate and simplify it with various functions, and its connection to the slope of a secant line. Student Book Pages Preparation and Planning Pacing 0 5 min Using the Review min Exercises Chapter Review of Prerequisite Skills 3
4 Initial Assessment When students understand Students can simplify complicated expressions involving exponents and radicals by applying the various rules. Students can find equations of lines given multiple combinations of conditions, i.e., slopes, perpendicular lines, points on the line, yintercept, et What You Will See Students Doing If students misunderstand Students misapply the rules of exponents, for instance adding exponents when they should be multiplying. Students can only find equations of lines when given basic information such as slope and yintercept, but cannot do slightly more complicated examples. Students can methodically factor a variety of different polynomials. Students can simplify rational expressions, particularly adding or subtracting by obtaining a common denominator. Students can rewrite numerical expressions involving radical denominators as equivalent expressions with whole number denominators. Students can determine an expression for average rate of change using the difference quotient and use it to make estimates of instantaneous rates of change. Students either factor incorrectly, or obtain correct factorizations through a tedious guess and check method. Students try to add across when adding rational expressions. Students do not simplify the resulting rational expression (not cancelling common factors or combining like terms) after applying the first operation. Students struggle to rewrite numerical expressions involving radical denominators as equivalent expressions with whole number denominators because they have difficulty identifying the conjugate radical and do not multiply two radicals together correctly. Students have difficulty determining an expression for average rate of change using the difference quotient because they substitute incorrectly and have difficulty with the algebra involved in simplification. Students struggle to use their expression to make estimates of instantaneous rates of change. 4 Calculus and Vectors: Chapter : Derivatives
5 . THE DERIVATIVE FUNCTION Section at a Glance GOAL Introduce the definition of the derivative function. Prerequisite Skills/Concepts Determining an expression for the difference quotient for a given function and point Estimating instantaneous rates of change/slopes of tangent lines Evaluating limits Specific Expectations Generate, through investigation using technology, a table of values showing the instantaneous rate of change of a polynomial function, f (x), for various values of x (e.g., construct a tangent to the function, measure its slope, and create a slider or animation to move the point of tangency), graph the ordered pairs, recognize that the graph represents a function called the derivative, f (x) dy or dx, and make connections between the dy graphs of f (x) and f (x) or y and dx Determine the derivatives of polynomial functions by simplifying f (x h) f (x) the algebraic expression lim h and then taking h 0 the limit of the simplified expression as h approaches zero f (x h) f (x) c i.e., determining lim h d h 0 Mathematical Process Focus Connecting Representing Reflecting MATH BACKGROUND SECTION OVERVIEW Student Book Pages Preparation and Planning Pacing 5 0 min Introduction 0 5 min Teaching and Learning min Consolidation Materials graphing calculator Recommended Practice Questions 5, 6, 8, 9, 0,,, 4 Key Assessment Question Question 7 Extra Practice Section. Extra Practice New Vocabulary/Symbols derivative normal differentiable dy f (x), dx Nelson Website Students should already know how to evaluate limits. Students should be familiar with the difference quotient and its connection to slope and average rate of change. Students will learn the definition of the derivative at a point and use it in calculations. Students will determine the derivative function from first principles. Students will use the derivative function to determine the slope of tangent lines at given points and find equations of tangent and normal lines. Students will use the derivative function to determine the instantaneous rate of change at a given point. Students will learn when a function is not differentiable..: The Derivative Function 5
6 Introducing the Section (5 to 0 min) Draw the graph of f (x) x on the board, overhead or graph and display it using The Geometers Sketchpad. Have the students determine the slope of the tangent line at x by using techniques from the previous chapter and techniques they remember from MHF4U, e.g., determining the limit of the difference quotient. When the class has finished, have each of the students pick an additional x value and determine the slope of the tangent line. Collect the data from the class and see if they can make any generalizations and/or predictions as to the slope of the tangent line at x 008. Students should see that the slope can be found by multiplying the x value by two. Tell the students that they will be investigating these types of generalizations in this section. If using Sketchpad, plot the point (, 4) on the graph and construct a second point on the graph of f (x). Use these two points to construct a secant line and measure its slope. Drag the point closer to the point of tangency and discuss the idea of the limiting process of the slope as the distance between the two points gets smaller and smaller. Define the derivative of f at the point x a. 6 Calculus and Vectors: Chapter : Derivatives
7 Teaching and Learning (0 to 5 min) In addition to presenting this section using the solved examples, after introducing students to the derivative at a point, give the students the cubic f (x) 4x 3 x 7x 0 and have them discover the derivative function using a quadratic regression on the graphing calculator. Have students form groups of three. Each group should be assigned a different point at which to determine the derivative either by successive approximations or from first principles (limits). Have each group present their x value and the slope of the tangent line on the board. As a whole class, gather the data and draw a scatter plot. Students should see that the points form what looks to be a parabola. Have the students run a quadratic regression on the data to determine the derivative function. Use the function to predict an additional point, and confirm this point algebraically. Finally, determine the derivative from first principles (limit) to confirm the regression model. Example presents a function and a point at which to determine the derivative. Present this problem to the whole class, and emphasize that the derivative is the slope of the tangent line at the given point. Draw a picture of the graph and its tangent line to reinforce this point. At the end of the example, the students are introduced to an alternate definition of the derivative at a point. Ask students to think about why the definitions are equivalent, and have them confirm the answer to this example by working with the alternate definition in their small groups. You can use the graphing calculator to confirm the calculated value by graphing the function and drawing a tangent line at x 3. The calculator will display the equation of the tangent line in the form y mx b, where m is the slope of the tangent line. Example introduces the notion of the derivative function. Much of the work for this was done in the introduction, so the teacher should be able to go over it relatively quickly. Have the students work on the Investigation in their small groups. This investigation allows students to discover the power rule that will be formally presented in the next section. Take up any questions with the whole class. Answers to Investigation A. a. f (x) 3x b. f (x) 4x 3 f (x) 5x 4 B. The coefficient of the derivative function is the same as the original power, and the power of the derivative function is one less than the original power. C. f (x) 39x 38 D. f (x) nx n.: The Derivative Function 7
8 3 Consolidation (30 to 35 min) Using the Solved Examples (Small Groups) Example 3 has students determine the derivative of f (t) t from first principles (limit). It is essential that students can do this using the limit definition and continue to connect it to the slope of the graph. Have students work this problem in their small groups without using their textbooks. Check their work and clear up and discrepancies. Have students discuss whether the pattern fits with what they discovered in the investigation. Students should see that it does. Emphasize to the whole class the domain of the original function and domain of the derivative function. Ask students to think about why the derivative does not exist at t 0, even though the function has a value at this point. Example 4 has students determine the equation of the tangent line at a point. The small groups should be able to handle this problem with books closed. The teacher may have to circulate to ensure that students can follow the steps in order, i.e., determine the derivative at the point, then use the point on the graph, together with the slope, to write the equation of the line. Have each group compare their work with that of the textbook when they are finished. Also, have each group graph the function and the tangent line on their calculator for further confirmation. Example 5 is very similar to Example 4 but asks students to determine a normal line instead of a tangent line. After defining normal, have the groups follow the same procedure as in Example 4. Before doing Example 6, discuss what can cause a function to not be differentiable at a point. Draw each of the three situations on the board: cusp, vertical tangent, and discontinuity. Emphasize that a function can be continuous but not differentiable, but if a function is differentiable, it must be continuous. Have the students discuss the differentiability of the function in Example 6 in their small groups. Take up any questions with the whole class. This example also illustrates that corners are points where a function is not differentiable. Answer to the Key Assessment Question dy 7. a. dx 7 dy b. dx (x ) dy 6x dx 8 Calculus and Vectors: Chapter : Derivatives
9 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students can determine the derivative at a point by using the limit definition. If students misunderstand Students can only estimate the derivative at a point by using successive approximations of average rate of change. Students can determine the derivative function by using the limit definition. Students can recognize when a graph is not differentiable and confirm that fact using one sided limits on either side of the point in question. Students can determine the equation of a tangent line and the equation of a normal line by using derivatives. Key Assessment Question #7 Students accurately determine all derivatives with proper notation from first principles. Differentiating Instruction How You Can Respond Students only guess at the derivative function by looking at patterns in the derivative at several points. Students equate differentiability with continuity. Students determine the slope of the tangent line or normal line to a graph, but cannot arrive at the equation of such a line. Students attempt to determine the derivatives by applying patterns discovered in the section. Students have difficulty with the substitutions in the difference quotient and have difficulty evaluating the resulting limit using techniques from Chapter. Students attempt to determine derivatives by determining the derivatives at points and examining patterns. EXTRA SUPPORT. Some students will struggle applying the limit definition to determine the derivative function right away. Have these students determine several derivatives at specific points (still using the limit definition) and look for patterns in their presentation. Students should see that where there was once a numerical value (3, 4, 5, et), we can simply replace it with a.. Some students will be able to determine the slope of a tangent line, but not the equation. Give these students some review problems in which they are given the slope and a point on the graph in order to strengthen their equationwriting skills. EXTRA CHALLENGE. Have students examine the derivative at several points on f(x) sin x, plot them, and attempt to determine the derivative function.. Have students examine the derivative at several points on f(x) x, plot them, and attempt to determine the derivative function..: The Derivative Function 9
10 . THE DERIVATIVES OF POLYNOMIAL FUNCTIONS Section at a Glance GOAL Begin to develop some rules for differentiation. Prerequisite Skills/Concepts The ability to express simple rational functions as powers using negative exponents The ability to express simple radical functions as powers using rational exponents Evaluating algebraic expressions for given values Determining the equation of a line based on given information Specific Expectations Verify the power rule for functions of the form f (x) x n, where n is a natural number [e.g., by determining the equations of the derivatives of the functions f (x) x, f (x) x, f (x) x 3, and f (x) x 4 f (x h) f (x) algebraically using lim h and graphically using slopes of h 0 tangents] Verify the constant, constant multiple, sum, and difference rules graphically and numerically [e.g., by using the function g(x) kf (x) and comparing the graphs of g (x) and kf (x); by using a table of values to verify that f (x) g (x) (f g) (x), given f (x) x and f (x h) f (x) g(x) 3x], and read and interpret proofs lim h of the h 0 constant, constant multiple, sum, and difference rules (student reproduction of the development of the general case is not required) Determine algebraically the derivatives of polynomial functions, and use these derivatives to determine the instantaneous rate of change at a point and to determine point(s) at which a given rate of change occurs Mathematical Process Focus Connecting Representing Problem Solving Reasoning and Proving MATH BACKGROUND Preparation and Planning Pacing 5 0 min Introduction 5 0 min Teaching and Learning 5 30 min Consolidation Materials graphing calculator Recommended Practice Questions 5, 6, 7, 8, 9, 0,,, 4, 0, Key Assessment Question Question 7 Extra Practice Section. Extra Practice New Vocabulary/Symbols constant function rule power function power rule constant multiple rule sum rule difference rule Nelson Website SECTION OVERVIEW Student Book Pages Students should already be familiar with determining the derivative function from first principles. Students will learn that the derivative of a constant function is zero. Students will justify then learn to apply the constant function, power, constant multiple, sum, and difference rules in order to determine derivatives of polynomial functions in an efficient manner. Students will use the derivative to determine slopes and equations of tangent lines of polynomial functions. Students will determine when the slope of a polynomial function is zero through algebraic means. Students will use the derivative to determine instantaneous rate of change in contextual situations and interpret the results. 0 Calculus and Vectors: Chapter : Derivatives
11 Introducing the Section (5 to 0 min) Have a student begin at one end of the room and walk to the other end. Instruct the student to begin slowly, speed up, slow to a stop at the halfway point, then move at a steady rate to the other end of the room. Have each student attempt to draw a displacement vs. time graph that models the student s distance from their starting point. It may be necessary to have the student repeat the motion several times. Take up suggestions and discuss the results until a fairly accurate graph is drawn. Finally focus on the portion where the student was at rest in the middle of the room. What is the slope of the graph during this interval? Write a function for just that portion of the graph. Students should see that this piece of the graph is a constant function and the slope is zero. Ask the students to think about the connection between slope and rate of change and to discuss the fittingness of the zero slope to the context of the demonstration (i.e., the rate of change, or speed, was zero). Teaching and Learning (5 to 0 min) Instead of presenting the properties followed by the proofs, students can discover these properties for themselves. For instance, have the students rely on their work from the previous section in order to determine the derivative function for f (x) 5 using first principles. Once they have completed this, have them discuss the nonsignificance of the number five. Students will see that the derivative of any constant function is zero. This can be further emphasized by graphing several constant functions and showing that these result in horizontal lines whose slopes are always zero at all points on the function. The work for the power rule was done in the previous section s Investigation, so it need not be repeated here. To discover the constant multiple rule, have students determine the derivative functions for f (x) x, x, 3x, and 4x using first principles. Finally, have students determine the derivative of a polynomial such as f (x) 3x 3 4x x 9 using first principles. Students should see that they could have applied the power and constant multiple rules to each individual term, which is essentially using the sum and difference rules. Students can also practice their understanding by writing questions that model Example 5 and Example 6. Have each pair of students write a quadratic function that has nice solutions. This quadratic will serve as the derivative function. (See Example 6.) Have the students take the derivative function and create the original function using what they know about the various rules in this section. The pairs can then switch their problem with another group, solve the other group s problem, and switch back for peer evaluation..: The Derivatives of Polynomial Functions
12 TechnologyBased Alternate Lesson If graphing software or graphing calculators are available, students can graph polynomials, draw tangent lines, and obtain their slopes via technology. Data can be collected very quickly in this manner. Once several (x, slope) pairs are obtained, students can create a scatter plot of the data, make an educated guess on the type of function that appears, and run regression with their graphing calculators in order to discover derivatives. Another alternative is to use The Geometer s Sketchpad. Polynomial functions can be created by choosing the Graph menu and selecting New Function and it then can be graphed by selecting the function, going to the Graph menu and choosing Plot Function. Selecting the function and choosing the Graph menu allows you to choose Derivative. This displays the derivative function of the selected function and its graph can then also be plotted. In this way, students can do many problems in a short amount of time and discuss the patterns that arise. 3 Consolidation (5 to 30 min) Using the Solved Examples (Pairs) Example gives students the constant function rule. This can and should be presented to the whole class in a very short amount of time. Graphical confirmation will help emphasize the meaning. Example is the most important example in the section for foundational purposes. Ask students to try to remember their results from the investigation in the previous section, i.e., the derivative function for f (x) x n. Once students have recalled it correctly, present the formal proof and have the class follow along in their textbook. Finally, have the class work in pairs at applying the power rule to the problems in Example. When students are finished, they should check their work with the solution in the book. It may be useful to go through a couple of additional questions with the class of the type given in b and Example 3 deals with the constant multiple rule, while Example 4 deals with the sum and difference rules. Present all three of these rules to the whole class, and give the pairs all of the problems in both examples. The pairs should work on determining the derivatives without using their textbook. Have them check their work with another pair, and then finally by looking in the book. Take up any questions with the whole class. Example 5 has students determine the equation of a tangent line by computing the derivative using the various rules in this section. This problem tends to give students some difficulties. Have the students begin the problem in pairs, but take up the example with the whole class when they are finished. Emphasize using graphing technology to check the work, but be sure that students can arrive at the answer without using the calculator. Example 6 uses the function from Example 5 and has students determine where the graph has a slope of zero. Having just found the derivative function, this is as simple as setting the derivative equal to zero. This process is Calculus and Vectors: Chapter : Derivatives
13 critical to understanding. Emphasize the connection between slope of a tangent and derivatives, then have the students read through the example in pairs and discuss the solution. Discuss the problem with the whole class. Answer to the Key Assessment Question 7. a. y 3 and 6x y 9 0 b. 0x y 47 0 and 4x y 0 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students can determine derivatives of polynomials without using the limit definition. Students can determine the equation of a tangent line at a given point on a polynomial graph. Students can determine algebraically where the slope of a tangent to a polynomial function is zero. Students can follow the algebraic proofs of the rules discussed in this section and work through examples from first principles to demonstrate the rules. Key Assessment Question #7 Students methodically determine both tangent lines that include the given points. Differentiating Instruction How You Can Respond If students misunderstand Students do not apply the appropriate rules to determine derivatives of polynomials in a timely manner. Students can determine the slope of a tangent line, but cannot apply the knowledge to determining the equation of the line. Students must examine a graph in order to guess where the slope is zero. Students can apply the various rules, but cannot explain why they are true. Students guess and check in order to determine the tangent lines. Students incorrectly use the given point as the point of tangency. EXTRA SUPPORT. Some students will have trouble expressing simple rational functions and radical functions as powers. Have students work on several review questions that deal with the rules x n and x m n x n n x m.. Some students may have no trouble getting the derivative function using the rules, but will not be able to set the derivative to zero in order to determine where the slope is zero. Have these students work on several review problems involving determining the zeros of polynomials. EXTRA CHALLENGE. Have students investigate what effect linear transformations have on the derivative functions. For instance, begin with the graph of y x, which has derivative y x. What is the derivative of y a(x b) c? What if the degree of the parent function was something other than?. Have students prove that the derivative function of y x is not y x x. Have them investigate this graph and try to develop the derivative function..: The Derivatives of Polynomial Functions 3
14 .3 THE PRODUCT RULE Section at a Glance GOAL Introduction of the product rule for differentiation. Prerequisite Skills/Concepts Expanding and simplifying polynomial expressions Constant, constant multiple, sum, and difference rules for derivatives of functions Specific Expectations Verify that the power rule applies to functions of the form f (x) x n, where n is a rational number [e.g., by comparing values of the slopes of tangents to the function f (x) x with values of the derivative function determined using the power rule], and verify algebraically the chain rule using monomial functions [e.g., by determining the same derivative for f (x) (5x 3 ) 3 by using the chain rule and by differentiating the simplified form, f (x) 5 3 x ] and the product rule using polynomial functions [e.g., by determining the same derivative for f (x) (3x )(x ) by using the product rule and by differentiating the expanded form f (x) 6x 3 4x 3x ] Solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions [e.g., by expressing f (x) x x as the product f (x) (x )(x ), radical functions [e.g., by expressing as the power f (x) (x 5) f (x) x 5 and other simple combinations of functions c e.g., f (x) x sin x, f (x) sin x cos x d Mathematical Process Focus Connecting Selecting Tools and Computational Strategies Reasoning and Proving MATH BACKGROUND Student Book Pages Preparation and Planning Pacing 5 min Introduction 0 5 min Teaching and Learning min Consolidation Recommended Practice Questions 6, 7, 8, 9, 0 Key Assessment Question Question 5 Extra Practice Section.3 Extra Practice New Vocabulary/Symbols product rule extended product rule the power of a function rule for positive integers Nelson Website SECTION OVERVIEW Students should already be familiar with determining derivatives of polynomials. Students will learn to apply the product rule to determine a derivative without expanding. Students will learn to apply the power of a function rule to determine derivatives without expanding. Students will learn to determine the derivative of rational functions by writing them as products. Students will apply their knowledge of derivatives and the various rules to solve problems involving velocity and other rates of change. 4 Calculus and Vectors: Chapter : Derivatives
15 Introducing the Section (5 min) Give students the following function: f (x) (x 3x ) (x 4x 5) and ask them to determine the derivative. Students should remember from the last section that they can combine the functions before determining the derivative. Ask the students if a similar strategy would hold for the product of two polynomials. Ask students to think about how they would prove or disprove such a property. Finally, give the students the function f (x) x written as f (x) x x. If the derivative of the product is the product of the derivatives, then the derivative should be f (x) ()(), not x. Teaching and Learning (0 to 5 min) The product rule can be taught in the reverse manner to how the textbook presents it. In other words, give the students two polynomials, f (x) and g(x), and have them compute the expression f (x)g (x) g(x)f (x). Tell the students that the resulting expression is a derivative of some simple combination of the two polynomials. It will not take students long to realize that the function under consideration is f (x)g(x). The power of a function rule is very easily discovered by giving students a function such as f (x) (3x x) 3. Have the students determine the derivative by expanding the function using the binomial theorem and using methods from the last section. Next have the students apply the derivative in the wrong way, i.e., f (x) 3(3x x). Have students compare the two expressions to first notice that they are not the same. Ask students to think about the difference between the two. If they cannot see the difference, have them factor the actual derivative. Many students will notice that the extra factor, (6x ), is the derivative of 3x x. From here, students can deduce the power of a function rule. TechnologyBased Alternate Lesson If a Computer Algebra System (CAS) is available, students can use it to calculate many derivatives of products in a very short amount of time. These systems are either available as computer software or as part of some advanced calculators that have symbolic manipulation. Give students examples of simple products at first, such as f (x) (x )(x ). Have the students change the to other values to observe patterns. The students can then move on to more complicated products such as f (x) (x 4 3x )(3x 6 4x x 7). Another alternative is to use The Geometer s Sketchpad. The product of two functions can be created by choosing the Graph menu and selecting New Function and it then can be graphed by selecting the function, going to the Graph menu and choosing Plot Function. Selecting the function and choosing the Graph menu allows you to choose Derivative. This displays the derivative function of the selected function and its graph can then also be plotted. The use of CAS or Sketchpad allows for many examples to be examined in a very timely manner..3: The Product Rule 5
16 3 Consolidation (30 to 35 min) Using the Solved Examples (ThinkPairShare) Example asks students to prove that the derivative of a product is not equal to the product of the derivatives. Present the product function from the example. Ask students to determine the derivative by expanding the original product. Then ask them to proceed as if one could simply multiply the separate derivatives. Finally, discuss with students why one counter example is sufficient to prove that a statement is false. Preceding Example is the statement and proof of the product rule. Students will need to see this proof done by the teacher. Once the students have been introduced to the idea, Example asks them to apply it to a very straightforward situation. Have the students use the ThinkPairShare strategy to solve the problem without using the textbook. Go over the example as a whole class when they are finished. Example 3 is only different in that the polynomials are a little more complex and that it asks students to evaluate the derivative at x. Have students proceed as they did in Example. Example 4 asks students to develop a product rule for a product of three functions by using the product rule for two functions twice. Students should be able to develop this rule using ThinkPairShare. If they struggle, go over the example with the whole class. Have each pair come up with a simple example that they can use to confirm their rule. The development of the Power of a Function Rule occurs prior to Example 5 and is critical because it is a precursor to the general chain rule. It has students investigate g (x), where g(x) 3 f (x) 4 n. Students should be led through the derivation for n. Then let them determine the solution for n 3 and n 4 in their pairs. In Example 6 students are asked to determine the derivative of a rational function by using the product and power rules. Once the power of a function rule has been developed, have each member of the pair select one of Example 5 and 6. The individual students should study the assigned example, take notes on the solution, and present the problem and solution to his or her partner without using the textbook. Take up any questions with the whole class. Example 7 is an application problem involving derivatives and velocity. Use this as an opportunity to reinforce the connection between a derivative, the slope of a tangent line, and instantaneous rate of change. Students should use a ThinkPairShare strategy to solve this problem with books closed. Answer to the Key Assessment Question 5. a. 9 b. 4 9 d. 36 e. f Calculus and Vectors: Chapter : Derivatives
17 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students use the product rule and power of a function rule as a shortcut for determining derivatives of products. Students can determine the derivative of a quotient by rewriting it as a product. If students misunderstand Students insist on multiplying the polynomials out in order to determine the derivative. Students forget to write the power when rewriting a quotient as a product. Students can present an algebraic argument with supporting examples for each of the rules in this section. Students can use derivatives to interpret real life situations involving velocity. Key Assessment Question #5 Students use the product and power rules to correctly determine all derivatives. Differentiating Instruction How You Can Respond Students move a power down and decrease it by to determine the derivative, i.e., they forget to multiply by the derivative of the inside. Students do not see the connection between velocity (rate of change) and derivatives. Students determine derivative by multiplying the polynomials out. Students determine the derivatives of the product by determining the product of the individual derivatives and/or misapply the power rule. EXTRA SUPPORT. Some students will have difficulty not simply multiplying derivatives. These students should spend more time expanding polynomials in order to determine the derivatives.. Some students will have a hard time keeping the various derivative rules straight. Encourage these students to make a summary sheet of derivative rules that they refer to as they work through the exercises until they become comfortable with them. EXTRA CHALLENGE. Have students investigate the derivative functions for y sin x and y cos x by calculating it at enough points to guess the function from a scatter plot. Then have them use the derivatives and the product rule in order to determine the derivative of y tan x.. Have students investigate the power of a function rule with exponents that are not integers and conjecture whether or not it is true..3: The Product Rule 7
18 MIDCHAPTER REVIEW Big Ideas Covered So Far f (a h) f (a) f (x) f (a) The derivative of a function f at a point (a, f (a)) is f (a) lim h, or f (a) lim x a, if the limit exists. A function is said to be differentiable at a if f (a) exists. A function is differentiable on an interval if it is differentiable at every number in the interval. f(x h) f(x) The derivative function for any function f (x) is given by f (x) lim h, if the limit exists. The value of the derivative at a point can be interpreted as the slope of the line tangent to the point or the instantaneous rate of change at the point. The derivative function can be determined using various rules, including: the constant function rule, the power rule, the constant multiple rule, the sum rule, and the difference rule. The rules for calculating derivatives of functions make the process easier than determining the derivative from first principles. The derivative of a product of differentiable functions is not the product of their derivatives. There are rules for determining the derivative of a function that is a product or a power of another function. Using the MidChapter Review h 0 Ask students if they have any questions about any of the topics covered so far in the chapter. Review any topics that students seem to be having trouble with. Have students complete as many of the exercises in class as possible and then complete any unfinished questions for homework. In order to gain greater insight into students understanding of the material covered so far in the chapter, you may want to ask them questions such as the following: Can you think of a reason to calculate a derivative using first principles? For a particular function? What about a general type of function? (The point is that the limit definition is useful for establishing a rule. Once the rule is determined, then the value of the limit approach is diminished because it is a far more tedious method for determining derivatives.) What does the sign of f (a) tell you about the function at x a? What does the derivative of a function represent? For review of the material in these exercises... Refer to...,, 3, 4, 5, 6, 7, 3, 4, 5, 6, 7, 8 8, 9, 0, h 0 x a Section. Section. Section.3 8 Calculus and Vectors: Chapter : Derivatives
19 .4 THE QUOTIENT RULE Section at a Glance GOAL Introduction of the quotient rule for differentiation. Prerequisite Skills/Concepts Expanding and simplifying polynomial expressions Constant, constant multiple, sum, difference, product and power of a function rules for derivatives of functions Specific Expectations Solve problems, using the product and chain rules, involving the derivatives of polynomial functions, sinusoidal functions, exponential functions, rational functions* e.g., by expressing f (x) x c x as the product f (x) (x )(x ) d, radical functions c e.g., by expressing as the power f (x) (x 5) f (x) x 5 d and other simple combinations of functions c e.g., f (x) x sin x, f (x) sin x cos x d *Even though the curriculum does not specify the development of the quotient rule, students who will take a postsecondary Calculus course will benefit from its exposure. Mathematical Process Focus Connecting Reasoning and Proving Selecting Tools and Computational Strategies MATH BACKGROUND SECTION OVERVIEW Student Book Pages Preparation and Planning Pacing 5 0 min Introduction 0 5 min Teaching and Learning min Consolidation Recommended Practice Questions 4, 6, 7, 8, 9, 0,,, 3, 4 Key Assessment Question Question 5 Extra Practice Section.4 Extra Practice New Vocabulary/Symbols the quotient rule Nelson Website Students should already be familiar with derivatives of polynomials, the product rule and the power of a function rule. Students will use the product rule to develop the quotient rule. Students will use the quotient rule to determine derivatives of rational functions. Students will determine equations of tangent lines to rational functions. Students will apply the quotient rule to problems involving instantaneous rates of change..4: The Quotient Rule 9
20 Introducing the Section (5 to 0 min) Remind students that they have been working on derivatives of various combinations of polynomials. They have already developed rules for the derivative of a product, sum, and difference of two polynomials. Ask the students to think about the one basic operation that is left. Present the students with the function f (x) x 3x x. Ask the students if they already have enough tools to evaluate this derivative. Students should recall from the previous section that they can write this as a product. Have the students determine the derivative individually, and check their work as a whole class. Teaching and Learning (0 to 5 min) Students can be taught to discover the quotient rule by rewriting the denominator using a power and applying a version of the power of a function rule. This is the strategy used in the previous section. Emphasize to students that the power of a function rule was only stated for positive integers, but this derivation strategy may be more straightforward with students. This section also provides a good opportunity to talk about vertical asymptotes of rational functions and the nondifferentiability of the function at these points. Give students a rational function to graph and ask them to determine the slope of the graph at various points. What happens to the derivative at the vertical asymptote? Students should see that the denominator of the original function shares a factor in common with the denominator of the derivative, and hence, where one function is undefined, so is the other. Students can also determine the derivatives of rational functions by doing long division first on the quotient. This does not avoid using the quotient rule, but it does provide a good opportunity to review long division. Give the students an example and have them determine the derivative by first using the quotient rule, and then by using long division. Ask them to discuss the relationship between the two answers as well as the pros and cons of each method. 0 Calculus and Vectors: Chapter : Derivatives
21 3 Consolidation (30 to 35 min) Using the Solved Examples (Small Groups) After presenting the derivation of the quotient rule with the whole class, have the class work in groups of 3 to apply the rule to the function in Example. This example is a straightforward application of the rule. Students may check their work with that in the book. Example has students determine the equation of a tangent line that requires the quotient rule. Students should work in their groups to solve this question. When they are finished, go over the technology solution as a whole class. Emphasize to students the importance of being able to solve this problem by using an algebraic approach. Example 3 has the students use derivatives to determine where the tangent line is horizontal. Students may need to be reminded that a horizontal tangent line has slope zero. From there, the small groups should be able to solve the problem and check their solution with other groups. Go over this example as a whole class when they are finished, and take up any questions or discrepancies. Answer to the Key Assessment Question 3 5. a. 4 7 b d. 3.4: The Quotient Rule
22 Assessment and Differentiating Instruction What You Will See Students Doing When students understand Students can explain the relationship between the quotient rule and the product rule. Students can apply the quotient rule to efficiently determine the derivatives of rational functions. If students misunderstand Students cannot apply the quotient rule, but must rewrite the function as a product. Students mix up the order in the numerator when using the quotient rule. Students can determine the equation of a tangent line to a rational function. Students can apply the quotient rule to problems involving rates of change. Key Assessment Question #5 Students determine all derivatives by hand using the quotient rule. Differentiating Instruction How You Can Respond Students can determine the slope of a function using the quotient rule, but cannot determine the equation of the tangent line. Students cannot interpret rates of change in real life applications in terms of derivatives. EXTRA SUPPORT. Some students will have difficulty remembering the quotient rule. Encourage these students to make a summary sheet of derivative rules that they refer to as they work through the exercises until they become comfortable with them. Give these students daily quizzes asking them to recall the formula until they can do it without looking.. Some students will consistently get an answer opposite to the correct answer when using the quotient rule. This is a result of subtracting in the numerator incorrectly. Show them the error they are making using a numerical example (e.g ). These students would benefit from memorizing the product and quotient rules presented using the same order in both. EXTRA CHALLENGE Students use the product rule instead of the quotient rule to solve the problems. Students attempt to determine the answers by using technology instead of algebraic means. ax b. Have students determine when the equation of the tangent line to f (x) cx d at the point x 3 crosses its graph. Does it always cross the graph? How many times? f(x) Q f(x) g(x) R. Have students develop a quotient rule for and for Q g(x). Are the two expressions different? h(x) h(x) R Calculus and Vectors: Chapter : Derivatives
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