Material for Difference Quotient

Transcription

1 Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05

2 Preface Te following difference quotient material starts wit te development of average slopes connecting points on a function. Ten tere is an algebra review focusing on factoring, rational expressions, radical notation and rational exponents. Examples are developed for eac of te average slope and algebra review topics. Ten te difference quotient topic is presented wit examples of various complexity. Te examples cosen use te concepts developed in te previous average slope and algebra review topics. Finally, in te summary te rationalization for computing difference quotients is sown by taking te next step and computing te derivative of te functions. Stepanie Quintal and Marvin Stick

3 Material for Difference Quotient Introduction Now tat we ave discussed te slope of a line and least squares regression to find te line of best fit, let s introduce function notation and average rate of cange. During tis course, we are using te notation wic represents a function. By definition, is a function of te independent variable x if for any x value tere is only one y value. Te y value is referred to as te dependent variable. Some examples of functions are sown below. a) b) c) d) For eac of te examples, one value of x in turn will generate only one value of y. Tere are some oter interesting points to mention about eac of te examples, including te domain and range. Te domain is te collection of possible x values tat a function can ave, and te range is te collection of generated y values. For example a), if we grap te line tere is no restriction on te values for te independent variable x, so te domain is all real numbers or. We don t use te symbol since x can never equal infinity. Te range or possible y values for example a) is likewise all real numbers, i.e.. For example b), grap te quadratic and you will no doubt be able to surmise intervals for te domain and range. Te independent variable x can again be anyting, so te domain is < x <. However, te smallest y value is, so te range of te function is. In te tird example c), y is a constant value regardless of te x value. It s like saying: No matter wo you are, I ll give you 7 extra points on your exam. Te domain of te function is < x < and te range is For te last example d), te possible values of te independent variable x are limited to x 0 since wen dealing wit real numbers we can t take te square root of a negative number. Likewise, te range of te function is y 0. Anoter way to write te domain and range is 0 x < and 0 y <. Some people use open and closed brackets to indicate intervals, but we ll concentrate on te interval notation just mentioned. Assume we are walking up a ill defined by te equation So, tis means as we walk up tis ill, te slope canges as we move. Tis equation is graped in figure.

4 Figure. Consider te points wic bot lie on te curve sown in figure y y. An average rate of cange x x is actually te slope of a secant line and te average rate of cange for te points wen x canges from to y y 4 0 becomes. A secant line cuts te curve at te two points. See x x if you can verify te computation.. Now let s see wat appens if te two x values get closer to eac oter. Using and, te average rate of cange is. Again, make sure you can verify tis computation. An important item to note is tat te average rate of cange or slope of te secant line decreased from to.. Wat appens if we pick a point still closer to? In tis last scenario, let s see wat appens wen we pick as te oter point. In tis case te y(.5) y().5 * average rate of cange becomes Summarizing te results of tese computations, te average slope started out to be, wen te x values were and, ten te slope became as te x values got a bit closer wen using te values and. Te last case sowed tat te slope of te secant line diminised to wen te x values got even closer using and. We are getting closer and closer to introducing te idea of instantaneous slope, i.e. te slope of tangent line as te difference in te x values approaces zero, but we re not quite tere as yet. Slopes of tangent lines will be were we start differentiating functions and doing some more involved calculus work. Algebra Review Before introducing te difference quotient, let s look at some rules of algebra needed to solve tese problems.. Factoring Problems ave been selected from Algebra and Trigonometry, nd ed. By Beecer, Penna and Bittinger, Pearson Education, Inc., 005, p.8.

5 Some steps to elp wit te factoring order of operations are:. Take out te GCF (greatest common factor).. Ceck for difference of squares. [Example:. Ceck for Quadratic Trinomial [Example: or Perfect Square Trinomial (Square of a Binomial). [Example: Note tat ( ) a+ b a + b unless eiter a0 or b0 4. Ceck for Sum of Cubes [Example: or Difference of Cubes. [Example: 5. Grouping: Split te equation in two and fully factor te two separate equations. Ten combine te two equations and factor tem togeter. 6. If none of tese work, te equation is fully factored. Factoring Example Factor First, let s try to take a greatest common factor from te expression. Te GCD of te two terms 7y and 4 is 7. Wen we pull a 7 out of 7y, we are left wit a y. Wen we pull a 7 out of 4, we are left wit a 6. Keep in mind tat an easy (and fast!) way to ceck work is to use te distributive property to multiply 7 by y plus 6. If your answer does not come out to te original expression, tere is a mistake somewere! Factoring Example Factor Most times wen tere are four terms in an expression, te expression sould be split in two and factored as two separate expressions. Let s try grouping. Te GCD of and is Like example, te GCD of 6x and 8 is 6. Next, we ave two expressions connected by an addition sign. Bot expressions ave a GCD of x plus. So, we ave

6 ( ) ( ) x x x x x x x x ( + 6)( + ).. Rational Expressions Problems ave been selected from Algebra and Trigonometry, nd ed. By Beecer, Penna and Bittinger, Pearson Education, Inc., 005, p.5-7. Some rules to elp wit factoring rational expressions (quotient of two polynomials) are:.. You can factor fractions, but you must factor te numerator and denominator separately.. To add/subtract fractions te denominator must be te same. 4. To find a common denominator, you must multiply te denominators by every unsared factor. 5. To multiply fractions, you multiply te numerator by te oter numerator(s) and te denominator by te oter denominator(s). 6. In order to divide fractions, you multiply te divisor by te reciprocal of te dividend. For example, te reciprocal of 7. Wen you multiply and divide fractions, you can opefully cancel out like factors (except wen eac factor is equal to zero). If one of te numerators sares a factor wit one of te denominators, you can reduce te fraction by canceling tose factors in te numerator and denominator. (Try to cancel as muc as possible because it makes te fraction muc simpler). 8. Always try to factor fractions! Rational Expressions Example Simplify te expression. Begin by factoring te numerator and denominator in bot of te fractions. a+ a 4 a+ a a a a + a 6 a a a a a a a a Next, combine bot numerators and bot denominators to make one fraction. Also, since tere is an and an on top and bottom, tey cancel out. 4

7 a+ a 4 a+ a a+ a+ a a a + a 6 a a a a a a a a a a Finally, since tere is an multiplied twice, it becomes. Radical Notation and Rational Exponents Problems ave been selected from Algebra and Trigonometry, nd ed. By Beecer, Penna and Bittinger, Pearson Education, Inc., 005, p First, let s look at te rules involving radicals.. If n is even, For example,. If n is odd, In tis case,. For example, Next, let s look at te rules for rational exponents.. For example,. Using te example before,. 4. a m n. An example is 8 m n a m n m n a a a +. An example is m 5. ( ) n mn a a. An example is ( ) ( ) Let s look at steps to rationalize te denominator. To rationalize te denominator, multiply te top and bottom by te conjugate. Consider te example Te conjugate of tis expression is so let s multiply te top and bottom of te fraction by tis expression. Rationalizing te numerator is very similar. To rationalize te numerator, multiply top and bottom by te conjugate of te numerator. 5

8 Radical Notation Example 4 Rationalize te denominator for Te conjugate is, we get Radical Notation Example 5 so multiplying te numerator and denominator of by Convert to radical notation and simplify a a b a 4 4 b b. Radical Notation Example 6 4 Simplify 8. ( ) ( ) Note tat in tis example we added te exponents wen evaluating since m n m n a a a +. Radical Notation Example 7 Convert to exponential notation. 6. Note tat in tis example we multiplied te exponents wen evaluating 6 since ( a m ) n mn a. Difference Quotient Let s now introduce te concept of te difference quotient. 6

9 Difference quotients are essentially average rates of cange but we use functional y y notation instead of notation, as seen previously. Te average rate of cange of x x te function wit respect to te independent variable x is called te difference f( x+ ) f( x) quotient and is defined as were represents te cange in x. Note tat some textbooks use x instead of to represent te cange in x. Also, since is in te denominator for te difference quotient, 0. Keep in mind tat, in general, Likewise, as we said earlier te algebraic expression ( ) a+ b a + b so long as 0 ( ) + +. Consider te following example. a or b 0. For tose wo like to use numbers, Difference Quotient Example 8 For find te simplified form of te difference quotient. Ten, find te value of te difference quotient wen and 0.. First, let s find Next, we use te FOIL metod to find f( x+ ) f( x) x+ Ten, and. Now, we factor te numerator. Te greatest common factor in te numerator is, so we pull out an from eac term in te numerator and see wat appens. ( + ) f( x+ ) f( x) x+ x x+. Te result x+ geometrically represents te slope of a secant for te function f( x) x at a particular value for x wit a gap. Assume x and 0., ten te slope of te secant line for f( x) x connecting te points x and x++0.. is x+()+0... Difference Quotient Example 9 Next, let s consider te example Find ( + ) ( ) f x f x. To start, let s find We substitute werever tere is an x. 7

10 So, ( + ) ( ) x + x x ( x x ) f x f x x + x x x + x x ( x ) x, 0. Te slope of te secant line for f( x) x x wen x and 0. is -x--() Difference Quotient Example 0 Let s look at a more involved example, Like te previous example, first let s find f ( x+ ) f ( x ) We see tat so x+ x. To simplify, we must subtract fractions, but we ave no common denominator for. We ave to find x+ x a common denominator and simplify. In tis case, te common denominator is f ( x+ ) f ( x ) x+ x x x+ x+ x x x+ x x+ ( x+ )( x) ( x+ )( x) 8

11 Now, we ave a common denominator ( x+ x ) and can subtract te fractions. Te result ( x+ x ) is f( x+ ) f( x). Wen x and 0., te slope of te secant ( x+ x ) line for f( x) is 0.8 rounded to decimal places. x ( x+ x ) ( + 0.). Difference Quotient Example Let s look at an example involving square roots to find te difference quotient. Consider te function f ( x) x. ( ) ( ) f x+ f x x+ x We can see tat f ( x+ ) x+, so. In order to furter simplify and remove te from te denominator, we need to rationalize te numerator. We multiply te top and bottom of te expression by te conjugate, or x+ + x. ( ) ( ) f x+ f x x+ x x+ + x x x + + x+ x x x ( + + ) x x ( + + ) ( x+ + x). Now te slope of te secant line for f( x) x at x and 0. is x+ + x. + rounded to decimal places. Difference Quotient Example Finally, consider f( x) x +. In tis example f( x+ ). We will ( x+ ) + ave to rationalize te denominator wen computing te difference quotient. Let s first begin wit 9

12 x+ ( x+ ) + f ( x+ ) f ( x) ( x+ ) + x+ ( ( x+ ) + )( x+ ) ). Make sure tat you can justify te steps required up to tis point. We now ave to multiply te numerator and denominator by te conjugate of x+ ( x+ ) + wic is x+ + ( x+ ) +. Tis results in x+ ( x+ ) + x+ ( x+ ) + f ( x+ ) f ( x) ( ( x+ ) + )( x+ ) ) ( ( x+ ) + )( x+ ) ) x+ + ( x+ ) + x+ + ( x+ ) + x+ (( x+ ) + ) ( ( x+ ) + x+ )( x+ + ( x+ ) + ) x x x x ( ( + ) + + )( + + ( + ) + ) Wow, wat a messy expression! Make sure you can justify te steps to get te above result. Don t worry, difference quotient problems tat you will see usually won t be tis demanding. Now wen x and 0., we can find te slope of te secant line to f( x) wic results in x + (.) (.) + rounded to decimal places. ( ) Summary Wen we started te discussion in te introduction, we spoke of average rates of cange wit x values getting closer to eac oter. Since a difference quotient is essentially an average rate of cange, but to a specified function, we can evaluate te slope of te secant lines at arbitrary x and values based on te domain of te function f( x ). Te derivative of a function f( x ) is te limit as 0 for te difference quotient and tis is called te slope of te tangent line to a curve f( x ) at some value x. Te formal ( + ) ( ) definition of a derivative for f( x ) is f '( x). For eac of te difference quotient examples we did, let s now find te derivatve. a) Difference quotient example 8: lim( x+ ) x+ 0 x so te derivative (slope of 0 0 f lim x f x te tangent line) of f( x) x is f '( x) x. b) Difference quotient example 9: lim( x ) x 0 x so te derivative (slope of te tangent line) of 0 is '( ) f( x) x x f x x.. 0

13 c) Difference quotient example 0: lim so te derivative 0 ( ) ( 0) x+ x x+ x x (slope of te tangent line) of f( x) is f '( x). x x c) Difference quotient example : lim so te 0 x+ + x x+ 0+ x x derivative (slope of te tangent line) of f( x) x is f '( x). x d) Difference quotient example : lim 0 ( x+ ) + x+ x+ + ( x+ ) + ( ) ( ) ( x+ 0) + x+ x+ + ( x+ 0) + (x+ ) x+ (x+ ) (x+ ) So te derivative (slope of te tangent line) of f( x) x + is f '( x) (x + ). Fortunately we will develop formulae based on te limit of te difference quotient to make finding derivatives of functions and teir applications muc easier tan te examples sown as we found te difference quotient and ten took te limit as ->0. Remember owever, tat witout going troug te difference quotient process, we would never ave been able to find derivatives and slopes of tangent lines to functions along wit te many applications tat are based on tis tecnique.