Computing Derivatives J. Douglas Child, Ph.D. Rollins College Winter Park, FL

Transcription

1 Computing Derivatives by J. Douglas Chil, Ph.D. Rollins College Winter Park, FL

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3 Objectives To use transformations to remove erivative symbols To simplify expressions Computing Derivatives Materials TI-89 or Voyage 00 Symbolic Math Guie (SMG) hanhel application Activity Overview These example problems provie ieas about using SMG to help stuents learn to solve erivative problems. The key transformations in SMG are the sum/ifference, constant (or scalar) prouct, prouct, quotient, an the chain rule. SMG also inclues transformations for erivatives of basic functions. Stuents can use SMG to learn successful strategies for computing erivatives. Necessary backgroun inclues the ability to classify functions as power, exponential, prouct, composition, an so on. Stuents also nee to know how to ecompose composite functions. Setting Up the TI-89 or TI Voyage 00 Before starting, make sure that the TI-89 or TI Voyage 00 has SMG installe. If the TI-89 or TI Voyage 00 oes not have SMG version.00 or higher installe, go to eucation.ti.com to ownloa the free app. If you are not familiar with SMG, the SMG User Guie an the SMG Guie Tour are available at the same link. Start SMG by pressing O, choosing SMG, an then pressing. Entering Derivative Problems Example Enter (3).

4 4 Activity for Symbolic Math Guie on the TI-89 an TI Voyage Press 1 an then press 1. The isplay shoul be similar to the one shown at the right. Press to set the isplay to enter erivative problems.. Type 3,x. Press until the winow isappears. The Derivatives of Basic Functions You begin to learn to compute erivatives by learning to recognize the erivatives of functions such as polynomials, sine, cosine, tangent, ln, log, an so forth. See Stuent Reference Sheet 1, Basic Derivatives, for a table of some of the basic erivatives that SMG knows. In aition, your teacher may choose examples from textbooks or other references for you to solve. An example set of basic erivatives problems is inclue in Stuent Work Sheet 1, Basic Problems. Master these problems before continuing to the next section. Example Compute (3) using SMG. Enter the problem as escribe in the previous section. 1. The problem appears on the right.

5 Computing Derivatives 5. Press. The isplay shoul be similar to the one shown at the right. 3. Press 1 to apply erivative of constant. 4. The complete solution for this simple problem contains three lines incluing the name of the transformation you chose. Derivatives of Sums an Differences ( f ± g)' f ' ± g' The erivative of a sum is the sum of the erivatives an the erivative of a ifference is the ifference of the erivatives. See Stuent Work Sheet, Derivatives of Prouct, Scalar Proucts, an Quotients, for practice using these formulas. Derivatives of Proucts ( f g)' f ' g + f g' SMG provies a special winow for applying the prouct formula. This winow gives you control over applying the prouct rule to terms mae of three or more factors an may help you not to apply the prouct rule when you shoul actually use the chain rule to ifferentiate a composition. Example x Compute ( x e ) using SMG. Make sure you use s to enter the exponential function. (Note: On the TI-89, press Ù.)

6 6 Activity for Symbolic Math Guie on the TI-89 an TI Voyage Press to list possible transformations for the first step. The function to ifferentiate is a prouct of x an an exponential function.. Press to select erivative of prouct. SMG isplays a winow to help you apply the prouct rule. 3. Decie how to choose f(x) an g(x), an then press ƒ to have SMG fill in the key information. Make sure that the substitution is the one you want. Change it if it is not. 4. Press until the ialog box isappears. 5. Compare SMG s result with yours. Resolve any ifferences. Press an choose basic erivatives to complete this problem. See Stuent Work Sheet, Derivatives of Prouct, Scalar Proucts, an Quotients, for practice using the prouct rule. Remember the goal is for you to be able to work these problems without SMG. Use SMG to make sure you correctly take each step. Your focus shoul be on choosing a goo transformation for the problem you are solving.

7 Derivatives of Scalar Proucts ( c f ) c ( f ) Computing Derivatives 7 Many stuents use the prouct rule when the scalar prouct rule, which is easier to use, applies. The two screens that follow show the ifference between using the prouct rule an using the scalar prouct rule when ifferentiating a constant times function. Choosing the simpler scalar prouct rule is especially important when ifferentiating complicate expressions. Begin by entering the problem. 1. Press ƒ.. Press. Choose basic erivatives. 3. Observe that the secon term is 0. One term will always be 0 when one of the factors is a constant. 4. Press C to move to the top of the problem. 5. Press Press Observe that the extra term never appears. No extra simplification is necessary. See Stuent Work Sheet, Derivatives of Prouct, Scalar Proucts, an Quotients, for aitional examples. Derivatives of Quotients ( f /g)' f ' g f g' g This important formula causes stuents some ifficulty because the erivative of the numerator is in a positive term while the erivative of the enominator is in a negative term. Practice writing the formula from memory until you have mastere it. Note that, once you have learne the chain rule in the next section, you can use the prouct rule to ifferentiate quotients. See Stuent Work Sheet 4, You Pick the Metho(s), problem 6. Stuent Work Sheet contains problems that you can use to practice the quotient rule. Example Compute (sin(x)/x,x) using SMG. Enter the problem an then follow along with pencil an paper anticipating SMG s results.

8 8 Activity for Symbolic Math Guie on the TI-89 an TI Voyage The problem appears on the right.. Press. The isplay shoul be similar to the one shown at the right. 3. The problem is to ifferentiate a quotient, so apply the quotient rule on your paper an then press Compare your answer with SMG s answer. Make sure you have the minus sign in front of the correct term. 5. You can finish removing the erivative symbols by applying erivative formulas for the basic functions sin(x) an x. Do this on your paper an then press an choose basic erivatives. Compare your answer with the result on the right or the one below an on the right.

9 Computing Derivatives 9 6. Finish by pressing to simplify the result. Derivatives of Compositions an the Chain Rule 3 10 Some example compositions ( x + 1), sin( x ), cos ( x), an x 7. Note that sin(x ) is not a prouct although it looks like one. In SMG, proucts are isplaye with a ot between factors like x ( x + 1). Look for the ot to make sure the expression contains a prouct instea of a composition, which will not have the ot. Example + Compute ( x + 1). Enter the problem. 1. To isplay choices, press. The isplay shoul appear similar to the one shown at the right.. The function to ifferentiate is a composition, so press 4: erivative of composition. 3. sqrt(x^+1) is a composition with outsie function f(u) = sqrt(u) an insie function x^+1. Press ƒ to confirm that this thinking is correct. You may type information yourself. In some cases, SMG will not know how to fill in the information.

10 10 Activity for Symbolic Math Guie on the TI-89 an TI Voyage Press until the winow isappears. To finish the problem, press, choose erivative of polynomial, an press to simplify the result. See Stuent Work Sheet 3 for practice solving these problems.

11 Computing Derivatives 11 Developing Successful Strategies To learn to compute erivatives: Learn the basic erivatives an how to choose the proper formula to use them. Stuents often confuse power an exponential functions. It is helpful to tie formulas to graphs. For example, have stuent compare the graphs of y = x an y = x. Then ask them to compare the slopes of tangents at x = 1, x = 10, an x = 100. SMG will place the proper transformation near the top of the list of transformations that result from pressing. Master each major formula. Ask stuents to apply specific formulas to particular problems. Learn to recognize sums, proucts, quotients, an compositions. Stuents often confuse scalar proucts an proucts as well as compositions an proucts. SMG places the scalar prouct transformation at the top when it applies this transformation. This serves as a prompt for the stuents. Learn to compute erivatives that require the use of more than one major rule. Drawing syntax trees may help. Syntax trees use in computer science serve as an alternate representation of expressions that help stuents visualize the structure of math expressions. Stuent Work Sheet 4, You Pick the Metho(s), provies stuents with practice in selecting an applying ifferent rules for a variety of problems. Teaching Notes 1. The use of names to ientify erivative rules provies an opportunity for stuents to learn the erivative formulas. Think about the formula for the chain rule before you choose it. When SMG shows you the formula, compare it with your version. Writing your version first may be especially helpful.. Uses of SMG: To give stuents an overview of SMG, emonstrate the use of SMG in class. Give stuents a goal to work towar. SMG provies a goal for each problem type. Press 1 ( [F7] on the TI-89) to fin the goal. For erivative problems, SMG says Apply erivative formulas to remove all inicate erivatives (an then simplify). The use of SMG suggests several types of problems that may help stuents evelop successful strategies for attacking compute the erivative problems.

12 1 Activity for Symbolic Math Guie on the TI-89 an TI Voyage 00 Problem Type 1: Give stuents an outline of a solution an have them fill in a solution. The outline coul be a sequence of SMG transformations that solve the problem. Problem Type : Have stuents evelop a plan for computing a erivative. The plan coul look like the transformations that SMG reports in the solution isplay. Some stuents benefit from structure trees to help them make solution plans. 3. Opportunities to learn with SMG: Before choosing a name transformation, the stuent coul write (or think) its formula (if it has one). Before pressing with TIME TO THINK moe ON, the stuent coul write the result of the transformation.

13 Stuent Reference Sheet 1 Basic Derivatives Name: Date: The following table contains a sample of the formulas available. Derivative of constant: ( C) 0 Derivative of ientity: ( x) 1 Power Rule: r ( r 1) ( x ) r x x x ( e ) e x x ( a ) a ln( a) (sin( x)) cos( x) (cos( x)) sin( x) (tan( x)) (sec( x)) (cot( x)) (csc( x)) (csc( x)) csc( x) cot( x) 1 (sin ( x)) 1 (cos ( x)) 1 1 x 1 1 x 1 1 (tan ( x)) 1+ x

14 Stuent Reference Sheet Key Transformations Name: Date: Key Transformations Derivative of scalar prouct ( c f ) c ( f ) Derivative of sum or ifference ( f ± g)' f ' ± g' Derivative of prouct ( f g)' f ' g + f g' Derivative of quotient f ' g f g' ( f / g)' g Derivative of composition [chain rule] Examples/Comments ( 3 ln( x)) ( x + ln( x)) ( x ln( x)) ln( ( x ( e x x) ) ) ( f ( u) ) f '( u) ( u) 1 (ln( f ) f ' f (ln( x ) 1 x (ln e x ) Basic erivatives Applies transformation for each basic erivative in the expression. ( 1/ x) + (sin( x))

15 Stuent Work Sheet 1 Basic Problems Name: Date: Use these problems to check your mastery of the basic formuals. Compute each of the erivatives below using SMG unless inicate otherwise. Basic Derivatives 1. (3) eriv of constant 1. Think: 3 is a constant an the erivative of a constant is 0.. Press an choose erivative of constant.. 1 x (1/x,x) -> -1/x^ Remember formula or think 1/x = x ( x ) 4. / 3 ( x ) 5. (ln( x)) 6. (log(x)) 7. x ( e ) 8. x ( ) Remember formula or think 1/ x = x

16 16 Activity for Symbolic Math Guie on the TI-89 an TI Voyage 00 Stuent Work Sheet Derivatives of Prouct, Scalar Proucts, an Quotients Name: Date: You shoul work several problems manually before using SMG to work the problems below. Derivative of a Prouct x 1. ( x e ). ( t t ( 1 t )) 3. ( x g( x)) erivative of prouct Note: You coul simplify sqrt(t)*(1-t) before entering this problem. How woul simplification help? Note: Do not enter a formula for g(x) when aske to. The solution will contain (g(x))/. 4. ( t 3 cos( t)) t Derivative of a Quotient x + x 5 x + 6 x e 1 x + tan( x) 7. x x 8. sin( x) + cos( x) Test your mastery of the quotient rule. a. The object being ifferentiate is a quotient. Try the quotient rule. b. Press an choose erivative of quotient. c. Write own your result of applying the quotient rule.. Press, fill in the ialog box, press until the ialog goes away, an compare SMG s result with your result. e. Resolve any ifferences.

17 Computing Derivatives 17 Derivative of Scalar Prouct 9. ( 3 x) (c f (x)) c ( f (x)) ( x) 1 [Press to clean-up.] 10. (7 x ) (c f (x)) c ( f (x)) (xr ) r x r 1 [Clean-up] 11. ( x) 1. ( y x) 13. ( y( x) x) Note: Do not enter a formula for y(x) when requeste; just press. Is this a scalar prouct?

18 18 Activity for Symbolic Math Guie on the TI-89 an TI Voyage 00 Stuent Work Sheet 3 Derivatives of Compositions Name: Date: Work several problems by han before using SMG to help you with the problems below. 1. ( x + 1). (ln( x 3 + 1) 3. t t t ( ln( x )) Note: Log properties might help you simplify this problem. 5. ( sin(sin(sin( x))) ) 6. ( ln( sin( x) ))

19 Computing Derivatives 19 Stuent Work Sheet 4 You Pick the Metho(s) Name: Date: Compute each of the erivatives below using SMG unless other instructions are provie ( x + 3x ) Version A erivative of sum or iff basic erivatives [clean-up] Version B [This is quicker an can be use once you have mastere ifferentiating polynomials.] erivative of polynomial [clean-up] x x. ( e ) erivative of sum or iff basic erivatives 3. (1 x + x) x 4. ( x e ) 5. t (t 1 t ) 6. 5e x x + 4 [Hint: You can think of this as a prouct an then use the chain + rule.]

20 0 Activity for Symbolic Math Guie on the TI-89 an TI Voyage 00 3 / 5 7. ( x ) 8. ( x 3 sin( x)) 9. (csc( t) + e t cot( t)) t erivative of sum or ifference. Select ( e t cot( t)) then select the erivative of the prouct. t basic erivatives. [Press to clean-up.] 10. (sec( x) cot( x)) Select sec( x) cot( x) then rewrite as 1/sin(x) Select 1/sin(x) then select A /(sin( B)) A csc( B) [Clean-up] (csc( x)) csc( x) cot( x) Note that simplifying the trigonometric expression before ifferentiating often simplifies the solution. 11. ( x 1 + x + 1) 1/ 3 1. ( ln( x ) )