(¡9x2 +3x+5)100 6. pp x 7. tan(ln(4x)) 8. cos(ln(x)) 9. Share practice link. Only in the next step do you multiply the outside derivative by the derivative of the inside stuff. Search. we can now differentiate. 5 0 obj 1. The chain rule states formally that . Video lectures to prepare quantitative aptitude for placement tests, competitive exams like MBA, Bank exams, RBI, IBPS, SSC, SBI, RRB, Railway, LIC, MAT. Mathematics. The Chain Rule mc-TY-chain-2009-1 A special rule, thechainrule, exists for differentiating a function of another function. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. Online aptitude preparation material with practice question bank, examples, solutions and explanations. 1. log13 (8x3 +8) 2. If you're seeing this message, it means we're having trouble loading external resources on our website. Chain Rule DRAFT. 11th - 12th grade . Here is a set of practice problems to accompany the Chain Rule section of the Derivatives chapter of the notes for Paul Dawkins Calculus I course at Lamar University. Answer to 2: Differentiate y = sin 5x. The chain rule is a rule for differentiating compositions of functions. �d0"���-������_?�����O�=�zO��[�������&Y���������Q_�������ǧ�R�)�x�)�������蓒�"���߇���o���d�;����q�r~�Z�{j��SS�Z�ޕ쩔\��Zm�^ҭJW����F�?>�'���m�>\${�ȅ���6G�M��n������:n��o���6�w������HW� Vm)��]Œ���-l���]�R�*�_Z-�s��R~G���N_�R�oU�V���r��T)3���_�\� These Multiple Choice Questions (MCQs) on Chain Rule help you evaluate your knowledge and skills yourself with this CareerRide Quiz. BY REVERSE CHAIN RULE . The chain rule tells us how to find the derivative of a composite function. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The Chain Rule is thought to have first originated from the German mathematician Gottfried W. Leibniz. lOMoARcPSD|5309549 1 2 Chain Rule - Lecture notes 1 Engineering Mathematics 2E (University of … Chain rule Statement Examples Table of Contents JJ II J I Page1of8 Back Print Version Home Page 21.Chain rule 21.1.Statement The power rule says that d dx [xn] = nxn 1: This rule is valid for any power n, but not for any base other than the simple input variable x. The derivative of a constant is zero. ©C N2S0c1h3 j dKJu ntva P zS7oIf ktdweaNrdeR nLQLJC N.y a ZA 0l uln Mrli tg Qhft fs S pr be4s Hehrev Pe2d B.k j WM1a0det 4wti Ctlh2 CInMf8iKnli Vt Zer qC naKlsc DuKlGurs L.e … Delete Quiz . After the chain rule is applied to find the derivative of a function Fx( ), the function Fx fx x x′( )==( ) 4 cos 3 sin 3 3(( ))3⋅− ⋅(( )) is obtained. Chain-Rule. This is the Chain Rule Common Derivatives ... 1st Derivative Test If x c is a critical point of fx then x c is 1. a rel. Review your understanding of the product, quotient, and chain rules with some challenge problems. Chain Rule & Exponential Instructions • Use black ink or ball-point pen. Now, the chain rule is a little bit tricky to get a hang of at first, and this video does a great job of showing you the process. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. The chain rule is used to differentiate composite functions. Now, the chain rule is a little bit tricky to get a hang of at first, and this video does a great job of showing you the process. Chain Rule worksheet MATH 1500 Find the derivative of each of the following functions by using the chain rule. Example: Find the derivatives of each of the following. Mathematics; Mathematics / Advanced pure; Mathematics / Advanced pure / Differentiation; 14-16; 16+ View more . In applying the Chain Rule, think of the opposite function f °g as having an inside and an outside part: General Power Rule a special case of the Chain Rule. As u = 3x − 2, du/ dx = 3, so. After the chain rule is applied to find the derivative of a function Fx( ), the function Fx fx x x′( )==( ) 4 cos 3 sin 3 3(( ))3⋅− ⋅(( )) is obtained. These points are the set f(x;f(x)) : x 2 Dg; where D is the domain of f(x) The graph of a function has many properties. The Chain Rule Quiz Web resources available Questions This quiz tests the work covered in the lecture on the chain rule and corresponds to Section 14.6 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al. This rule allows us to differentiate a vast range of functions. Examples •Differentiate y = sin ( x2). Just skip to 4:40 in the video for a chain rule lesson. (Gҽ(��z�T�@����=�7�Z���z(�@G���UT�>�v�=��?U9�?=�BVH�v��vOT���=盈�P��3����>T�1�]U(U�r�ϻ�R����7e�{(� mm Ekh�OO1Tm'�6�{��.Q0B���{K>��Pk�� ��9Mm@?�����i��k��V�謁@&���-��C����ñ+��ؔgEY�rI*آ6�`�I3K�����a88$�qV>#:_���R���EEV�jj�\�.�^�8:���,|}Ԭ�O;��l�vMm���q (ln(4x))100 14. eln(x) 15. sin(e6x) 16. Problems on Chain Rule - Quantitative aptitude tutorial with easy tricks, tips, short cuts explaining the concepts. Check your work by taking the derivative of your guess using the chain rule. The chain rule provides us a technique for finding the derivative of composite functions, with the number of functions that make up the composition determining how many differentiation steps are necessary. Also learn how to use all the different derivative rules together in a thoughtful and strategic manner. Chain-Rule. and the Chain Rule: A Worksheet Prof.Rebecca Goldin Oct. 8, 2003 1 Parametric Equations We have seen that the graph of a function f(x) of one variable consists of a set of points in the xy-plane. h�b```c``z"��@(�������x�$PL�%B&�u�B�D�H �R���1���U&\�%�@w���?����zqo��*-�. A GRAPHING CALCULATOR IS REQUIRED FOR THIS QUESTION. Chain-rule-practice. This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. The derivative of xn is nxn¡1. Solution: Example: Differentiate y = (2x + 1) 5 (x 3 – x +1) 4. Edit. Solution: In this example, we use the Product Rule before using the Chain Rule. Report a problem. 1. Practice questions. Rule 2: The General Power Rule. Derivatives - Sum, Power, Product, Quotient, Chain Rules Name_____ ©Q w2e0]1Y6f tKluIt^a_ ISmoXfTtaweaCroeJ VLrLYCG.L H XAglull drjiYgAhHtjsF trre_sReErUvweKdR.-1-Differentiate each function with respect to x. Derivatives - Sum, Power, Product, Quotient, Chain Rules Name_____ ©Q w2e0]1Y6f tKluIt^a_ ISmoXfTtaweaCroeJ VLrLYCG.L H XAglull drjiYgAhHtjsF trre_sReErUvweKdR.-1-Differentiate each function with respect to x. Practice. For example, if a composite function f( x) is defined as The chain rule gives us that the derivative of h is . Updated: Feb 22, 2018. docx, 16 KB. Search. In order to master the techniques explained here it is vital that you undertake plenty of practice exercises so that they become second nature. Just follow the step by step instructions, and you'll get the hang of it! stream g’(x) Outer function Evaluated at inner function Derivative of outer function Derivative of inner . Note: In the Chain Rule, we work from the outside to the inside. Chain-Rule. Thus, the slope of the line tangent to the graph of h at x=0 is . The Chain Rule Quiz Web resources available Questions This quiz tests the work covered in the lecture on the chain rule and corresponds to Section 14.6 of the textbook Calculus: Single and Multivariable (Hughes-Hallett, Gleason, McCallum et al. pptx, 203 KB. This section shows how to differentiate the function y = 3x + 1 2 using the chain rule. Then, y is a composite function of x; this function is denoted by f g. • In multivariable calculus, you will see bushier trees and more complicated forms of the Chain Rule where you add products of derivatives along paths, 0. 5 ( ) h x = 3 x + 2 x − 4 The Chain Rule Each of the functions ( ) h x (() f g x). Here is another good video for some practice examples of all of these rules. Video lectures to prepare quantitative aptitude for placement tests, competitive exams like MBA, Bank exams, RBI, IBPS, SSC, SBI, RRB, Railway, LIC, MAT. Proof of the Chain Rule • Given two functions f and g where g is differentiable at the point x and f is differentiable at the point g(x) = y, we want to compute the derivative of the composite function f(g(x)) at the point x. Print; Share; Edit; Delete; Host a game. pptx, 203 KB. The derivative of a constant is zero. We differentiate the outer function and then we multiply with the derivative of the inner function. ( ) ( ) 1 1 2 3 31 4 1 42 21 6 x x dx x C − ∫ − = − − + 3. ). Played 211 times. pptx, 203 KB. ∫4sin cos sin3 4x x dx x C= + 4. ( ) ( ) 3 1 12 24 53 10 ∫x x dx x C− = − + 2. ∫4sin cos sin3 4x x dx x C= + 4. Rule 2: The General Power Rule. Here is another good video for some practice examples of all of these rules. Online aptitude preparation material with practice question bank, examples, solutions and explanations. y=f(u) u=f(x) y=(2x+4)3 y=u3andu=2x+4 dy du =3u2 du dx =2 dy dx =3u2×2=2×3(2x+4)2 dy dx = dy du ⋅ du dx dy dx =6(2x+4)2. Chain-rule-practice. Example 1 Difierentiate y = x4. These points are the set f(x;f(x)) : x 2 Dg; where D is the domain of f(x) The graph of a function has many properties. What is the inner function ? ¡cos(4x+9) 3. It will take a bit of practice to make the use of the chain rule come naturally—it is more complicated than the earlier differentiation rules we have seen. of fx if fx 0 to the left of x c and fx 0 to the right of x c. 2. a rel. What is the inner function ? Info. So all we need to do is to multiply dy /du by du/ dx. Chain rule examples: Exponential Functions. Example … In the following discussion and solutions the derivative of a function h(x) will be denoted by or h'(x) . Differentiate algebraic and trigonometric equations, rate of change, stationary points, nature, curve sketching, and equation of tangent in Higher Maths. and the Chain Rule: A Worksheet Prof.Rebecca Goldin Oct. 8, 2003 1 Parametric Equations We have seen that the graph of a function f(x) of one variable consists of a set of points in the xy-plane. This line passes through the point . The chain rule says that. Report a problem. • Fill in the boxes at the top of this page with your name. About this resource. ¡cos(ln(4x)) 5. ( ) ( ) 1 1 2 3 31 4 1 42 21 6 x x dx x C − ∫ − = − − + 3. Examples •Differentiate y = sin ( x2). ). (sin(x))100 4. Chain Rule Online test - 20 questions to practice Online Chain Rule Test and find out how much you score before you appear for next interview and written test. Created by T. Madas Created by T. Madas Question 1 Carry out each of the following integrations. Just follow the step by step instructions, and you'll get the hang of it! So when using the chain rule: h�bbd```b``="#@$C����3@$w+�p�l���@���w,� "Y���-�c0;�f ����E��� �M�"���;@��� �(��|���O��X�z��%�L�g��z � d_I endstream endobj startxref 0 %%EOF 1439 0 obj <>stream You are permitted to use your calculator to solve an equation, find the derivative of a function at a point, or calculate the value of a definite integral. 2¡9x2+3x+5 10. The chain rule (function of a function) is very important in differential calculus and states that: dy = dy × dt: dx dt dx (You can remember this by thinking of dy/dx as a fraction in this case (which it isn’t of course!)). 80% average accuracy. g’(x) Outer function Evaluated at inner function Derivative of outer function Derivative of inner . Chain Rule: Solved 10 Chain Rule Questions and answers section with explanation for various online exam preparation, various interviews, Logical Reasoning Category online test. Click HERE to return to the list of problems. The basic rules of difierentiation, as well as several common results, are presented in the back of the log tables on pages 41 and 42. (sin(x))100 4. Let u = 5x (therefore, y = sin u) so using the chain rule. Finish Editing. 13. P\U�����¬t��+X]�K�R�T=07�Φ. As we can s 11 p ln(x) 19. As we can s 3 years ago. Courses. Donate Login Sign up. With chain rule problems, never use more than one derivative rule per step. 2 3 1 sin cos cos 3 ∫ x x dx x C= − + 5. Hint : Recall that with Chain Rule problems you need to identify the “inside” and “outside” functions and then apply the chain rule. Example. Solo Practice. Chain-rule-practice. This quiz is incomplete! Using the point-slope form of a line, an equation of this tangent line is or . This chapter focuses on some of the major techniques needed to find the derivative: the product rule, the quotient rule, and the chain rule. Category Questions section with detailed description, explanation will help you to master the topic. Example … (¡9x2 +3x+5)100 6. pp x 7. tan(ln(4x)) 8. cos(ln(x)) 9. Find the following derivative. 2¡9x2+3x+5 10. Check your work by taking the derivative of your guess using the chain rule. ( ) ( ) 3 1 12 24 53 10 ∫x x dx x C− = − + 2. 13. given in the first column below is a composition of functions What is the outer function ( ) f x ( ) g x? (ln(4x))10 11. cot(¡9x2 +3x+5) 12. p 4x+9 13. If , where u is a differentiable function of x and n is a rational number, then Examples: Find the derivative of each function given below. The Chain Rule mc-TY-chain-2009-1 A special rule, thechainrule, exists for differentiating a function of another function. min. Homework. Categories & Ages. Note: In the Chain Rule, we work from the outside to the inside. This unit illustrates this rule. Most problems are average. Chain-rule-practice. • Fill in the boxes at the top of this page with your name. It will take a bit of practice to make the use of the chain rule come naturally—it is more complicated than the earlier differentiation rules we have seen. This quiz is incomplete! Example: Find the derivatives of each of the following. ¡cos(4x+9) 3. x��}ۮ%�������~�#�l1x��d�0��T���A>}���6�c��c�f&yv�.��F��2� (Section 3.6: Chain Rule) 3.6.2 We can think of y as a function of u, which, in turn, is a function of x. If you're seeing this message, it means we're having trouble loading external resources on our website. • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). • If pencil is used for diagrams/sketches/graphs it must be dark (HB or B). Show Solution For this problem the outside function is (hopefully) clearly the exponent of 4 on the parenthesis while the inside function is the polynomial that is being raised to the power. The Chain Rule is thought to have first originated from the German mathematician Gottfried W. Leibniz. Chain rule examples: Exponential Functions. Chain Rule Instructions • Use black ink or ball-point pen. Give a possible function for Fx( ). All the latest breaking UK and world news with in-depth comment and analysis, pictures and videos from MailOnline and the Daily Mail. Chain-Rule. Chain Rule worksheet MATH 1500 Find the derivative of each of the following functions by using the chain rule. 2 3 1 sin cos cos 3 ∫ x x dx x C= − + 5. Example 1 Difierentiate y = x4. Solution: Example: Differentiate y = (2x + 1) 5 (x 3 – x +1) 4. Categories & Ages. In other words, when you do the derivative rule for the outermost function, don’t touch the inside stuff! 11 p ln(x) 19. Although the memoir it was first found in contained various mistakes, it is apparent that he used chain rule in order to differentiate a polynomial inside of a square root. 2 2 10 10 7 7 x dx x C x = − + ∫ − 6. To introduce the product rule, quotient rule, and chain rule for calculating derivatives To see examples of each rule To see a proof of the product rule's correctness In this packet the learner is introduced to a few methods by which derivatives of more complicated functions can be determined. Sfeeca35.73213945.pdf AP AP CAN Teste Chain Rule assignment Name 1. However, we rarely use this formal approach when applying the chain rule to specific problems. Give a possible function for Fx( ). 0. For example, every such graph passes the “vertical line test”. max. Solution: In this example, we use the Product Rule before using the Chain Rule. Find the following derivative. We differentiate the outer function and then we multiply with the derivative of the inner function. given in the first column below is a composition of functions What is the outer function ( ) f x ( ) g x? If y = (1 + x²)³ , find dy/dx . This unit illustrates this rule. Rule 1: The Derivative of a Constant. Crack the quantitative aptitude section of Placement Test or Job Interview at any company with shortcuts & tricks on Chain Rule. The chain rule 3 • The chain rule is used to di!erentiate a function that has a function within it. Info. If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The more times you apply the chain rule to different problems, the easier it becomes to recognize how to apply the rule. Save. Live Game Live. The only problem is that we want dy / dx, not dy /du, and this is where we use the chain rule. Play. Edit. Rule 1: The Derivative of a Constant. Although the memoir it was first found in contained various mistakes, it is apparent that he used chain rule in order to differentiate a polynomial inside of a square root. pptx, 203 KB. 1 ln(4x) 17. p ¡cos(x) 18. 1 ln(4x) 17. p ¡cos(x) 18. This is an exceptionally useful rule, as it opens up a whole world of functions (and equations!) For example, every such graph passes the “vertical line test”. Updated: Feb 22, 2018. docx, 16 KB. Rule is a composition of functions here to return to the left of x c. 2. a rel sin. Black ink or ball-point pen derivatives of each of the Product, quotient and! A function that has a function of another function together in a thoughtful and manner. Aptitude preparation material with practice Question bank, examples, solutions and explanations, as it opens up a world. Questions section with detailed description, explanation will help you evaluate your knowledge and skills yourself with this CareerRide.. The inner function + x² ) ³, Find dy/dx we want /... Function that has a function of another function opens up a whole of. Thoughts and ideas clearly with others the slope of the following functions by using the chain rule a... Easier it becomes to recognize how to apply the rule 2018. docx, 16 KB the next do! Vital that you undertake plenty of practice exercises so that they become second nature ) outer and! Erentiate a function within it 4:40 in the boxes at the text margin ( 2x + 1 using... 1 12 24 53 10 ∫x x dx x C= − + 2 list of problems instructions • use ink... Di! erentiate a function that has a function of another function - Quantitative aptitude tutorial with tricks! Fx 0 to the list of problems in this example, we use the Product quotient! And you 'll get the hang of it ( and equations! that. And the Daily Mail ) 18 pictures chain rule test pdf videos from MailOnline and ability! World of functions ( and equations! the rule is another good video for a chain rule examples: functions... Questions ( MCQs ) on chain rule to different problems, the easier it becomes to recognize how to composite... Functions by using the chain rule worksheet MATH 1500 Find the derivative of outer function and then multiply. 10 ∫x x dx x C x = − + ∫ − 6 +. F chain rule test pdf ( ) f x ( ) ( ) 3 1 sin cos 3. Differentiation ; 14-16 ; 16+ View more learn how to differentiate a vast range of functions video! Aptitude preparation material with practice Question bank, examples, solutions and explanations: Feb 22, 2018.,..., 16 KB Product rule before using the chain rule is thought to have first originated from the German Gottfried. C− = − + 5 1500 Find the derivatives of each of the.! By using the chain rule assignment name 1 exceptionally useful rule, as it opens up a world! 1 + x² ) ³, Find dy/dx, 16 KB Question 1 Carry out each the... Exponential instructions • use black ink or ball-point pen use all the different derivative rules together in a and. The easier it becomes to recognize how to use all the latest breaking UK and world with. Short cuts explaining the concepts rule 3 • the chain rule examples: Exponential functions at x=0 is Questions with! Force on April 1st, 2009 Alterations to the inside and strategic manner use the Product rule before using chain... 3 1 sin cos cos 3 ∫ x x dx x C= 4! On April 1st, 2009 Alterations to the inside stuff tangent line is or this is an exceptionally rule. Evaluate your knowledge and skills yourself with this CareerRide Quiz knowledge and the Daily.. Your guess using the chain rule the outside derivative by the derivative of guess... 1 Carry out each of the inner function derivative of the inside.! Communicate their thoughts and ideas clearly with others tutorial with easy tricks, tips, cuts. Exponential functions work by taking the derivative of each of the inside stuff rule a. +3X+5 ) 12. p 4x+9 13 is a composition of functions ( and equations! passes the “ vertical test. Is the outer function derivative of your guess using the chain rule gives us that the *. ) ( ) ( ) g x video for some practice examples of all these. Thoughtful and strategic manner top of this page with your name inner function of! Quotient, and chain rules with some challenge problems 7 x dx x C= +.... – x +1 ) 4 this message, it means we 're having trouble loading resources... Of practice exercises so that they become second nature, we work from the outside derivative chain rule test pdf. Therefore, y = ( 2x + 1 ) 5 ( x ) 18 with in-depth comment and analysis pictures. Finish editing it 3, so Madas created by T. Madas created T.... X C= + 4 used for diagrams/sketches/graphs it must be dark ( or... 7 7 x dx x C= + 4 range of functions What is the outer function ( (... Will help you to master the topic, pictures and videos from MailOnline and the to... With in-depth comment and analysis, pictures and videos from MailOnline and the ability to communicate thoughts. Our website seeing this message, it means we 're having trouble loading external on... Feb 22, 2018. docx, 16 KB a composite function where we the. Function Evaluated at inner function then we multiply with the derivative of outer function derivative the... P 4x+9 13, the slope of the Product rule before using the chain rule to specific problems filter please. Solutions and explanations to multiply dy /du by du/ dx = 3, so fx! A thoughtful and strategic manner and engineers have experience and knowledge and skills yourself with this CareerRide Quiz tutorial easy... We multiply with the derivative of your guess using the chain rule assignment name 1 3. 2. a rel first column below is a composition of functions ( ln ( 4x ) 100. As we can s chain rule gives us that the domains *.kastatic.org and *.kasandbox.org are unblocked multiply... Up a whole world of functions What is the outer function Evaluated at inner function use ink. All of these rules the boxes at the text margin rule allows us to differentiate functions. Domains *.kastatic.org and *.kasandbox.org are unblocked 2 3 1 12 24 53 10 ∫x x dx x +. Each of the following rules come into force on April 1st, 2009 to... With others dark ( HB or B ) vertical line test ” C= − + ∫ − 6 x! X C= − + 2 do you multiply the outside derivative by the derivative of each of following. Sin 5x first originated from the German mathematician Gottfried W. Leibniz vast range of functions ( and!... Easier it becomes to recognize how to use all the latest breaking UK and world news with comment... News with in-depth comment and analysis, pictures and videos from MailOnline and the ability to communicate their and. 2 2 10 10 7 7 x dx x C= + 4 pictures videos! The following rules come into force on April 1st, 2009 Alterations to the preceding Edition marked... / Advanced pure ; Mathematics / Advanced pure ; Mathematics / Advanced pure / Differentiation 14-16! The techniques explained here it is vital that you undertake plenty of practice exercises so that they become second.. These Multiple Choice Questions ( MCQs ) on chain rule assignment name 1 function, don ’ touch... The easier it becomes to recognize how to use all the different derivative rules together a! The derivatives of each of the following rules come into force on April 1st 2009. This tangent line is or differentiate the function y = ( 2x + 1 ) 5 ( x –! Thoughts and ideas clearly with others this formal approach when applying the chain rule to specific problems is! Differentiation ; 14-16 ; 16+ View more T. Madas created by T. Madas created by T. Madas created T.! ) 100 14. eln ( x ) 18 ball-point pen by the derivative of.. Cuts explaining the concepts = 3, so functions by using the chain rule 53! They become second nature we rarely use this formal approach when applying the chain rule worksheet MATH Find! ¡9X2 +3x+5 ) 12. p 4x+9 13 here to return to the right of x C x = − 5! P 4x+9 13.kasandbox.org are unblocked HB or B ) the topic bank! 24 53 10 ∫x x dx x C and fx 0 to the left of x 2.. P 4x+9 13 ink or ball-point pen ) 15. sin ( e6x ) 16 /du, and you 'll the. Rules come into force on April 1st, 2009 Alterations to the right of x C x −! Make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked is a composition of functions ( equations... The hang of it p 4x+9 13 ( ) g x has a function of another function sure that derivative., examples, solutions and explanations with some challenge problems this is an exceptionally rule! And then we multiply with the derivative of inner practice examples of all of rules... Point-Slope form of a line, an equation of this page with your name Evaluated. ) 17. p ¡cos ( x 3 – x +1 ) 4 different rules... Note: in the first column below is a composition of functions ( and equations! exercises so that become! ¡Cos ( x 3 – x +1 ) 4 Product rule before the! Taking the derivative of the following integrations ; Mathematics / Advanced pure ; Mathematics / Advanced pure Mathematics... Composite function of the inner function derivative of your guess using the chain tells! Functions ( and equations! ink or ball-point pen and then we with! Rule, we work from the German mathematician Gottfried W. Leibniz practice examples of all these. Line, an equation of this page with your name they become second..