Purpose The purpose of this task is to explore the relationship between various exponential functions and their tangents. Students will gain knowledge in developing, testing, and proving valid conjectures. Assessment Type: Mathematical Investigation Task Weighting 20% Task Summary Consider the graphs of the exponential functions f(x)=e^x and g(x)=e^(-x) and their tangents drawn at the points where x=1, as shown below. The two tangents are intersecting the x and y-axes, and they are also intersecting each other at the point labelled as A. Part A – Identifying the relationship between the slopes of the two tangents Use an algebraic approach to find the exact equation of the two tangents to the graphs of f(x)=e^x and g(x)=e^(-x) , at x=1. Following a similar approach or technology, determine the exact equation of the two tangents at x=a for a= 2, 3, 4. Using results obtained, complete the table below relating the slopes of those two tangents. Value of a Equation of the tangent to Slope of the tangent to Product of the two slopes f(x)=e^x g(x)=e^(-x) f(x)=e^x g(x)=e^(-x) 1 2 3 4 Describe any patterns observed in the table. Outline a conjecture for the equation of the tangent to these functions, and another conjecture for the product of the two slopes. Prove your conjectures. Part B – Investigating the relationship between the y-intercepts of the two tangents Determine the y-intercepts of each set of equations of tangent identified in Part A. Summarise your results in a table to investigate the products of the y-intercepts for each set of equations. Formulate a conjecture and prove it. Part C – Exploring further relationships between exponential functions and their tangents Following a similar process outlined in Part A and B above, explore the effect of transformation on f(x)=e^x and g(x)=e^(-x) may have on the conjecture found previously. Alternatively, you may like to consider various conjectures on exponential functions such as y=b^x and y=b^(-x) or transformation of these functions. Assessment Guide In constructing your investigation, you need to use the following structure: Introduction Define the problem that has been presented to you (the ‘what’) Describe clearly and concisely how this task is applicable to your learning and to the real world (the ‘why’- the significance of the task) List all methods you are going to demonstrate to solve this problem, including any use of technology (the ‘how’) Mathematical Calculations & Analysis Follow all the instructions provided with your problem. Provide answers to all questions. Use headings where appropriate. Provide relevant diagrams or graphs that could help illustrate any concepts you are explaining. Explain each step in your mathematical reasoning with notes or short sentences. Justify all choices you make when answering questions. Clearly outline the steps in developing, testing, and proving your conjectures. Discuss the reasonableness and limitations of the mathematical results in your investigation. Conclusion Summarise your findings by: Outlining any observations or findings you have made. Clearly describe and illustrate your results. Suggest any alterations you could make to improve the accuracy of your conjectures. Appendices Includes any additional information you would like to include that strengthens your justification or documents process that is not directly essential in your analysis section. You are expected to provide a bibliography for any resources you have used outside of the Mathematical Methods course during this task. Submission Requirements Modified Format (including text type) Written report ? Word Count / Length Maximum 15 A4 pages Minimum font size 10 Deadlines Date Submission Requirements Checkpoint #1 6/5/22 Part A + Part B fully completed. Due TBC Assessment Design Criteria What you will be assessed on Task Specific Clarification To meet the level descriptors make sure you: Concepts and Techniques Knowledge and understanding of concepts and relationships. Selection and application of mathematical techniques and algorithms to find solutions to problems in a variety of contexts. Application of mathematical models. Use of electronic technology to find solutions to mathematical problems. Complete, concise, and accurate solutions to the questions given in this investigation. Appropriate selection of techniques to investigate, formulate and develop conjectures. Efficient use of technology (Efofex or other graphing packages, and graphics calculators) to find equations of tangent to various exponential functions. Reasoning and Communication Interpretation of mathematical results. Drawing conclusions from mathematical results, with an understanding of their reasonableness and limitations. Use of appropriate mathematical notation, representations, and terminology. Communication of mathematical ideas and reasoning to develop logical arguments. Development and testing of valid conjectures with proof. Clear, efficient, and precise interpretation of the mathematical results. All choices of functions used clearly justified. In-depth discussion of any limitations to the conjecture developed or techniques used to determine the slope of the tangent and the reasonableness of the results. Report constructed in the appropriate format. All graphs clearly labelled. Correct notation proficiently used for all calculations. Mathematical arguments clear and easy to follow. Steps in developing, testing, and proving your conjectures, must be clearly outlined. Concepts and Techniques Reasoning and Communication A Comprehensive knowledge and understanding of concepts and relationships. Highly effective selection and application of mathematical techniques and algorithms to find efficient and accurate solutions to routine and complex problems in a variety of contexts. Successful development and application of mathematical models to find concise and accurate solutions. Appropriate and effective use of electronic technology to find accurate solutions to routine and complex problems. Comprehensive interpretation of mathematical results in the context of the problem. Drawing logical conclusions from mathematical results, with a comprehensive understanding of their reasonableness and limitations. Proficient and accurate use of appropriate mathematical notation, representations, and terminology. Highly effective communication of mathematical ideas and reasoning to develop logical and concise arguments. Effective development and testing of valid conjectures, with proof. B Some depth of knowledge and understanding of concepts and relationships. Mostly effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine and some complex problems in a variety of contexts. Some development and successful application of mathematical models to find mostly accurate solutions. Mostly appropriate and effective use of electronic technology to find mostly accurate solutions to routine and some complex problems. Mostly appropriate interpretation of mathematical results in the context of the problem. Drawing mostly logical conclusions from mathematical results, with some depth of understanding of their reasonableness and limitations. Mostly accurate use of appropriate mathematical notation, representations, and terminology. Mostly effective communication of mathematical ideas and reasoning to develop mostly logical arguments. Mostly effective development and testing of valid conjectures, with substantial attempt at proof. C Generally competent knowledge and understanding of concepts and relationships. Generally effective selection and application of mathematical techniques and algorithms to find mostly accurate solutions to routine problems in a variety of contexts. Successful application of mathematical models to find generally accurate solutions. Generally appropriate and effective use of electronic technology to find mostly accurate solutions to routine problems. Generally appropriate interpretation of mathematical results in the context of the problem. Drawing some logical conclusions from mathematical results, with some understanding of their reasonableness and limitations. Generally appropriate use of mathematical notation, representations, and terminology, with reasonable accuracy. Generally effective communication of mathematical ideas and reasoning to develop some logical arguments. Development and testing of generally valid conjectures, with some attempt at proof. D Basic knowledge and some understanding of concepts and relationships. Some selection and application of mathematical techniques and algorithms to find some accurate solutions to routine problems in some contexts. Some application of mathematical models to find some accurate or partially accurate solutions. Some appropriate use of electronic technology to find some accurate solutions to routine problems. Some interpretation of mathematical results. Drawing some conclusions from mathematical results, with some awareness of their reasonableness or limitations. Some appropriate use of mathematical notation, representations, and terminology, with some accuracy. Some communication of mathematical ideas, with attempted reasoning and/or arguments. Attempted development or testing of a reasonable conjecture. E Limited knowledge or understanding of concepts and relationships. Attempted selection and limited application of mathematical techniques or algorithms, with limited accuracy in solving routine problems. Attempted application of mathematical models, with limited accuracy. Attempted use of electronic technology, with limited accuracy in solving routine problems. Limited interpretation of mathematical results. Limited understanding of the meaning of mathematical results, and their reasonableness or limitations. Limited use of appropriate mathematical notation, representations, or terminology, with limited accuracy. Attempted communication of mathematical ideas, with limited reasoning. Limited attempt to develop or test a conjecture.