WRITING PROJECT 1

Due 11/12/15 1. State the Mean Value Theorem. 2. Mean Value Theorem Example Choose 1 of 2 of the following (A,B), then perform the following steps(a,b,c). A. f(x)=sin^(-1)?(x) on [0,1/2] B. f(x)=x+1/x on [1,3] a. Determine whether the MVT applies to the function on the given interval. b. Find the point(s) that are guaranteed to exist by the MVT. c. Create a graph on a graphing device (such as Desmos.com) and mark the secant line connecting the endpoints. Indicate the coordinates (c,f(c) ), and mark the tangent line, where the function satisfies the conclusion of the\ MVT. 3. Rolle’s Theorem For the function f(x)=x^3-5x^2+6x+2 on [0,3] a. Determine whether the MVT applies to the function on the given interval. b. Find the point(s) that are guaranteed to exist by the MVT. c. Create a graph on a graphing device (such as Desmos.com) and mark the secant line connecting the endpoints. Indicate the coordinates (c,f(c) ), and mark the tangent line, where the function satisfies the conclusion of the\ MVT. d. Why does the slope equal zero? e. For a general function that satisfies the MVT, if f(a)=f(b), verify what will we set f^' (c) equal to every time. 4. CounterExamples a) Draw a picture of a function that satisfies: -Condition (1) of MVT is True -Condition (2) of MVT is False -Conclusion of MVT is False b) Draw a picture of a function that satisfies: -Condition (1) of MVT is True -Condition (2) of MVT is False -Conclusion of MVT is True c) Draw a picture of a function that satisfies: -Condition (1) of MVT is False -Condition (2) of MVT is True -Conclusion of MVT is False d) Draw a picture of a function that satisfies: -Condition (1) of MVT is False -Condition (2) of MVT is True -Conclusion of MVT is True e) Draw a picture of a function that satisfies: -Condition (1) of MVT is False -Condition (2) of MVT is False -Conclusion of MVT is False f) Draw a picture of a function that satisfies: -Condition (1) of MVT is False -Condition (2) of MVT is False -Conclusion of MVT is True 5. Proofs Choose 2 of 3 of the following (A,B), then prove it. A. If f^' (x)=0 for all xin (a,b), then f is constant on (a,b). B. If f^' (x)>0 for all xin (a,b), and f is continuous on [a,b], then f(b)>f(a). C. If f and g are both continuous on [a,b] and differentiable on (a,b), where g(a)?g(b), then there exists c in (a,b) such that (f^' (c))/(g^' (c) )=(f(b)-f(a))/(g(b)-g(a) ). Please work in groups of 2 – 4. Reports must be very neat, well-organized, and stapled. They should be written in complete sentences and typed. You should use graphing software for any graphs, but can hand draw any charts, derivatives or equations or diagrams if needed. All handwriting in PEN. Summary of typing/handwriting Section 1: all typed Section 2: a) typed b) typed or handwritten c) computer generated graph, highlights can be handwritten Section 3: a) typed b) typed or handwritten c) computer generated graph, highlights can be handwritten d) typed e) typed or handwritten Section 4: a-f) label the graphs, computer generated axes, function can be handwritten Section 5: typed or handwritten What you need to turn in (all stapled together!): -A cover page with a title, and the names and signatures of everyone in your group. By signing the report you are affirming that you and the other students listed shared the work on this project. -Your answers RUBRIC Form : 1. Clearly (re)state the problem to be solved (including all the essential details)? 2. Answer the question that was originally asked? 3. Give acknowledgment where it is due? Content: 4. Define all variables, terms and notation used? 5. Clearly label diagrams, tables, graphs or other visual representations of the math? 6. Contain correct mathematics? Presentation: 7. Use correct spelling, grammar and punctuation? 8. Look neat?