Posted: November 10th, 2015

Math Project

WRITING PROJECT Due

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.

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

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?

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