This student did well to pick up the inflection point on the function near x=2. Many students missed these inflection points (they're rather subtle), or mistakenly called the "corners" inflection point.
Also note that s/he wanted to call the absolute min a local min, then thought better of it!
I told you that I'd ask for a definition of rational and irrational numbers - did you not believe me? Some of you confused rational numbers with rational functions.
The theorem makes specific reference to rational numbers because we understand how to raise a number to a rational power. It does indeed make sense to a raise a number to an irrational power, but we're not ready for "logarithms" yet!
Way to factor out an x2 from top and bottom! Too many of you attempted to use a "conjugate" (whatever that is in this case!).
I wanted to see the issues of
- Domain and intercepts
- Symmetry/Periodicity
- Asymptotes (slant!)
- Intervals of increase, decrease, and local maxima and minima
- Concavity and points of inflection
As for the "context", this student forgot to add numbers along the x-axis, to give me some idea of where the roots are located, etc.
Also, the inflection point is misidentified as x=1. Otherwise, good job!
I was surprised to see the number of plots that weren't right - as I told you at the beginning, those students who know how to use their calculators do better than those who don't....
This student does a nice job of laying out the issues of Newton's method. One thing to improve: it would be nice if the cubic graph looked a little more cubic! Sorry that the values of the function made the two iterates and their tangent lines hard to draw - my bad.
The only thing this student forgot in problem 6 was the conclusion: the best isosceles triange is actually an equilateral triangle! It would have also been nice to see the value of b as well.
Nice use of the Mean Value Theorem. Guilty! It would have been nice had this student also mentioned the requirements for the mean value theorem - e.g. differentiable on the interval. But I was just glad to see the MVT.
Nice job! Keep up the good work.