True or False. If false give a counter-example.
- If $f(x)$ is continuous at $x=2$ and $f(2)$ is the maximum
y-value of the function then $f(x)$ is differentiable at $x=2$.
- You take a boat trip from New York to London and follow a
smooth, but curvy course. At some point on your journey you are traveling
parallel to the direct straight line course.
- On a trip from Dallas to Austin you go through Waco at 10 p.m. and
Temple at 11 p.m. (50 miles apart). Between 10 and 11, at some point, you
were driving exactly 50 $mph$.
- If $f(x)$ is defined on $ [0,1] $ and continuous and differentiable
on $(0,1)$ then there exists a point $x_0$ in $ [0,1] $ such that
$f'(x_0) = f(1)-f(0).$
Tags: diff-at-a-point, MVT, abstract-function, true-false, proof, using-a-theorem, creating-counterexamples, v--a, t1, c2
This problem works well in class. The problem starts out with a question that gets them to think about the implication between continuity and differentiability. Coming up with counter examples reinforces the concepts they have studied. They begin to create a mental framework about how they understand continuity and differentiability. Parts b and c are qualitative questions that require understanding of the MVT. You see students drawing diagrams and discussing why the MVT leads to the answer. Some students may say that the MVT doesn’t apply for part b because the curvature of the earth is not negligible across the Atlantic. Keep this in mind.
– James Epperson 1998-08-09