Course:MATH110/Archive/2010-2011/003/Lectures/Lecture 35

First clicker question

Consider the function:

$f(x)={\begin{cases}\quad x^{2}&{\text{if }}x{\text{ is rational,}}\\\,\,-x^{2}&{\text{if }}x{\text{ is irrational,}}\\\quad {\text{undefined}}&{\text{if }}x=0.\end{cases}}$

Then,

A) There is no $a$ for which $\lim _{x\rightarrow a}f(x)$ exists.

B) There may be some $a$ for which $\lim _{x\rightarrow a}f(x)$ exists, but it is impossible to say without more information.

C) $\lim _{x\rightarrow a}f(x)$ exists only when $a=0$.

D) $\lim _{x\rightarrow a}f(x)$ exists for infinitely many $a$.

Answer

The correct answer is C and a good sketch of the graph of the function should provide enough information for this.

Second clicker question

A drippy faucet adds one milliliter to the volume of water in a tub at precisely one second intervals. Let $f$ be the function that represents the volume of water in the tub at time $t$.

A) $f$ is a continuous function at every time $t$.

B) $f$ is continuous for all $t$ other than the precise instants when the water drips into the tub.

C) $f$ is not continuous at any time $t$.

D) not enough information to know where $f$ is continuous.

Answer

The correct answer is B and a good sketch of the graph of the function should provide enough information for this.

Third clicker question

Your mother says “If you eat your dinner, you can have dessert.” You know this means, “If you don’t eat your dinner, you cannot have dessert.” Your calculus teacher says, “If $f$ is differentiable at $x$, $f$ is continuous at $x$.” You know this means:

A) If $f$ is not continuous at $x$, $f$ is not differentiable at $x$.

B) If is not differentiable at $x$, $f$ is not continuous at $x$.

C) Knowing $f$ is not continuous at $x$, does not give us enough information to deduce anything about whether the derivative of $f$ exists at $x$.

Answer

The correct answer is A. You can simply think that the sentence: "if something is a candy, then it is food" implies that "if something isn't food, it can't be candy" and not "if something isn't candy, then it isn't food". You should also be able to produce sketches of graphs of functions which are continuous at a point, but not differentiable there. You should convince yourself, that the opposite is impossible to perform.