Consider the function given by

$f(x)=\{\begin{array}{ccc}sin(x)& if& x>\\ x-& if& x\end{array}$

$($a$)$ Identify the real numbers at which $f(x)$ is discontinuous. Hint: You should justify why $f(x)$ is discontinuous at certain values of $x$ and why $f(x)$ is continuous everywhere else! Remember, we can think of discontinuity as a hole or gap.

$($b$)$ Identify the points of the horizontal and vertical asymptotes of $f(x).$ Hint: Limits? Remember, an asymptote is a line that a curve approaches but never touches.

Consider the function given by

$f(x)=\{\begin{array}{ccc}sin(x)& if& x>\\ x-& if& x\end{array}$

$($a$)$ Identify the real numbers at which $f(x)$ is discontinuous. Hint: You should justify why $f(x)$ is discontinuous at certain values of $x$ and why $f(x)$ is continuous everywhere else! Remember, we can think of discontinuity as a hole or gap.

$($b$)$ Identify the points of the horizontal and vertical asymptotes of $f(x).$ Hint: Limits? Remember, an asymptote is a line that a curve approaches but never touches.

Mean Value Theorem

If a function $f$ is continuous on the closed interval $a,b$ and differentiable on the open interval $(a,b),$ then there exists at least one point cin$(a,b)$ such that:

$f^{\text{'}}(c)=\frac{f(b)-f(a)}{b-a}$

$($c$)$ What does the Mean Value Theorem say about $f(x)$ on the interval $2,3?$ Hint: Given the interval we are working with, which function are we going to use?

$f(x)=sin(x)$

$f^{\text{'}}(x)=cos(x)$

Consider the function given by

$f(x)=\{\begin{array}{ccc}sin(x)& if& x>\\ x-& if& x\end{array}$

$($a$)$ Identify the real numbers at which $f(x)$ is discontinuous. Hint: You should justify why $f(x)$ is discontinuous at certain values of $x$ and why $f(x)$ is continuous everywhere else! Remember, we can think of discontinuity as a hole or gap.

$($b$)$ Identify the points of the horizontal and vertical asymptotes of $f(x).$

Hint: Limits? Remember, an asymptote is a line that a curve approaches but never touches.

Mean Value Theorem

If a function $f$ is continuous on the closed interval $a,b$ and differentiable on the open interval $(a,b),$ then there exists at least one point cin$(a,b)$ such that:

$f^{\text{'}}(c)=\frac{f(b)-f(a)}{b-a}$

$($c$)$ What does the Mean Value Theorem say about $f(x)$ on the interval $2,3?$

Hint: Given the interval we are working with, which function are we going to use?