### Subsection2.8.8Exercises

###### 1

Graph sine and cosine on the same axes, from $x=-2\pi$ to $x=2\pi\text{.}$ Mark the points where $\sin x$ has a horizontal tangent. What do these points correspond to, on the graph of cosine?

###### 2

Graph sine and cosine on the same axes, from $x=-2\pi$ to $x=2\pi\text{.}$ Mark the points where $\sin x$ has a tangent line of maximum (positive) slope. What do these points correspond to, on the graph of cosine?

###### 3

Differentiate $f(x)=\sin x + \cos x +\tan x\text{.}$

###### 4

For which values of $x$ does the function $f(x)=\sin x + \cos x$ have a horizontal tangent?

###### 5

Differentiate $f(x)=\sin^2 x + \cos^2 x\text{.}$

###### 6

Differentiate $f(x)=2\sin x \cos x\text{.}$

###### 7

Differentiate $f(x)=e^x\cot x\text{.}$

###### 8

Differentiate $f(x) = \dfrac{2\sin x + 3 \tan x}{\cos x + \tan x}$

###### 9

Differentiate $f(x) = \dfrac{5\sec x+1}{e^x}\text{.}$

###### 10

Differentiate $f(x)=(e^x+\cot x)(5x^6-\csc x)\text{.}$

###### 11

Differentiate $f(\theta)=\sin\left(\frac{\pi}{2}-\theta \right)\text{.}$

###### 12

Differentiate $f(x)=\sin(-x)+\cos(-x)\text{.}$

###### 13

Differentiate $s(\theta)=\dfrac{\cos \theta + \sin \theta}{\cos \theta - \sin\theta}\text{.}$

###### 14(✳)

Find the values of the constants $a$ and $b$ for which

\begin{equation*} f(x) = \left\{ \begin{array}{cc} \cos(x) & x\le 0\\ ax + b & x \gt 0\end{array} \right. \end{equation*}

is differentiable everywhere.

###### 15(✳)

Find the equation of the line tangent to the graph of $y=\cos(x)+2x$ at $x=\dfrac{\pi}{2}\text{.}$

###### 16(✳)

Evaluate $\displaystyle \lim_{x\to 2015}\left( \dfrac{\cos(x)-\cos(2015)}{x-2015}\right).$

###### 17(✳)

Evaluate $\displaystyle \lim_{x\to \pi/3}\left( \dfrac{\cos(x)-1/2}{x-\pi/3}\right).$

###### 18(✳)

Evaluate $\displaystyle \lim_{x\to \pi}\left(\dfrac{\sin(x)}{x-\pi}\right).$

###### 19

Show how you can use the quotient rule to find the derivative of tangent, if you already know the derivatives of sine and cosine.

###### 20(✳)

The derivative of the function

\begin{equation*} f(x)=\left\{\begin{array}{ll} ax+b& \mbox{for }x \lt 0\\ \frac{6\cos x}{2+\sin x+\cos x}& \mbox{for }x\ge 0 \end{array}\right. \end{equation*}

exists for all $x\text{.}$ Determine the values of the constants $a$ and $b\text{.}$

###### 21(✳)

For which values of $x$ does the derivative of $f(x) = \tan x$ exist?

###### 22(✳)

For what values of $x$ does the derivative of $\dfrac{10\sin(x)}{x^2+x-6}$ exist? Explain your answer.

###### 23(✳)

For what values of $x$ does the derivative of $\dfrac{x^2+6x+5}{\sin(x)}$ exist? Explain your answer.

###### 24(✳)

Find the equation of the line tangent to the graph of $y=\tan(x)$ at $x=\dfrac{\pi}{4}\text{.}$

###### 25(✳)

Find the equation of the line tangent to the graph of $y=\sin(x)+\cos(x)+e^x$ at $x=0\text{.}$

###### 26

For which values of $x$ does the function $f(x)=e^x\sin x$ have a horizontal tangent line?

###### 27

Let

\begin{equation*} f(x)=\left\{\begin{array}{ccc} \frac{\sin x}{x}&,&x \neq 0\\ 1&,&x=0 \end{array}\right. \end{equation*}

Find $f'(0)\text{,}$ or show that it does not exist.

###### 28(✳)

Differentiate the function

\begin{equation*} h(x) = \sin(|x|) \end{equation*}

and give the domain where the derivative exists.

###### 29(✳)

For the function

\begin{equation*} f(x) =\left\{\begin{array}{ll} 0 & x\le 0\\ \frac{\sin(x)}{\sqrt{x}} & x \gt 0\end{array}\right. \end{equation*}

which of the following statements is correct?

1. $f$ is undefined at $x = 0\text{.}$
2. $f$ is neither continuous nor differentiable at $x = 0\text{.}$
3. $f$ is continuous but not differentiable at $x = 0\text{.}$
4. $f$ is differentiable but not continuous at $x = 0\text{.}$
5. $f$ is both continuous and differentiable at $x = 0\text{.}$
###### 30(✳)

Evaluate $\lim\limits_{x\rightarrow 0} \dfrac{\sin x^{27}+2x^5 e^{x^{99}}}{\sin^5 x}\text{.}$