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?
Subsection 2.8.8 Exercises
Exercises — Stage 1
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?
Exercises — Stage 2
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 \(x=\dfrac{\pi}{2}\text{.}\)
Exercises — Stage 3
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?
- \(f\) is undefined at \(x = 0\text{.}\)
- \(f\) is neither continuous nor differentiable at \(x = 0\text{.}\)
- \(f\) is continuous but not differentiable at \(x = 0\text{.}\)
- \(f\) is differentiable but not continuous at \(x = 0\text{.}\)
- \(f\) is both continuous and differentiable at \(x = 0\text{.}\)
30 (✳)
Evaluate \(\lim\limits_{x\rightarrow 0} \dfrac{\sin x^{27}+2x^5 e^{x^{99}}}{\sin^5 x}\text{.}\)