1
Give the domains of each of the following functions.
\begin{align*}
\mbox{(a) } f(x)\amp=\arcsin(\cos x) \amp
\mbox{(b) } g(x)\amp=\arccsc(\cos x)\\
\mbox{(c) } h(x)\amp=\sin(\arccos x)
\end{align*}
Subsection 2.12.2 Exercises
Exercises — Stage 1
2
A particle starts moving at time \(t=10\text{,}\) and it bobs up and down, so that its height at time \(t \geq 10\) is given by \(\cos t\text{.}\) True or false: the particle has height 1 at time \(t=\arccos(1)\text{.}\)
3
The curve \(y=f(x)\) is shown below, for some function \(f\text{.}\) Restrict \(f\) to the largest possible interval containing \(0\) over which it is one--to--one, and sketch the curve \(y=f^{-1}(x)\text{.}\)
4
Let \(a\) be some constant. Where does the curve \(y=ax+\cos x\) have a horizontal tangent line?
5
Define a function \(f(x)=\arcsin x + \arccsc x\text{.}\) What is the domain of \(f(x)\text{?}\) Where is \(f(x)\) differentiable?
Exercises — Stage 2
6
Differentiate \(f(x)=\arcsin\left(\dfrac{x}{3}\right)\text{.}\) What is the domain of \(f(x)\text{?}\)
7
Differentiate \(f(t)=\dfrac{\arccos t}{t^2-1}\text{.}\) What is the domain of \(f(t)\text{?}\)
8
Differentiate \(f(x)=\arcsec(-x^2-2)\text{.}\) What is the domain of \(f(x)\text{?}\)
9
Differentiate \(f(x)=\dfrac{1}{a}\arctan\left(\dfrac{x}{a}\right)\text{,}\) where \(a\) is a nonzero constant. What is the domain of \(f(x)\text{?}\)
10
Differentiate \(f(x)=x\arcsin x + \sqrt{1-x^2}\text{.}\) What is the domain of \(f(x)\text{?}\)
11
For which values of \(x\) is the tangent line to \(y=\arctan (x^2)\) horizontal?
12
Evaluate \(\ds\diff{}{x}\{\arcsin x + \arccos x\}\text{.}\)
13 (✳)
Find the derivative of \(y=\arcsin \!\big(\frac{1}{x}\big)\text{.}\)
14 (✳)
Find the derivative of \(y=\arctan \big(\frac{1}{x}\big)\text{.}\)
15 (✳)
Calculate and simplify the derivative of \((1+x^2)\arctan x\text{.}\)
16
Show that \(\ds\diff{}{x}\left\{\sin\left(\arctan(x) \right)\right\} = (x^2+1)^{-3/2}\text{.}\)
17
Show that \(\ds\diff{}{x}\left\{\cot\left(\arcsin(x) \right)\right\} = \dfrac{-1}{x^2\sqrt{1-x^2}}\text{.}\)
18 (✳)
Determine all points on the curve \(y=\arcsin x\) where the tangent line is parallel to the line \(y=2x+9\text{.}\)
19
For which values of \(x\) does the function \(f(x)=\arctan(\csc x)\) have a horizontal tangent line?
Exercises — Stage 3
20 (✳)
Let \(f(x) = x + \cos x\text{,}\) and let \(g(y) = f^{-1}(y)\) be the inverse function. Determine \(g'(y)\text{.}\)
21 (✳)
\(f(x) = 2x-\sin(x)\) is one--to--one. Find \(\big(f^{-1}\big)'(\pi-1)\text{.}\)
22 (✳)
\(f(x) = e^x+x\) is one--to--one. Find \(\big(f^{-1}\big)'(e+1)\text{.}\)
23
Differentiate \(f(x)=[\sin x +2]^{\arcsec x}\text{.}\) What is the domain of this function?
24
Suppose you can't remember whether the derivative of arcsine is \(\dfrac{1}{\sqrt{1-x^2}}\) or \(\dfrac{1}{\sqrt{x^2-1}}\text{.}\) Describe how the domain of arcsine suggests that one of these is wrong.
25
Evaluate \(\displaystyle \lim_{x\to 1}\left( (x-1)^{-1}\left(\arctan x - \frac{\pi}{4}\right)\right).\)
26
Suppose \(f(2x+1)=\dfrac{5x-9}{3x+7}\text{.}\) Evaluate \(f^{-1}(7)\text{.}\)
27
Suppose \(f^{-1}(4x-1)=\dfrac{2x+3}{x+1}\text{.}\) Evaluate \(f(0)\text{.}\)
28
Suppose a curve is defined implicitly by
\begin{equation*}
\arcsin(x+2y)=x^2+y^2
\end{equation*}
Solve for \(y'\) in terms of \(x\) and \(y\text{.}\)