# Topic 8 Calculus of Inverse Functions

## 8.1 Inverse Functions

Maple package `Student[Calculus1]`

provides the following command

`InversePlot(f(x), x = a..b);`

which graphs the original function \(f(x)\) and the inverse function \(f^{-1}(x)\) together over the interval \([a, b]\).

You will see clearly that the graphs of a function and its inverse are symmetric with respect to the line \(y=x\).

**Example 8.1 **

1. Graph the function \(f(x)=x^3-2\), its inverse function, and the line \(y=x\) over the interval \([-2,2]\).

- Find the inverse function.

*Solution. *
One way to plot the function and its inverse together is to use the following command which is supported by the package `Student[Calculus1]`

.

` InversePlot(x^3-3, x = -2..2)`

Here is the output in Maple

Another way to plot the function \(f\) and its inverse \(g\) together uses the `plot`

function.

` plot([f(x), g(x), x], x = -2 .. 4, y = -5 .. 5, color = [red, black, blue])`

To find the inverse function, we replace \(f(x)\) by \(y\), then switch \(x\) and \(y\), and solve for \(y\). In Maple, you may use the command `solve(equation/inequality, variable)`

to solve an equation or an inequality (even system of equations/inequalities).

In this example, we may find the inverse function by type in the following command. Note I have switch \(x\) and \(y\).

`solve(x=y^3, y)`

To find the derivative of the inverse function of a function \(f\) at a given point \(x=a\), we may apply the formula \[(f^{-1})'(a)=\dfrac{1}{f'(f^{-1}(a))}.\] In Maple, we may use the following commands to calculate the value of the derivative function.

- Calculate the derivative of the function \(f\).

```
diff(f(x), x)
```

- Find \(f^{-1}(a)\) which is the solution of the equaiton \(f(x)=a\).

```
solve(f(x)=a, x)
```

- Plug in the formula to evaluate.

```
eval(subs(x=f^{-1}(a), 1/f'(x)))
```

**Example 8.2 **
Find \((f^{-1})'(0)\), where \(f(x)=\cos(x)\) and \(0\leq x\leq \pi\).

*Solution. *
Find the derivative of \(f\)

`diff(cos(x), x)`

Find the value of \(f^{-1}(0)\)

`solve(cos(x)=0, x)`

Apply the formula

`eval(subs(x=Pi/2, -1/sin(x)))`

Using Maple, we find \((f^{-1})'(0)=-1\).

**Exercise 8.1 **
1. Graph the function \(f(x)=3+2\sin x\), its inverse function, and the line \(y=x\) over the interval \([-2,2]\).

- Find the value \((f^{-1})'(5)\).

## 8.2 Logarithmic and Exponential Functions

### 8.2.1 Basic properties and graphs

The natural logarithmic function \(y=\ln(x)\) is defined by \(\ln(x)=\int_1^x\frac{1}{t}\mathrm{d} t\).

The natural exponential function \(y=e^x\) is defined as the inverse function of \(y=\ln(x)\).

From the definition, we have very important identities \[ \ln(e^x)=x\qquad \text{and}\qquad e^{\ln x}=x. \]

Using those two identities, we may define general exponential functions and general logarithmic function, and deduce the Law of Logarithms and Law of Exponents.

For any positive number \(b\ne 1\), we have \(b^x=(e^{\ln b})^x=e^{x\ln b}\).

For any positive number \(b\ne 1\), we define \(y=\log_bx\) to be the inverse function of \(y=b^x\)

By solving \(x=b^y\) for \(y\), we find that \(\log_bx=\dfrac{\ln x}{\ln b}\). This identity is called the change of base property.

How do graphs of logarithmic functions and exponential functions look like?

**Example 8.3 **
Graph the following functions together.
\[
y=\ln x, \qquad y=e^x, \qquad y=2^x, \qquad y=\log_2x, y=x.
\]

*Solution. *
In Maple, the logarithm \(\log_bx\) is given by `log[b](x)`

. When \(b=e\), you simply use `ln(x)`

for \(\ln x\). When \(b=10\), you may also use `log(x)`

or `log10(x)`

for \(\log_{10}x\).

The exponent \(b^x\) is given by `b^x`

in Maple. When \(b=e\), you may also use `exp(x)`

to represent \(e^x\).

To graph the functions together with different colors, we use the following command

`plot([ln(x), exp(x), 2^x, log[2](x), x], x=-5..5, color=[blue, green, purple, yellow, red])`

Here is the output

**Exercise 8.2 **
Graph the following functions together.
\[
y=\log_3x, \qquad y=3^x, \qquad y=(1/3)^x, \qquad y=\log_{1/3}x.
\]

Find the pairs that are symmetric to each other with respect to a certain line.

**Exercise 8.3 **
Graph the following functions together.
\[
y=0.5^x, \qquad y=2^x, \qquad y=5^x.
\]

Describe the monotonicity (increasing/decreasing) of the functions?

Fix an input \(x\). Describe how \(y\)-values change when bases changes from small number to bigger number?

**Exercise 8.4 **
Graph the following functions together.
\[
y=\log_{0.5}x, \qquad y=\log_2x, \qquad y=\log_{5}x.
\]

Describe the monotonicity (increasing/decreasing) of the functions?

Fix an input \(x\). Describe how \(y\)-values change when bases changes from small number to bigger number?

### 8.2.2 Differentiation and integration of logarithmic and exponential functions

In Maple, one way to do differentiation and integration is to use the `Calculus Palette`

on the left side.

The other way is to use the commands `diff(f(x), x)`

, `int(f(x), x)`

, and `int(f(x), x=a..b)`

.

Supported by the `Student[Calculus1]`

package, Maple also provides the tutor commands `DiffTutor()`

and `IntTutor()`

which can show step-by-step solution of differentiation and integration.

Note you may also access tutor commands from the `Start`

page (click the home button in the toolbar and look for Calculus).

**Example 8.4 **
Find \(y'\), where \(y=\ln \left(x^{3}+5x+1\right)\).

*Solution. *
Using `diff`

:

`diff(ln(x^3+5*x+1), x)`

We get \[ y'=\dfrac{3x^{2}+5}{x^{3}+5 x+1}. \]

Type in (assume that `with(Student[Calculus1])`

was run)

`DiffTutor(ln(x^3+5*x+1), x)`

and hit enter you will see

By click `Next Step`

or `All Steps`

you will see detailed solution with rules used.

**Example 8.5 **
Evaluate the integral
\[
\int\dfrac{e^x-1}{e^x+1}\mathrm{d} x.
\]

*Solution. *
Using `int`

:

`int((exp(x)-1)/(exp(x)+1), x)`

We get \[ \int\dfrac{e^x-1}{e^x+1}\mathrm{d} x=2 \ln \left(\mathrm{e}^{x}+1\right)- x+C. \]

Type in (assume that `with(Student[Calculus1])`

was run)

`IntTutor((exp(x)-1)/(exp(x)+1), x)`

and hit enter you will see

By click `Next Step`

or `All Steps`

you will see detailed solution with rules used.

**Exercise 8.5 **
Find the derivative \(\frac{\mathrm{d} y}{\mathrm{d} x}\), where \(y=\ln|\cos x|\)

**Exercise 8.6 **
Find the derivative \(\frac{\mathrm{d} y}{\mathrm{d} x}\), where \(y=x^{\cos x}\)

**Exercise 8.7 **
Evaluate the integral
\[
\int \frac{\left(e^{4x}+e^{2x}\right)}{e^{3x}} d x
\]

**Exercise 8.8 **
Evaluate the integral
\[
\int 2^{3x} d x
\]

## 8.3 Solve differential equations

In Maple, you may solve the equation \(y'(x)=k y(x) + c\) (which is called an ODE) using the command `dsolve({ics, eq})`

, where `ics`

stands for initial condition \(y(0)=c\) and `eq`

stands for the differential equation. Without the `ics`

, `dsolve`

will provide a general solution.

**Example 8.6 **Find the function \(f(x)\) which satisfies the differential equation \(f'(x)=k f(x)\) with \(f(0)=5\) and \(f(2)=3\).

*Solution. *
Use the following command

`dsolve({f(0)=5, f'(x)=k f(x)})`

we get \(f(x)=5e^{kx}\).

To find \(k\), we solve the equation \(3=5e^{2k}\) by

`solve(3=5*e^(2*k), k)`

which shows that \(k=\frac{\ln3-\ln5}{2}\approx -0.255\).
Here we use `evalf(%)`

(% represents the previous result) to get the approximation.

So the function \(f\) is given by \[ f(x)=5e^{\frac{x(\ln3-\ln5)}{2}}\approx 5e^{-0.255x}. \]

**Exercise 8.9 **
Find the function \(y\) which satisfies the differential equation \(y'(x)=k y(x)\) with \(y(0)=2\) and \(y(5)=11\).

## 8.4 Inverse Trigonometric Functions

### 8.4.1 Domains and Ranges

To define the inverse function, the original function must be a one-to-one function. For a trigonometric function, we have to restrict the function over a specific domain to ensure that the function is one-to-one. For simplicity, we pick domains near the origin for trigonometric functions. To be more precise, we consider the following trigonometric functions: \[ y=\sin x,\quad -\pi/2\leq x\leq \pi/2 ~~\text{and}~ -1\leq y\leq 1; \]

\[ y=\cos x,\quad 0\leq x\leq \pi ~~\text{and}~ -1\leq y\leq 1; \]

\[ y=\tan x,\quad -\pi/2< x< \pi/2 ~~\text{and}~ -\infty<y<\infty. \]

Their inverse functions are

\[ y=\arcsin x,\quad -\pi/2\leq y\leq \pi/2 ~~\text{and}~ -1\leq x\leq 1; \]

\[ y=\arccos x,\quad 0\leq y\leq \pi ~~\text{and}~ -1\leq x\leq 1; \]

\[ y=\arctan x,\quad -\pi/2< y< \pi/2 ~~\text{and}~ -\infty< x<\infty. \]

To see the graphs of the functions, we may use the `plot(f(x), x = a..b, opts)`

command. Or, to graph \(f\), \(f^{-1}\) and \(y=x\) together, we may also use the command `InversePlot(f(x), x = a..b, opts)`

supported by the package `Student[Calculus1]`

.

**Example 8.7 **
Graph the following functions together.
\[
y=\tan x, \qquad y=\arctan x, \qquad y=x.
\]

*Solution. *
#load the package ``Student[Calculus1]".
with(Student[Calculus1])
#plot the functions
InversePlot(tan(x),x=-Pi/2..Pi/2)

Here is the output

**Exercise 8.10 **
Graph the following functions together over an appropriate domain.
\[
y=\cot x, \qquad y=\mathrm{arccot} x, \qquad y=x.
\]

What’s the domain and range of \(y=\mathrm{arccot} x\)?

### 8.4.2 Differentiation and integration of inverse trigonometric functions

In the section {#Differentiation and integration of logarithmic and exponential functions}, we learned to how use Maple to learn differentiation and integration.

Now let find derivatives and integrals of some inverse trigonometric functions.

**Example 8.8 **
Find \(y'\), where \(y=\mathrm{arccot} x\).

*Solution. *
Surely, we may use `diff`

or `DiffTutor`

to find the derivative.

Here let’s me introduce to you another command `implicitdiff`

.

We’ve learned that (see {#Inverse Functions}) to find the inverse function, we switch \(x\) and \(y\) and then solve for \(y\). When finding the derivative, we don’t have to solve for \(y\) instead, we want \(y'\) which is implicitly defined by an equation. In this case, we have \(x=\mathrm{arccot}y\).

Enter the following commands in Maple, you will find \(y'=-\frac{1}{x^2+1}\).

```
implicitdiff(x=cot(y), y, x)
subs(cot(y) = x, %)
```

where `%`

is a ditto operator that allows you to refer to a previously computed result in Maple.

**Exercise 8.11 **
Find the derivative of \(y=\mathrm{arcsec} x\)

**Exercise 8.12 **
Find the derivative of \(y=\mathrm{arccsc} x\)

For integrals of inverse trigonometric function, you may need the method of integration by parts.

Use `DiffTutor`

to find antiderivatives of inverse trigonometric functions.

**Exercise 8.13 **
Evaluate the integral
\[
\int \arcsin x \mathrm{d} x
\]

**Exercise 8.14 **
Evaluate the integral
\[
\int \arctan x \mathrm{d} x
\]

**Exercise 8.15 **
Evaluate the integral
\[
\int \mathrm{sec} x \mathrm{d} x
\]

## 8.5 L’Hospital’s Rule

In Maple, supported by the package, `Student[Calculus1]`

, the command `LimitTutor`

can show step-by-step solutions of evaluating limits.

**Example 8.9 **
Evaluate the limit
\[
\lim\limits_{x\to \infty}(1+x)^{1/\ln(x)}
\]

*Solution. *
# load the package Student[Calculus1].
with(Student[Calculus1])
#Find the limit step-by-step using LimitTutor
LimitTutor((1+x)^(1/ln(x)), x = infinity)

**Exercise 8.16 **
Estimate the limit
\[
\lim\limits_{x\to 1}\frac{x^2-2x+1}{x^2-x}
\]
by graphing and verify your estimation.

**Exercise 8.17 **
Evaluate the limit
\[
\lim\limits_{x\to \infty} x-\ln(x).
\]

**Exercise 8.18 **
Evaluate the limit
\[
\lim\limits_{x\to \infty} x\tan(\frac1x).
\]