If the function $\,y=f(x)\,$ has an inverse function $\,x=f^{-1}(y)\,,$ then we can find the derivative of $\,f^{-1}(y)\,$ like this:
$$(f^{-1})'(y)=\frac{1}{f'(x)}\,.$$
A
Sine has in the interval $\,\left[-\frac{\pi}2\,,\,\frac{\pi}2\,\right]\,$ an inverse function denoted $\arcsin\,.$
Determine the derivative of $\arcsin\,.$
hint
$\,\arcsin’(y)=\ldots\,$. Your goal is to rewrite the right-hand-side to an expression in terms of $y$ (meaning in terms of $\sin$).
Show that the derivative of the tangent function for every $\,t\,$ in the domain is given by the expression
$$\tan'(t)=\,1+\tan^2(t)\,.$$
hint
Use the product rule and the chain rule.
B
Tangent has, on the open interval $\,]-\frac{\pi}2\,,\,\frac{\pi}2\,[\,$, an inverse function called arctangent denoted $\arctan\,.$ Determine $\,\mathrm{arctan}’(x)\,$ for every $\,x\,$ in the interval.
answer
$$\mathrm{arctan}'(y)=\frac{1}{1+y^2}$$
After computing this result you can switch variable from $y$ to $x$ if you wish.
Exercise 6: Derivatives Directly from the Definition
A
Show that the real function $\,f\,$ given by $\,f(x)=x^2\,$ is differentiable for every $\,x_0 \in \reel$ with the derivative
$$\frac{d}{dx} f(x_0)=2x_0\,.$$
Then repeat with another function of your choice.
hint
Substitute the expression for $\,f\,$ into the expression
Exercise 7: Quadratic Equations with Real Coefficients
A
Solve the equations below both in $\reel$ and in $\mathbb C\,.$
$\,2x^2+9x-5=0$
$\,x^2-4x=0$
$\,x^2-4x+13=0$
answer
1) Within both $\Bbb R$ and $\Bbb C$ the solutions are
$$-5\,\,\,\mathrm{and}\,\,\,\frac 12\,.$$
2) Within both $\Bbb R$ and $\Bbb C$ the solutions are
$$0\,\,\,\mathrm{and}\,\,\,4\,.$$
3) No real solutions. Within $\Bbb C$ the solutions are
$$2-3i\,\,\,\mathrm{and}\,\,\,2+3i\,.$$
B
Solve the equation
$$2(x+1-i)(x+1+i)=0$$
and show that it is of the type: quadratic polynomial equation with real coefficients.
hint
Find the solutions by use of the rule of zero product.
answer
$$x=-1+i\quad\text{and}\quad x=-1-i$$
To investigate the polynomial type and its coefficients, expand the polynomial fully by multiplying the brackets together.
Exercise 8: Quadratic Equations with Complex Coefficients
A
Find the solutions to the equation
$$z^2-(1+5i)z=0\,.$$
hint
First factorize the left-hand-side.
hint
The factorized form becomes
$$z(z-(1+5i))=0.$$
Now use the rule of zero product.
answer
$$0\,\,\,\mathrm{and}\,\,\,1+5i$$
B
Show that the following quadratic equation has a purely imaginary discriminant. Then solve the equation.
$$z^2+(2+2i)z-2i=0\,.$$
hint
Here factorization and then the rule of zero product is not usable since there’s no common factor. You must rely on the quadratic solution formula in its full form. Be careful with the imaginary discriminant.
hint
With an imaginary discriminant you cannot take the square root of it in the solution formula. Instead, solve the binomial equation $w^2=\text{discriminant}$ and substitute a value of $w$ into the solution formula.
Show that the function
\begin{equation}
f_{1}(x)=
\begin{cases}
|x|/x& \text{if}\, x \neq 0 \newline
0& \text{if}\, x = 0
\end{cases}
\end{equation}
is not an epsilon function.
hint
Which requirements must a function fulfill for it to qualify as an epsilon function?
hint
What does the function go towards when moving the variable rightwards, from $-\infty$ towards $0$? What does it go towards, when moving leftwards, from $+\infty$ towards $0$?
B
Show that the function
\begin{equation}
f_{2}(x)=1-\cos(x)
\end{equation}
is an epsilon function.
C
Show that the complex function
\begin{equation}
f_{3}(x)=i\e^{ix}-i
\end{equation}
is an epsilon function.
