4Week 39

A function $f(x)$ is given by the expression $f(x) = 6x^3 - 12x^2 - 54x + 108$ for $x > 3$. Assume that the inverse function $g(x)$ of $f(x)$ exists. Find the derivative for the inverse function, $\frac{\mathrm{d} f^{-1}}{\mathrm{d} x}$ in the point $x = f(4) = 84$.

A function $f(x)$ has the graph shown below. In the point $(x, y) = (5,4)$ the function has a tangent described by the equation $y = \frac{5}{4} x - \frac{9}{4}$. Find the value of the derivative of the inverse function at $x = 4$.

Answer the questions below, given $f(x) = 2x^3$.

1. Calculate the value of $\frac{\mathrm{d} f^{-1}}{\mathrm{d} x}$ for $x = f(2) = 16$ using $f'(x)$.
2. Find an explicit expression for the inverse function $g(x) = f^{-1}(x)$.
3. Find an expression for $g'(x)$ and calculate $g'(16)$.
4. Did you get the same results in question 1 and 3?

Calculate the limits below.

1. $\displaystyle\lim_{x\to \infty} \frac{\ln(e^x + 3)}{2x}$
2. $\displaystyle\lim_{x\to \infty} \frac{\sqrt{x^2+5}}{5}$
3. $\displaystyle\lim_{x\to \infty} \frac{\ln x}{\sqrt{x}}$
4. $\displaystyle\lim_{x\to 2} \frac{x^2 - 4}{x - 2}$
5. $\displaystyle\lim_{x\to \infty} 2x^{-2}e^{-2x}$
6. $\displaystyle\lim_{x\to \infty} 2x^{2}e^{-2x}$

Can l'Hôpital's rule be applied to calculate this limit? $$\lim_{x\to 2} \frac{e^x - 2}{(x - 2)^3}$$ If yes, find the limit.

Calculate the limit $$\lim_{x \to 2} \frac{3x^2 - 3x - 6}{2x^2 + 2x - 12}$$

Here we look at the probability that two persons in a group of people have their birthday on the same date. We are interested in how this probability changes as a function of the number of people in the group, $n$.Assuming there are 365 days in a year and that all dates are equally likely as a birthday, the probability that two people have their birthday on the same date, $p(n)$, can be approximated as $$p(n) = 1- e^{\frac{-n^2+n}{730}}.$$ The function is shown in the plot below.

1. What is $\lim\limits_{n\to \infty} p(n)$?
   What is $\lim\limits_{x\to -\infty} e^{x}$?

   What is $\lim\limits_{x\to \infty} \frac{-x^2+x}{730}$?
   Assume that $p(n)$ can be approximated by the function $h(n) = 1- \frac{(1+n)^{\frac{1}{730}}}{e^{\frac{n^2}{730}}}$
2. What is $\lim\limits_{n\to \infty} h(n)?$
   Use l'Hôpital's rule.

   You only need to use l'Hôpital's rule once.
3. Did you get the same result in both limits? Was the result what you expected, intuitively?

Show that the amount dough needed for a cylindrical pizza with radius $z$ and thickness $a$ can be written as the following function of two variables. $$V(z, a) = \pi z^2 a$$ Fun fact

Given the function above, answer the following questions. It is natural to express the answers to the first two questions in units of $\pi\,\mathrm{cm}^3$ (yes, that's units of pie!)

1. How much dough is needed for a (Chicago style) pizza with a radius of 10 cm and a thickness of 1 cm?
2. How much dough is needed for a pizza (perhaps from New York or Naples) with a radius of 20 cm and a thickness of 2 mm?
3. If the thickness is doubled, by what factor is the amount of dough multiplied?
4. If the radius is doubled, by what factor is the amount of dough multiplied?
5. Show that $V(tz, ta) = t^k V(z, a)$ and determine the value of $k$.

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A function of two variables is given by the expression $$f(x, y) = \sqrt{4-x^2-y}.$$

1. What is the domain for $f(x, y)$?
2. Sketch the domain.
3. Calculate the function values $f(1,3), f(-2,-10)$ and $f(4,1)$. Can they all be computed? If not, why not?

A function of two variables is given by the expression $$f(x, y) = \frac{\sqrt{6-x-y^2}}{\sqrt{(x^2+y^2) - 1}}$$

1. Express the domain for $f(x, y)$ using one or more inequalities.
   For one of the inequalities it is easier to express $x$ as a function of $y$, rather than the other way around.
2. Sketch the domain.