# Tutor profile: Natascha G.

## Questions

### Subject: German

When you're in a grocery store and you'd like to use the restroom, how do you ask?

This is a trick question because unlike the US, most grocery stores don't have public restrooms. But if you still want to try, ask this "Dürfte ich bitte Ihre Toilette benutzen?" - May I use your restroom, please? "Dürfen" is a verb that means "to be allowed to" and can be used in the context of "may" if it is used in the present tense (Darf ich...) or the form of subjunctive I (Dürfte ich). If you use it in present tense this would be the sentence: "Darf ich bitte Ihre Toilette benutzen?" It has the same meaning, just sounds a bit less formal. But it can definitely be used, it is just based on your preference. "ich" means I, 'bitte" means please, "Toilette" means toilet, restroom or bathroom, and "benutzen" means to use. "Ihre" means your, but it is the formal way of adressing someone. It can also be used if you are talking to multiple people. If you are talking to a single friend you would use "dein" (male/neutral noun following) or "deine" (female noun following). "Die Toilette" is a female noun, so you would use "deine". The sentence would look like this: "Darf ich bitte deine Toilette benutzen?" If you are lucky and you finally find a public restroom, I bet you that there will either be a sign or an employee in front of it saying, "1 Euro bitte!"

### Subject: Pre-Calculus

Create two piecewise functions that would create a heart in a graph.

The answers can vary based on your creativity, but here is a basic solution: $$-x;[-1,0]$$ $$y=$$ { $$x;(0,1]$$ $$\sin{(\pi x+\pi)}+1; [-1,0]$$ $$y=$$ { $$\sin{(\pi x)}+1; (0,1]$$

### Subject: Calculus

In 2008 the website Reddit had 10,926 subreddits. In 2017 the number of subreddits grew to 1,179,342. Growth of Reddit's subreddits is exponential and is represented by this formula: $$ \frac{dS}{dt}=kS $$ where $$S(t)$$ is the number of subreddits at a certain time $$t$$, and $$k$$ is the growth constant. 1. Find the growth rate 2. Estimate the number of subreddits in 2014

1. To find the growth rate you first need to find k. You do this by integrating the differential equation: $$ \frac{dS}{dt}=kS $$ Separate the equation by having $$S$$ on the same side as $$dS$$ and divide by $$dt$$ to have it on the other side to be able to integrate the equation $$ \frac{dS}{S}=k dt$$ Integrate both sides: $$\int{\frac{1}{S}dS}=\int{kdt}$$ $$\ln{|S|}=kt+C$$ Solve for $$S$$: $$\mathrm{e}^{ln|S|}=\mathrm{e}^{kt+C}$$ $$|S|=\mathrm{e}^{kt+C}$$ $$|S|=\mathrm{e}^{C}\mathrm{e}^{kt}$$ $$\mathrm{e}^{C}$$ is a constant and can therefore just be written as $$C$$ $$|S(t)|=C\mathrm{e}^{kt}$$ Let's say 2008 is year 0 in our equation. You know that at $$t=0$$ the number of subreddits was 10,926. $$10,926=C\mathrm{e}^{k(0)}$$ $$10,926=C$$ Now that you found $$C$$, and you have to find $$k$$. You know that in 2017 Reddit had 1,179,342 subreddits. Since we said the year 2008 is $$t=0$$ $$2017-2008=9$$ Therefore $$S(9)=1,179,342$$ $$1,179,342=10,926\mathrm{e}^{9k}$$ $$\frac{1,179,342}{10,926}=\mathrm{e}^{9k}$$ $$\ln{|\frac{1,179,342}{10,926}|}=9k$$ $$k=\frac{1}{9}\ln{|\frac{1,179,342}{10,926}|}$$ $$k≈0.52017$$ ANSWER: The growth rate of subreddits is approximately 52.017%. 2. Find the approximate number of subreddits in the year 2014. Since 2008 is $$t=0$$ $$2014-2008=6$$ in 2014 $$t=6$$ You plug $$t=6$$ in the previously found equation: $$S(t)=10,926\mathrm{e}^{0.52017t}$$ $$S(6)=10,926\mathrm{e}^{(0.52017)(6)}$$ $$S(6)≈247,692.91$$ ANSWER: The approximate amount of subreddits in 2014 was 247,692.91. (This mathematical example is not completely accurate, and the number of subreddits in 2014 was way higher, almost 600,000, if you're interested)

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