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Patricia S.

University research scientist with science, math, coding expertise

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Pre-Calculus

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Question:

How do I find the x- and y-intercepts of a quadratic function?

Patricia S.

Answer:

Consider the generalized function: $$y = ax^2+bx+c$$ In order to find where this function crosses the x- and y-intercepts, let's think about what an intercept is. The x-intercept is the position on the x-axis where the function crosses the x-axis. This means that at this position, the y-value is 0. We can use that to find the x-intercept. So... at x-intercept: y = 0 At y-intercept, x = 0 We'll start with the y-intercept because it is simpler. To find the y-intercept of any equation, plug x = 0 in: $$y = a (0)^2 + b(0) + c = c$$ The y-intercept is c. Correct. Now let's do the x-intercept. To find the x-intercept, plug y = 0 in: $$0 = ax^2 + bx + c$$ We can solve for x using the quadratic formula. This formula gives us the "roots" of the quadratic equation, where the roots are the x-intercept(s). $$x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}$$ There could be 0, 1, or 2 x-intercepts of this equation. 0: The function never crosses the x-axis. 2: The function crosses the x-axis in 2 places. 1: The function crosses one time (probably a linear function, or quadratic where a=0).

LaTeX

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Question:

Why should I use LaTeX instead of Microsoft Word for writing my lab report?

Patricia S.

Answer:

LaTeX has some built-in features that will make your life easier and allow you to focus on the content of your lab report and not the formatting. Here is a feature that you should consider using: Define a new command! One time I wrote an entire report which calculated the number of samples at the peak of a distribution. This was first referred to as `N_peak`. $$N_{peak}$$ Later, my group decided instead to refer to this value with `N-hat`: $$\hat{N}$$ which meant that I had to search through my paper for every instance and replace the former notation with the newer notation. This change of notation seems to happen at least once in every lab report or paper I write. I finally learned that there is a better way! You can define your own custom commands at the beginning of your LaTeX document. Place the following text in the preamble (before \begin[document]) \newcommand{\Npeak} {\ensuremath{N_{\textrm{peak}}}\xspace} This line tells LaTeX to make a new command, `\Npeak`, which will insert the text N_{peak}. `\ensuremath` tells LaTeX that this is an equation, `\textrm{}` tells LaTeX to use text formatting for 'peak', and `\xspace` tells LaTeX how to handle spacing issues before the next word. Now, in my report, I can refer to this variable using the command `\Npeak`. When the time comes to change my notation, I can change the command in the preamble to: \newcommand{\Npeak} {\ensuremath{\hat{N}}\xspace} Now when I recompile LaTeX, all of the instances of N_peak will be changed to hat-N.

Calculus

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Question:

What is the difference between an integral and a sum?

Patricia S.

Answer:

An integral can be thought of as a sum across continuous data. Let's revisit the difference between continuous and discrete data. If you have a graph of $f(x)$ vs. $x$, continuous data would be that in which the data is represented by a line, while discrete data is represented by points. One way would be to go outside every hour and take the temperature and put that on a plot. This would give you a set of discrete data: 8 AM: 30 degrees 9 AM: 32 degrees 10 AM: 33 degrees And so on. You have individual measurements that you would plot as points on a plot. Another way would be to use a temperature logger that measures continuously throughout the day so that you know the temperature at every time. This would produce a set of continuous data, or, a smooth curve of data. Let's consider a function $y = f(x)$ from $x = 0$ to $10$. The x-axis of data in a continuous distribution would be represented by: $$0<x<10$$ While data in a discrete distribution from 0 to 10 would be: $$x = 0, 1, 2, ..., 9, 10$$ The y-axes for a continuous distribution would be: $$y = f(x), 0<x<10 $$ And for a discrete distribution: $$y = f(x), x = 0,1,2,...,9,10$$ $$y = f(0), f(1),f(2),...,f(9),f(10)$$ You can imagine again that if you were to draw the distribution of $y$ vs $x$, you would use a smooth line for the continuous distribution and a set of points for the discrete distribution. Now if we want to take a total over these two distributions, we get to the integral vs. sum question. An integral calculates the total over a continuous distribution: $$\int_0^{10} f(x)dx$$ While the sum calculates the total over a discrete distribution. $$\sum_{i=0}^{i=10} f(x_i)$$ If you are calculating a sum or integral, the first question you should ask yourself is: Is my data discrete or continuous?

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