What is a function?
A function can be viewed as a black-box which takes an input value and produces some output. The black-box contains some instructions on how to use and manipulate the input value in order to produce some output value. It is important to note that each every unique input produces only 1 output. Let's consider a simple function: f(x) = x^2 + 1. If we insert a value of 1 into our black box, the following operation is performed on the value 1 (e.g. (1)^2 + 1 --> 2). We see that that the box will output the value 2 for this input value. *I will continue to show this for a couple more examples, and plot these input/output values on a graph*. As I mentioned earlier, a function must follow the rule that each input produces only 1 output, to test this we draw a vertical line in our graph - we note that our vertical line should only intersect our function line only once!
How do derivatives work and why are they important?
Say you are driving your car and we were to plot the velocity of your car at equal time intervals over a total interval of 120 seconds. We note that at 0 seconds, your speed is 1km/h, and similarly your speed increases by 1km/h every second. If we modeled this mathematically, we observe that your velocity, v(t), is equal to the product of time, t, and 1km/h (i.e. v(t) = 1*t). If plot this we observe a linear line. Now say we would like to know the acceleration of the car at every point in time over the 0-120 second interval. Intuitively we can observe that acceleration is how fast the velocity changes over a very small interval of time. Looking at our graph at a time, t=5seconds, we observe that acceleration turns out to be the slope of the tangent line to our graph (*Draws a tangent line on graphical example*) - you will notice in this example, that the tangent line everywhere is equal to the original slope of our function. (*I will continue to plot the corresponding acceleration graph with respect to the original velocity function*). Now I will show you an example where our velocity is not linear (i.e. v(t) = t^2). Now I will show you the value of the slope of each tangent line at each point on the v(t) graph and graph our corresponding a(t) graph. From this we see that the value of the slope of the tangent line to each point on the velocity graph has a corresponding value on the a(t) graph - here we see that a(t) is the derivative of the v(t) graph. *Next I would proceed to discuss the formal definition of the derivative at a given point using limits
What are the characteristics and possible applications of capacitor elements? Please provide an intuitive high-level explanation of these elements.
Capacitor elements store electrical energy by creating an electrical field between the two plates of the capacitor - in other words, capacitors store electrical charges on the two separated plates. In practical use, we use capacitor elements for a couple purposes: (i) A capacitor can be though of as a large reservoir of charges that will attempt to maintain a fixed voltage across itself - circuits concerned with steady voltage supplies will exploit this property of capacitors (e.g. Power Electronic Converters, Voltage Regulators); (ii) A capacitor has different properties at different frequencies - in other words its resistive effects change at different frequencies (i.e. Z = -j/wC) - based on this equation, we see that at low frequencies, the impedance of a capacitor will reach a large impedance (approaches infinity), therefore blocking low-frequency current from travelling through the capacitor. By the same concept, we see that high-frequency currents will observe the capacitor as a low impedance path (approaches zero), therefore acting as a short circuit at higher frequencies. Circuit designs that care about frequency behaviors will exploit capacitors for their frequency properties (e.g. Filters, Microwave Networks, RF Transmission).