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Businesses can determine how much product they should make in order to avoid over expenditures. The greatest y-value of a 2nd derivative (absolute maximum extreme value point) will answer the question of what is the perfect amount of product. One must create a formula, set it equal to zero, derive, and finally find the max EVP.
Here lies the secret to all digital recording, including CDs and DVDs. Suppose you want to use a digital recording device to record yourself singing in the shower. The song comes out as a continuous function. The digital recording device can't record what you sound like at every moment in time, but it can record little bits of what you sound like several times a second. Since the song is a continuous function and continuous functions are nice , the several-times-a-second recording contains enough information for a computer to reproduce more-or-less what you sounded like the whole time you were singing.
In cases where something may need to be constructed or determined relative to another measurement, by substituting known values and using implicit differntitation, we can determine reted rates. In some cases the use of porportions is also necessary.
Derivative helps in the real world of:
Business (optimization)
Engineering (related rates)
Audio/Video recording (physics)
A tangent is simply a line that passes parallel to a curve at one point.
Derviatives function as a way to solve for the tangent slopes of graphs. This slope can be defined also as the rate of change.
f'(x) represents the numeric value of the slope of in a particular differentiated function.
f"(x) is resposible for determining concavity.
The point at which the function changes direction in a second derivative
*When observing a graph in its entirety, local extreme value points, also known as relative extreme value points can be any turning point, however the points with the greatest and lowest y-values are uniquely known as the absolute extreme value points.
In regards to continuity, there are two ways to describe a function; continous or discontinous. A function is continuous if it can be drawn without picking up the pencil; otherwise, it is discontinuous.
When the variable of a function is not isolated, it is in its implicit form, therefore implicit differention is the way in which we are able to dervive a function in this format.
A differentiable function of one real variable is a function whose derivative exists at each point in its domain. If a graph's slope of its tangent exists, it is differentiable and this slope has a numeric value.