Real World Example of a Composite Function

One example of a composite function is the cost of life insurance as one ages. First one’s life insurance cost goes up depending on the age of the buyer, that in turn is dependent on the current date. This is a simple example where f(x) = the cost and x is the age, and g(x) = the age of the man dependent on the date: x. Then by taking f(g(x)), where x is the date, one gets a real life composite function. Insurance companies probably have a similar algorithm, but it most likely far more complex.

 

Question: Do calculators take derivatives of equations differently?

Real World Example of Exponential Growth

Bacterial growth follows exponential growth. For example if a bacteria take 24 hours to divide, then at time = 0 there would be one, t=1 there would be 2, t=2 there would be 4, etc. What it means for an equation to be exponential in the real world is that the current number of the grower will have an impact on the growth. Thus the more things that can grow the more that will, and the longer the growth has been taking place then the more things there will be to grow. Thus, what this ends up meaning is that at any given moment the rate of change at time=t will be proportional to the the function itself. This is simply because as the function, i.e. the number of things growing, goes up, then there will be more things to which will grow, causing the rate of growth to go up. Vice Versa as well.

Question: Are there any other unique derivatives like e^x, that also apply to the real world?

Real World Example of a Derivative

One simple example of a real world derivative would be the speedometer inside the car. This device shows the instantaneous rate of change, i.e. speed. More specifically it shows the speed fairly accurately that one is traveling at that moment. It is technically the derivative of the distance traveled via the car. Though the speedometer most likely uses a system based on its specific components inside the vehicle to calculate speed, the concept that of one’s speed that is truly the derivative. The speedometer is just one example of how speed is actually a real world instance of a derivative. This another example would also be the instantaneous rate of change of the speed itself, which would yield the acceleration. Both are used in the real world.

Question: What is a real world example of a derivative of the acceleration of an object? Is there such a concept that it can be explained, or is it more theoretical without much real world application?

What is a real world example of a limit? Also, ask 1 question about exponentials, logarithms or limits.

The universal law of gravitation obeys a limit and is therefore a real world example. The acceleration of gravity is inversely proportional to the square of the distance between any two objects. Thus no matter how far one is from any object one will always be influenced by its gravity. The strength of gravity may be extremely small at far distances, but it will never be equal to zero. Since the strength of gravity decreases with distance, it is thus ever approaching zero, but will never be able to reach 0. It will just continue to get infinitely smaller. Thus the limit for the formula of the acceleration of gravity is 0 as the distance increases to infinity.

Question:  I ran into Euler’s formula and was intrigued; why does Euler’s formula equal 0? How is it possible that e to any power equals 0?

What is the difference between the product of two functions f(x)⋅g(x) and the composition of them f(g(x))?

The difference between the product of two function and composition of them is fairly big. When one multiplies two functions, that is just that, multiplication. However a composition requires that the inner function be substituted for “x” in the outer function. In such a case of f(g(x)), then g(x) must be plugged into every “x” in the function f(x). Such a composition yields a very different result than does the product of these two functions. With the product, one has to perform multiplication and is quite simply done. With the composition one must substitute the inner function for all the outer function’s “x” so that g(x) is f(x)’s new “x”; and more often than not, one is left with a function that is quite different from the product.  The difference between the two is great and each is completed differently.

Question: How did Logarithms initially become implemented into Math?

Trigonometry is about the Geometry of a Circle

Trigonometry is not necessarily the geometry of a circle, but every trigonometric function of an angle can be constructed/defined geometrically from the unit circle. It shows that trigonometry is linked to circles from the very start. There are other uses and definitions, however since all the functions can be defined on a unit circle, it shows that all the functions describe the geometry of the circle. So although Trigonometry is not used solely on circles, it still uses functions that have their basis in circles and so is technically always related to circular geometry. Since the relation is there it is a fair statement to say Trigonometry is about the Geometry of a Circle, but it is important to know that this is not the only application of this field.

 

Question: Can Trigonometry be said to be about anything else besides circles?