Finding a function for rate of change

In summary, the conversation discusses the need to calculate the rate of change for a given function and the desire to find a function that describes the instantaneous rate of change for a specific curve. The concept of derivatives is mentioned as the key to finding this rate of change, with examples provided using the functions arctan and tan. The suggestion to use a website called Wolfram Alpha for help with mathematical problems is also mentioned.
  • #1
PennyPuzzleBox
1
0
Hello!

I was wondering if somebody could help me develop an answer to the question below.

I would like to calculate a rate of change of the following function:

θ = arctan (a * (tan β)) where "a" is a constant.

By the way, the shape of the curve -- angle alpha as a function of angle Beta --resembles an arm of a parabola. Angle Beta increases with angle alpha, but less so with increasing values.

I would like to derive a function that would describe the instantaneous rate of change at each point of that curve. It would be great if the function would say: hey, this is an arm of a parabola! :)

Unfortunately I took math a very long time ago... I believe I would
need to find the derivative of the function? Could you kindly help me out with this?
thnx

Penny
 
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  • #2
The whole point of the "derivative" is that it is the rate of change of a function. The derivative of arctan(x), with respect to x, is [itex]1/(1+ x^2)[/itex]. The derivative of tan(x) is [itex]sec^2(x)[/itex].

So, by the "chain rule", with f= arctan(u) ,u= a tan(x), [itex]df/du= 1/(1+ u^2)[/itex] and [itex]du/dx= a sec^2(x)[/itex], so [itex]df/dx= (df/du)(du/dx)= (1/(1+ u^2))(a sec^(x)= (a sec^2(x))/(1+a^2tan^2(x)[/itex].
 
  • #3
^ What HallsOfIvy said

If you're looking for an easy way to find derivatives and do other mathsy things, look up wolfram alpha. It's a website that is quite excellent.
 

1. What is the rate of change?

The rate of change, also known as the slope, is a measure of how a dependent variable changes in relation to an independent variable. It represents the steepness of a line on a graph and can be calculated by dividing the change in the dependent variable by the change in the independent variable.

2. Why is finding a function for rate of change important?

Finding a function for rate of change is important because it allows us to model and predict how one variable will change in response to changes in another variable. This is crucial in many fields such as economics, physics, and engineering.

3. What are the steps for finding a function for rate of change?

The steps for finding a function for rate of change are:

  • Identify the independent and dependent variables
  • Collect data points for the two variables
  • Plot the points on a graph
  • Draw a straight line that best fits the data points
  • Calculate the slope of the line using the rate of change formula
  • Write the function in the form of y = mx + b, where m is the slope and b is the y-intercept

4. How can we use the function for rate of change in real life?

The function for rate of change can be used in a variety of real-life situations. For example, in economics, it can help businesses determine the most profitable price point for their products. In physics, it can be used to calculate the speed of an object at a given time. In engineering, it can aid in designing structures that can withstand certain forces.

5. Are there any limitations to finding a function for rate of change?

Yes, there are limitations to finding a function for rate of change. It assumes that the relationship between the two variables is linear, meaning that the rate of change is constant. However, in real-life scenarios, this may not always be the case. Additionally, the data used to create the function may not be accurate, leading to an inaccurate representation of the relationship between the variables.

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