What's the purpose of finding derivatives? I don't understand why you would need to find a derivative, which is the tangent of the slope of a point on a graph, to find something like "How far did Bob fall". Confusing. Please help. Thank you!
If you had a graph of distance versus time for example you might want to know the speed at some specific time. That is the slope at that time
s = dx/dt
again the magnitude of acceleration is the time derivative of speed
|a| = ds/dt = d/dt(dx/dt) = d^2x/dt^2
note that I said distance and speed and magnitude of acceleration. Position , velocity, and acceleration also have direction so they are vectors with both magnitude and direction. To measure speed you need a speedometer. To measure velocity you need a speedometer and a compass.
Of course if you want to do physics or hydrodynamics or design bridges you need much more.
Now as for Bob falling
acceleration = -g = d^2h/dt^2
velocity = dh/dt = -gt + initial velocity Vi
height h = -gt^2/2 + Vi t + Hi the initial height
Ohh; I see. Thanks!
Finding derivatives has various practical applications in mathematics, physics, engineering, economics, and many other fields. While it may not be immediately evident how derivatives are connected to finding distances or solving real-world problems, I can explain the underlying concept.
The derivative of a function measures how that function changes at any given point. By finding the derivative, you can determine the rate at which a quantity is changing, whether it be time, distance, temperature, or any other measurable quantity. This makes derivatives an essential tool for analyzing and understanding the behavior of functions.
To clarify your example, let's consider Bob falling. Imagine that Bob is falling from a height and we want to know how far he has fallen at a specific time. We can model Bob's position as a function of time, such that his position at any given time is given by a mathematical equation.
Using calculus, we take the derivative of this position function, which yields the derivative function representing Bob's velocity. The velocity function tells us how fast Bob is falling at any moment in time.
Next, we can take the derivative of the velocity function to obtain the acceleration function. This function represents how fast Bob's velocity is changing as he falls. It tells us whether Bob is speeding up or slowing down during his descent.
By integrating the acceleration function, we can find the velocity function. And by integrating the velocity function, we can determine the position function, which describes Bob's distance fallen from the starting point at any given time.
So, in summary, finding derivatives allows us to analyze how a quantity is changing with respect to another quantity, whether it be time, distance, or any other variable. This knowledge and understanding of rates of change enable us to solve a wide range of problems, including determining distances, predicting future outcomes, optimizing processes, and more.