iRaid said:Homework Statement
Hello, I'm having trouble understanding this, seemingly simple, concept. Any help or input is appreciated.
Evaluate the following derivatives:
$$\frac{d}{dt} u(t-1)u(t+1)$$
$$\frac{d}{dt} r(t-6)u(t-2)$$
$$\frac{d}{dt} sin(4t)u(t-3)$$
Homework Equations
The Attempt at a Solution
1) $$\delta(t-1)u(t+1)+u(t-1)\delta(t+1)$$
2) $$r(t-6)\delta(t-2)+u(t-6)u(t-2)$$
3) $$sin(4t)\delta(t-3)+cos(4t)u(t-3)$$
Now, I don't really understand what to do from here...
Ray Vickson said:Before trying to differentiate the function ##f(t) = u(t-1)u(t+1)## you should first see if you can re-write ##f(t)## in a simpler form. For example, what does the graph of ##f(t)## look like?
Do the same type of preliminary analysis on all the other functions you are given.
iRaid said:Sorry, I'm honestly not sure what 2 singularity functions multiplied together would look like. I know that they're both step functions starting at t=-1 and stepping up at t=1.
Surely you can multiply 1s and 0s! If t< -1 then t-1< -2 and t+ 1< 0 so both u(t-1) and u(t+ 1) are 0. u(t-1)u(t+1)= 0. If -1< t< 1 then t-1< 0 but t+1> 0 so u(t-1)= 0 and u(t+ 1)= 1. u(t-1)u(t+1)= 0. if t> 1 then t-1> 0 and t+ 1> 2> 0 so both u(t-1) and u(t+ 1) are 0. u(t-1)u(t+1)= 1.iRaid said:Sorry, I'm honestly not sure what 2 singularity functions multiplied together would look like. I know that they're both step functions starting at t=-1 and stepping up at t=1.
Singularity functions are mathematical functions that are defined differently at different points, often resulting in a discontinuity at a specific point. They are commonly used to model physical phenomena such as a sudden change in force or displacement.
Derivatives of singularity functions are the mathematical operations that determine the rate of change of a singularity function. They are useful in understanding the behavior of a function at a specific point and can represent physical quantities such as velocity or acceleration.
The derivatives of singularity functions can be calculated using the standard rules of differentiation. However, since these functions are often discontinuous, special techniques such as the Heaviside step function or Dirac delta function may be required to accurately calculate the derivative at a specific point.
Derivatives of singularity functions have various applications in physics and engineering, such as modeling the behavior of a mass-spring system or analyzing the stress and strain in a material undergoing sudden changes. They are also used in signal processing, control systems, and robotics.
One limitation of using derivatives of singularity functions is that they can be mathematically challenging to work with due to their discontinuous nature. Additionally, in practical applications, it may be difficult to accurately measure or simulate the sudden changes represented by these functions. Care must also be taken when interpreting the results, as they may not always align with physical reality.