Final answer:
The question is about using the limit definition of the derivative in calculus to find a function's rate of change and to verify that it satisfies a differential equation. This involves understanding the concepts of first and second derivatives, integrating to find potential, and applying limits in various physical contexts, such as electric fields and potentials.
Step-by-step explanation:
The student's question is about using the limit definition of the derivative to find the rate of change of a function or to verify a solution to a differential equation. The limit definition states that the derivative of a function f(x) at a point x is the limit as h approaches zero of the difference quotient, [(f(x + h) - f(x))/h]. This concept is fundamental in calculus and is applied in various ways, such as when calculating the velocity of a body (the first derivative of position) or acceleration (the second derivative of position).
To verify that a given function is the solution to a differential equation, one can take its derivatives, plug them back into the equation, and check if the equation holds. Moreover, calculus topics also include integrating to find potential from the electric field and using limits to solve for quantities in physics, such as dealing with charges, fluid pressures, and the potentials of charged rods.
In problem 97. (a), finding the potential of a charged rod in the limit as x becomes much larger than L and showing that it coincides with that of a point charge demonstrates how calculus methods can simplify complex problems or confirm that simplified models behave as expected at certain limits.