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Why do we need to find if a "limit" exists when differentiating?

Why is the limit usually "h=0"?

If I differentiate a function, I would get a gradient function. What does this mean? What does the gradient function do for me?

Thanks!


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Best answer

Wow, three tough questions in one!

Differentiation only works when the curve exists from both sides, otherwise you can't really speak of a "gradient" (which is essentially what differentiation is aiming to achieve). This is why you need to find a limit - which is essentially a calculation done by approaching it from one side or the other (e.g. because you can't divide by zero, let's see what happens when we divide by 0.1, then 0.01, then 0.001, and see what it converges to?)

The example in the brackets above explains why the limit is usually h=0, because you can avoid dividing by zero with a limit - which is exactly what you are doing with differentiation. You are calculating a gradient of a single point, which is the same as taking a normal straight line gradient between two points, except you are shrinking the "run" part of "rise"/"run" to zero (dividing by zero). A limit allows you to realistically achieve that calculation without dividing by zero.

The gradient function reads out the gradient of any point of the original function. e.g. y=x^2, dy/dx=2x. At x=1, the gradient of y=x^2 is dy/dx=2*1=2


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