Tangents and the Derivative at a Point (Part I)

Tangents and the Derivative at a Point (Part I)

Topic coming next is the tangents and the derivative at a point of a curve.

Now we have an example of finding the instantaneous velocity of a certain object at a certain time point, t. In order to find this instantaneous velocity, we actually look at quotients of this form: where the function f tells us the position of the piece of rock at time t.

Now, in general, we can look at ratios of this shape for any function and consider the limit when h tends to zero.

Well, sometimes the limit does not exist and sometimes it does. If the limit exists, we denote this value by f'(x-naught). And we call this f'(x-naught) the derivative of a function at a point x-naught.

The derivative f'(x-naught) is actually the slope of the tangent line to a curve at a given point, x-naught. The y coordinate of that point is f(x-naught).

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Let us consider the function f equal to x squared, and let us attempt to compute the derivative of this function at the point x equals minus one.

The derivative should be equal to the ratio here. The function at -1+h, -the function at -1 and then divide h. And then we consider when h tends to zero, the limit of this ratio here. So relying on formula for the function x squared, this part here becomes -1+h squared.

Then the expression becomes h times the sum of the squares of its parts.

In the bottom here, we have h. Now we can expand the squares in the numerator and cancel this minus one squared. After that, we can do the cancellation of h and arrive at this expression.

The function f(x) = |x| is clearly equal to zero when x = 0, and it is less than one for all other values of x. When h tends to zero, this is clearly equal to minus two. So the limit of this ratio when h tends to zero exists and this is equal to the derivative of the function at minus one.

We want to look at the derivative of the function at x = 0. The graph here shows quite clearly that the function has a very sharp corner at this point, which suggests that there is no tangent at this point.

Saying in another way, the function f is not differentiable at x equals to zero. How can we prove that?

The definition of f’(0) is this: if h tends to zero, then the ratio f(x)/h has a limit. We will show that this limit does not exist.

When h tends to zero, the ratio is negative. That means h is on the left-hand side of zero. H is negative. So this is equal to the absolute value of h minus zero over h. Now when h is negative, we have an absolute value of minus h.

Okay, the numerator is minus h, and the bottom is h. We do cancellation to get minus one. So the limit equals minus one when h tends to zero from the left side.

Similarly, we can take the limit as h approaches zero from the right-hand side. In this case, since h is positive, the absolute value of h is just h itself, and we have the same formula on

both sides of the inequality. When h is positive, taking the absolute value of h gives us just h itself, and again for the bottom h.

You can cancel h from both sides of the equation, and you get one; that means that the limit is equal to one. As h approaches zero from either side of the equation, it is negative one from the left-hand side and positive one from the right-hand side.

Since the limit does not exist, the derivative of f at x = 0 does not exist. In other words, this graph has no tangent at the point x = 0.

After two examples on differentiation, we can now summarize what we have been discussing. First of all, the main thing to consider is the limit of this ratio as h tends to zero.

This is basically the ratio of the change in a function at the point x equals naught, divided by the change in its variable. And we are considering this limit when h tends to zero. If this limit exists, we call it the derivative of the function at x equals zero.

Geometrically, this is equal to the slope of the tangent line to the curve at x = 0. Or simply called the slope of this function at x = 0.

The limit of f(x) as x approaches x zero is the rate of change of f with respect to x at x equal to x zero.