Lecture 1. Geometric and physical meaning of the derivative
Today, we will talk about the meaning of the derivative. What tasks is it used for? The topic of the lecture is “Geometric and physical meaning of the derivative.”
Let’s consider the problem of the movement of a material point. So, a point moves along a certain curve according to the law S(t), that is, each time point corresponds to the value of the distance that the point passed. So, at time t0, the point was in the position М0, and after a certain time interval Δt, it was in the place of the point M on the specified curve.
What distance did the point travel during Δt? So, you need to subtract the first reading at time t0 from the last counter reading. So, during the time Δt, the point passed the distance ΔS(t0).
How fast was the point moving? It is clear that you need to divide the distance traveled by time, this will be the average speed of the point. But, interestingly, we see on the right side of this equality the ratio of the increment of the function to the increment of the argument. We remember that the derivative was calculated as the limit of such a fraction when Δt tends to zero, but then it is logical to assume that S‘(t0) is the limit of this fraction, it is the actual speed of a point at time t0. If we talk about the function f in general, we now understand that the derivative of the function f characterizes the rate of change of the function.
Let’s turn to the geometric meaning of the derivative. It is related to the concept of tangent to a curve. The curve γ, a fixed point М0 is given on the plane. What is a tangent? Try to answer this question. I’m not sure you can do it. It is sometimes said to be a straight line that intersects the curve γ at a single point, but there are infinitely many such lines. It is wrong.
So, where do we start? Take an arbitrary point M on the curve other than М0 and draw a secant М0М. We start moving the point M to the point М0 along the curve. What will happen to the secant? Look, its position changes, and the closer the point M to the point М0 is, the closer it begins to be to the position of a certain straight line M0T. We will call this line the tangent. Look, this is the limit position of the secant.
So, a straight line М0T is called a tangent if this is the limit position of the secant М0М when the point M moves to the point М0 along the curve. It should be noted that there may not be a tangent. Look, if the point M moved to the point М0 on the right, we would be in the position М0Т1 (the limit position of this secant), and on the left, we would get the limit position of the secant М0Т2. In this case, we say that there is no tangent.
And the existence of a tangent implies the most important meaning. So, the theorem is: if a function f is differentiable at х0, at the point (x0, f(х0)) the graph of the function has a tangent whose equation is specified as follows (see the video). Let’s prove this theorem.
So, what do we have? We have a fixed point М0 on the graph of the function, its coordinates are (х0, f(x0)). M is an arbitrary current point on the graph. We need to prove that this curve, the graph of the function, has a tangent – a straight line М0Т.
So, what do we do? We denote Δx, x, x0. X is the current coordinate of point M. We get that x = x0 + Δx. As a result, the point M (we can write it down) has coordinates (x0 + Δх, f(x0 + Δх)). If two coordinates on a straight line are known, the angular coefficient of this straight line is easily determined, of course, Δx is not equal to 0 in this case. The angular coefficient of the secant М0М is found as the ratio of the increment of the function at the point x0 to the increment of the argument.
We will now have to use a well-known equation from elementary mathematics – the equation of a straight line, which is given by the angular coefficient and the point through which it passes. So, this equation has the form (see the video). Let’s use this equation and make the equation of the secant М0М. Remember, we know the coordinates of the point М0, and we have just found the angular coefficient of this line to the secant. So, the secant equation in this case has the form (see the video).
Now, to get a tangent, we must move the point M to the point М0 along the curve, along the graph of the function. It is obvious that x will tend to zero. What will happen to the secant equation? Look, it will only change the angular coefficient. The point М0 remains fixed, only the angular coefficient changes, and if Δx tends to zero, the limit value of the angular coefficient is the value of the derivative at the point x0.
What else should we do? To replace the secant to this limit value. We get: the limit position of the secant is given by the following equation (see the video). By definition, this is a tangent. The formula is proved.
Another important term is a normal. What is the normal to a curve? This is a straight line drawn through the point of contact perpendicular to the tangent. To write the normal equation, again we need to use the fact from elementary mathematics: two straight lines with angular coefficients are perpendicular if the product of the angular coefficients is minus one. Well, since we know the tangent equation, the normal passes through the same point (x0, f(x0)), only the coefficient changes, and if the value of the derivative is not zero, we get the following normal equation (see the video).
A logical question arises: what if the derivative is zero? How do we use this equation? In this case, this equation cannot be used. If the line is horizontal (it is tangent), the perpendicular line will be vertical, passing through the same point, and then we get the following conclusions. If the derivative is zero, the tangent and normal are straight lines that pass through the tangent point parallel to the coordinate axes.
So, let’s draw some conclusions. First, since we found out in the course of the proof that the angular coefficient of the tangent is f‘(x0), we always remember that the geometric meaning of the derivative is as follows: the derivative is the angular coefficient of the tangent drawn to the graph of the function at the point (x0, f(х0)). The second important point is that since the angular coefficient of any straight line, not just a tangent, is defined as the tangent of the angle of inclination of this straight line to the Ox axis, this fact also remains true for the derivative of the function at the point х0.