Lecture 2. Differentiation of a function given parametrically
When we spoke about how to define a function, we generally had one method – analytical – the most important one. We have been familiar with it since school. But it does not limit all the diversity. There were other ways. There are actually many more ways to define a function. Today, we will find out how to define functions parametrically and implicitly (as it is usually called). And, accordingly, since we study differential calculus, we will also speak about calculating the derivatives of such functions.
So, to begin with, let’s determine what a function defined parametrically is. Let a system of two functions of the variable t is given. x = φ(t), y =ψ(t), where t runs through values from a certain interval from α to β. If we take a specific value t from this interval, we calculate two numbers x and y, but a pair of numbers x, y defines a point M on the plane with such coordinates. So, each value of t from the interval from α to β corresponds to a point on the coordinate plane. This means that if t runs through the values from α to β, points appear on the plane in turn, which connect to form a curve. So, this system of functions defines a parametric curve, and t is called a parameter.
If the function φ is invertible, look, it means that t can be expressed in terms of x, and substituting in y, we get that y becomes the function of the variable x. Thus, this system of functions also defines a function in the usual sense of the word, the function of the variable x. We say such a function is given by a given system of functions parametrically. So, the function system defines the curve and the function.
Let’s look at some examples.
Example 1. Parametric equation of a circle. So, a curve – a circle on a plane – can be defined by a system of functions. It is easy to see that if we square each of the equations and then find the sum of the left parts, we get x2+y2. On the right, by virtue of the basic trigonometric identity, we get r2. In fact, we see that the points of the plane form a circle centered at the origin of radius r. How are x, y, t and r related? This can be seen in this illustration. So, the t-parameter is the angle between the direction of the axis Оx and the ray OM, which connects the origin point and the point (x, y).
Another curve is an ellipse. An ellipse is a set of points on a plane defined parametrically by the following equation (see the video). If we exclude the parameter from this equation, it is easy to see that x and y satisfy the following equation (see the video). We get the equation of the ellipse in the coordinate system Оху. In fact, there is another way to define an ellipse, such as the geometric location of points, and the method for constructing an ellipse is shown in this image (see the video).
Another very interesting curve, the cycloid, is a curve that is defined parametrically by a system of functions where t runs through all the real values. How can we imagine a cycloid? A circle rolls along a straight line, and a point is fixed on the circle. The trajectory of this point is called a cycloid. Look at this illustration (see video). The arc that appears is called a cycloid. It consists of arches.
There are many other interesting curves that are defined parametrically.
What is special about functions defined parametrically and curves defined parametrically? When you change the parameter, a new point appears, movement appears, and, therefore, the direction on the curve. You should also keep in mind that if a function is defined analytically in the usual form y=f(x), it can be easily parameterized, and there are infinitely many ways to introduce a parameter. This means that the parametric method of defining is broader than defining the analytical function. We can always assume that x is the parameter t, and then from y = f(x) we can easily pass to the parametric task x = t, y = f(t). We can assume that x = t - 1, so we get a different way to define the same function.
How can we differentiate a function defined parametrically? Let the function be defined parametrically by a system of functions. To find its derivative at the point x0, which is the value of the function φ at some point t0, we use the definition of the derivative – this is the limit of the ratio of the increment of the function to the increment of the argument. What can we do? We divide the numerator and denominator by Δt. As a result, we get the value of the derivative of the function ψ in the numerator, and the value of the derivative of the function φ in the denominator at the point t0. Of course, provided that these functions are differentiable at t0 and the denominator value is not zero.
Thus, the value of the derivative is determined by the formula (see the video). We often say that we must divide y’(t) as a function of a variable from t by x’(t). This is the derivative function that we’ve just found. We can assume that it is defined parametrically as well.
At what point is the value of the derivative calculated? At the point x, the value of which remains the same – φ(t). How did we find the y value? At what point? φ(t). At what point did we find the value of the derivative? At the same point – φ(t). Therefore, to calculate the second-order derivative, we use the same rule. To find the second derivative, we will have to divide the derivative of y by the derivative of x.
Let’s take the following example. So, the task is to find the first- and second-order derivatives of a function given parametrically by a system of functions (see the video). t runs through all real values. There are no restrictions on t here.
So, we find the first-order derivative by the formula, we divide y’(t) by x’(t). We calculate the derivative of the sine 4t, in the denominator 2t - 1, and get the result. So, the derivative of the function is defined parametrically, as a result. The argument is 2t - 1. The value of the derivative is the same value of 2 cosine 4t that we’ve just found.
Therefore, the second derivative is calculated again by the same rule. We calculate the value of y’ as a function of the variable t in the numerator and divide by x’(t), calculating the derivative, we get the answer. If we continue to do this, we can get derivatives of the second, third, or any other order.
Draw your attention to the following point. What is a function that is defined implicitly? We consider the circle x2 + у2 =1. This is the set of points on the plane, which, generally speaking, is not a graph of a function. Why? We can draw a vertical line so that it intersects the circle at two points. Look, it means that the argument x corresponds to two values of y. This is not a function. But if we remove some points on the circle and leave only those parts so that any vertical line intersects the remaining set at no more than one point, we will get a function that satisfies any point on the plane. This function is defined implicitly. If we leave the lower part of the circle, it will be the same function given implicitly by the same equation.
In general, there are infinitely many functions. Only two of these functions are continuous. The graphs of these continuous functions are the upper semicircle or lower semicircle.
How can we find the derivative of this function? So, what questions arise? If we look at this equation, the first thing we noticed is that this equation is given implicitly by infinitely many functions. But for these functions, what is the domain of definition, how are the values calculated, how many of them are continuous, we said two, and how do we calculate their derivatives? So, the domain of definition is those points x for which there exists y such that the pair x, y satisfy this equation. How is the value calculated? In fact, this question is ambiguous. One of the two values can be selected as the value. If there are 2 points on the circle, we can take any of them.
Well, to find out how derivatives are calculated, let’s solve this problem. So, the task is to find the derivative of the function given by the implicit equation (see the video). What do we do? We differentiate the right and left parts, and we assume that x is an argument, and y is the same function defined implicitly, and y is a function of the variable x. We will take this into account when differentiating. Look, the derivative of x2 – 2x, and the derivative of у2 is a function, it is 2у*у’, the derivative of one is 0, and what do we get? It is easy: y’ is calculated, it is expressed from here, and we get this – - x / y. Thus, y’ depends not only on x, but also on y. And we understand which point y we have chosen, the value at the point x. So, the value of the derivative is determined not only by the argument, but also by the value of the function at the point x, which is selected as the function value.