![tangent line calculator tangent line calculator](https://www.geogebra.org/resource/wybfxwef/Mec5E653yr8BIAr5/material-wybfxwef.png)
![tangent line calculator tangent line calculator](https://i.ytimg.com/vi/g3XPNsSt36I/maxresdefault.jpg)
We know that ∛8 = 2 and 8.1 very close to 8. Use the tangent line approximation to find the approximate value of ∛8.1. We can understand this from the example below. i.e., The equation of the tangent line of a function y = f(x) at a point (x 0, y 0) can be used to approximate the value of the function at any point that is very close to (x 0, y 0). The concept of linear approximation just follows from the equation of the tangent line. Step - 4: Find the equation of the tangent using the point-slope form y - y 0 = m (x - x 0).Step - 3: Substitute the point (x 0, y 0) in the derivative f '(x) which gives the slope of the tangent (m).Step - 2: Find the derivative of the function y = f(x) and represent it by f'(x).Step - 1: If the y-coordinate of the point is NOT given, i.e., if the question says the tangent is drawn at x = x 0, then find the y-coordinate by substituting it in the function y = f(x).To find the tangent line equation of a curve y = f(x) drawn at a point (x 0, y 0) (or at x = x 0): Slope of the tangent line, m = (f '(x)) (x 0, y 0 )īy substituting m, x 0, and y 0 values in the point-slope form y - y 0 = m (x - x 0) we can get the tangent line equation. Let us consider the tangent line drawn to a curve y = f(x) at a point (x 0, y 0). We know that the equation of a line with slope 'm' that is passing through a point (x 0, y 0) is found by using the point-slope form: y - y 0 = m (x - x 0). Note that we may have to use implicit differentiation to find the derivative f '(x) if the function is implicitly defined. (f '(x)) (x 0, y 0 ) is the value obtained by substituting (x, y) = (x 0, y 0) in the derivative f '(x).f'(x) is the derivative of the function f(x).The slope of the tangent line of y = f(x) at a point (x 0, y 0) is (dy/dx) (x 0, y 0 ) (or) (f '(x)) (x 0, y 0 ), where Therefore, the slope of the tangent is nothing but the derivative of the function at the point where it is drawn. We know that this is nothing but the derivative of f(x) at x = x 0 (by the limit definition of the derivative (or) first principles). i.e., the slope of the tangent line at P can be obtained by applying h → 0 to the slope of the secant line. Slope of secant line = / (x 0 + h - x 0) = / hįrom the above figure, we can see that if Q comes very close to P (by making h → 0) and merges with P, then the secant line becomes the tangent line at P. Then the slope of the secant line using the slope formula is, i.e., P and Q are at a distance of h units from each other. Also, let us consider a secant line passing through two points of the curve P (x 0, f(x 0)) and Q (x 0 + h, f(x 0 + h)). Let us consider a curve that is represented by a function f(x). A secant line may also pass through any two points of the curve without the need to touch the curve at each of the two points. The above line PQ can also be called the secant line. Here is the tangent line drawn at a point P but which crosses the curve at some other point Q. Here is an example.Īgain, the tangent line of a curve drawn at a point may cross the curve at some other point also. Here is a typical example of a tangent line that touches the curve exactly at one point.Īs we learned earlier, a tangent line can touch the curve at multiple points. Here, we can see some examples of tangent lines and secant lines. The following shows a secant line PQ but which is NOT a tangent either at P or at Q.
![tangent line calculator tangent line calculator](https://www.mathepower.com/en/graphen/tangenteschneidet.png)
In that case, the line is called a secant line. If a line passes through two points of the curve but it doesn't touch the curve at either of the points then it is NOT a tangent line of the curve at each of the two points. We can see the tangent of a circle drawn here. The tangent line in calculus may touch the curve at any other point(s) and it also may cross the graph at some other point(s) as well. The point at which the tangent is drawn is known as the "point of tangency". The tangent line of a curve at a given point is a line that just touches the curve (function) at that point.