Tangent of Points Worksheets

How to Find the Tangent of Points - Let us discuss the definition of the tangent. A tangent is a line that touches a single point of the curve lying on the plane. In hindsight, the gradient (slope) of the curve is equal to the gradient of the tangent to the curve. Let's take an example where the equation of the curve is y = x³ and you need to find the tangent which passes through the line (1, -4). Let's resolve this in a sequential manner. The first step is to derivate the equation: y^' = 3x². Now, we know that y' is equal to the slope, while the points of the slope can be considered as (x, x³) , (1, -4). (-4 - x³) / (1-x) = 3x², -4 - x³ = 3x² - 3x³, 2x³ - 3x² - 4 = 0. x = 2. The considered points will then become (2, 8). To find the equation of the tangent line, we can either use the two-point form or one-point slope form. In this example, we'll take the two-point slope form to find the tangent's equation, where the points are (1, -4) and (2, 8). (y - y1) / (y2 - y1) = (x - x1) / (x2 - x1) | (y - (-4)) / (8-(-4)) = (x-1) / (2-1) | (y + 4) / 12 = (x-1) / 1 | y + 4 = 12x - 12 | y = 12x - 12 - 4 | y = 12x - 16.