VIDEO: Calculate tangent and secant

In order to calculate the tangent or secant of a certain function, you need at least one or more functions in addition to the function. two x values. These must therefore be specified. There may already be a, resp. two intersection points are given, so there is no need to calculate the y-value or of the y values. The function of the tangent is: t (x) = mx + n. Here, m is the rise and n is the point of intersection with the y-axis.

How do you calculate the tangent of a certain function?

For a better understanding the function f with f (x) = 2x²-3x + 5 is given as an example. The tangent should be created at the point x = 2.

1. You now calculate the increase m of the function at the point x = 2. The rise of the function is also the derivative of this, i.e. f '(x) = 4x-3.
2. If you now insert the x-value in f ', you have the rise of the function AND the tangent at the point x = 2: f' (2) = 5.
3. In the next step, look for the y-value at the point x = 2, i.e. f (2). To do this, insert the value 2 in the function f (x). You get the result f (2) = 7.
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4. Now you only need the point of intersection with the y-axis to set up the function t (x). Since m, y and x are already known, these values ​​are now inserted into the function t (x) = mx + n and converted to n: Variables inserted 7 = 5x2 + n | - (5x2), shifted after n n = -3.
5. You now get the function t (x) = 5x-3.

How do you calculate the secant of a certain function?

We stay with the example function f with f (x) = 2x²-3x + 5 and the x-value x₁ = 2. However, since we need at least two x-values ​​for the secant calculation, a second with x₂ = 0.5 specified. The secant g is calculated as follows:

1. As with the tangent calculation, you start by determining the slope m. However, since you now have two points of intersection with the function, a calculation using the derivative, as in the case of the tangent, is not possible. To calculate these intersection points, proceed exactly as in step 3 of the tangent calculation, but now insert two x-values ​​in f and thus also get two y-values. The rise m is then determined using the gradient triangle. This is the formula for m_s : f (x₂) - f (x₁) / x₂ - x₁, i.e. 4 - 7 / 0.5 - 2. Accordingly, the increase in the secant m_s = 2. The secant runs through the points P (2/7) and Q (0.5 / 4) (points of intersection of f and g).
2. Next, as with the tangent calculation, you need the point of intersection with the y-axis n. Insert the determined values ​​m and y into the secant function g: 7 = 2x2 + n, resp. 4 = 2x0.5 + n. Now switch back to n and you get n = 3.
3. You now have the necessary information again to set up the function of the secant. So insert this into the function g (x) = mx + n. The function of the secant is therefore g (x) = 2x + 3.