Question

Find an equation of the tangent line to the function y=2x at the point P(1, 2).SolutionWe will be able to find an equation of the tangent line as soon as we know its slope m. The difficulty is that we know only orie point, P, on f, whereas we need two points to compute the slope. But observe that we can compute anapproximation to m by choosing a nearby point Q(x, 2x) on the line (as in the figure below) and computing the slope mo of the secant line PQ. (A secant line, from the Latin word secans, meaning cutting, is a line that cuts(intersects) a curve more than once.]

Answer 1

To find:

The equation of the tangent line to the function y = 2x at point (1,2).

Solution:

The given function is y = 2x. Now, take a point through which the given function is passing through i.e., (2,4).

Now, find the slope of the line:

$\begin{gathered} m=\frac{4-2}{2-1} \\ m=\frac{2}{1} \\ m=2 \end{gathered}$

The slope of the function is m = 2. The given function represents a line whose slope is same at every point.

Now, at (1,2), the slope is 2. So, the equation of the tangent line is:

$\begin{gathered} y-y_1=m(x-x_1) \\ y-2=2(x-1) \\ y-2=2x-2 \\ y=2x \end{gathered}$

Thus, the equation of the tangent line is y=2x.