for the given function, find (a) the equation of the secant line through the points where x has the given…

for the given function, find (a) the equation of the secant line through the points where x has the given values and (b) the equation of the tangent line when x has the first value. y = f(x)=x² + x; x = 1, x = 3 a. the equation of the secant line is y = □

for the given function, find (a) the equation of the secant line through the points where x has the given values and (b) the equation of the tangent line when x has the first value. y = f(x)=x² + x; x = 1, x = 3 a. the equation of the secant line is y = □

Answer

Explanation:

Step1: Find function values at given x - values

First, find (y) when (x = 1) and (x=3) for (y = f(x)=x^{2}+x). When (x = 1), (y_1=f(1)=1^{2}+1=2). When (x = 3), (y_2=f(3)=3^{2}+3=9 + 3=12).

Step2: Calculate the slope of the secant line

The slope (m) of the secant line passing through ((x_1,y_1)=(1,2)) and ((x_2,y_2)=(3,12)) is given by the formula (m=\frac{y_2 - y_1}{x_2 - x_1}). (m=\frac{12 - 2}{3 - 1}=\frac{10}{2}=5).

Step3: Use the point - slope form to find the equation of the secant line

The point - slope form of a line is (y - y_1=m(x - x_1)). Using the point ((1,2)) and (m = 5), we have: (y-2=5(x - 1)) Expand the right - hand side: (y-2=5x-5). Add 2 to both sides to get the slope - intercept form: (y=5x-3).

Answer:

(y = 5x-3)