Question

question 9 (3 points)
determine the equation of the tangent to ( f(x)=x^{2}+x ) at the point (1, 2).
please insert and image of your solution below.

Explanation:

Step1: Find the derivative of the function

The derivative of \(f(x)=x^{2}+x\) using the power rule \((x^n)^\prime = nx^{n - 1}\).
\(f^\prime(x)=(x^{2}+x)^\prime=(x^{2})^\prime+(x)^\prime = 2x + 1\)

Step2: Evaluate the derivative at \(x = 1\)

Substitute \(x = 1\) into \(f^\prime(x)\) to get the slope \(m\) of the tangent line.
\(m=f^\prime(1)=2(1)+1=3\)

Step3: Use the point - slope form of a line

The point - slope form is \(y - y_1=m(x - x_1)\), where \((x_1,y_1)=(1,2)\) and \(m = 3\).
\(y - 2=3(x - 1)\)

Step4: Simplify the equation

Expand and simplify \(y - 2=3x-3\).
\(y=3x-3 + 2\)
\(y=3x-1\)

Answer:

The equation of the tangent line is \(y = 3x-1\)