Question

let ( f ) be a differentiable function with ( f(1)=3 ). the graph of ( f ), the derivative of ( f ), is shown above. which of the following statements is true about the line tangent to the graph of ( f ) at ( x = 1 )? a the tangent line has slope 2 and passes through the point ( (1,3) ). b the tangent line has slope 2 and passes through the point ( (1,2) ). c the tangent line has slope 0 and passes through the point ( (1,3) ).

Explanation:

Step1: Recall tangent - line property

The slope of the tangent line to the graph of \(y = f(x)\) at \(x=a\) is given by \(f^{\prime}(a)\).

Step2: Find the slope at \(x = 1\)

From the graph of \(f^{\prime}\), when \(x = 1\), \(f^{\prime}(1)=2\).

Step3: Use the point - slope form information

We know that \(f(1) = 3\), so the tangent line to the graph of \(y=f(x)\) at \(x = 1\) passes through the point \((1,3)\) with slope \(m = f^{\prime}(1)=2\).

Answer:

A. The tangent line has slope 2 and passes through the point (1,3).