Question

1. let ( f ) be a differentiable function such that ( f(3)=2 ) and ( f^{prime}(3)=5 ). if the tangent line to the graph of ( f ) at ( x = 3 ) is used to find an approximation to a zero of ( f ), that approximation is (a) ( 0.4 ) (b) ( 0.5 ) (c) ( 2.6 ) (d) ( 3.4 ) (e) ( 5.5 )

Explanation:

Step 1: Find the equation of the tangent line

The equation of the tangent line to the graph of \(y = f(x)\) at the point \((a,f(a))\) is given by \(y - f(a)=f^{\prime}(a)(x - a)\). Here, \(a = 3\), \(f(3)=2\) and \(f^{\prime}(3)=5\). So the equation of the tangent line is \(y-2 = 5(x - 3)\).
Simplify it to \(y=5x-15 + 2=5x-13\).

Step 2: Find the \(x\) - intercept of the tangent line

To find the zero of the function (using the tangent - line approximation), we set \(y = 0\) in the equation of the tangent line.
Set \(0=5x-13\). Then \(5x=13\), and \(x=\frac{13}{5}=2.6\).

Answer:

C. \(2.6\)