1. let ( f ) be a differentiable function such that ( f(3)=2 ) and ( f^{prime}(3)=5 ). if the tangent line…

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 )

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 )

Answer

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)