Problem 45b

Given information

According to the given question, we are asked to find the value of v'(5) that is the value of derivative of function v(x) at x = 5.

Since, v(x) = f(x)/g(x), so we can infer that to find derivative of v(x) we need to use the Quotient rule of differentiation.

And also to find the functions f(x) and g(x) we need to find the equations of line near x = 5 from the given graph.

Step-by-step explanation

Now, first of all we need to find the equations of f(x) and g(x) near x = 5.

For equation of f(x), we use the two point point form equation of line, that is:

..............(1)

where (x1,y1) and (x2,y2) are points on the graph.

For f(x) we have two pints near x = 5 as (2,4) and (5,3)     {from red curve}

Using equation (1) we find equation of line f:

So, equation of f(x) is given as: y = -x/3 + 14/3.................(2)