Rate of Change Question

Question

Examine the following graph, which includes function f(x), as well as a line that passes through (0,0) and is tangent to f(x) at (0.5, 0.65). Then, answer the question.

What is the instantaneous rate of change for f(x) at x=0.5?

Enter your answer as a decimal, like this: 42.5.

Answer 1

From the graph we can see that the line passes through the point (x,y) = (3,4) and the origin (0,0)

We can compute the sloe of this line, m using the equation (y₂ - y₁) / (x_2--x₁) = (4-0) / (3-0) = 4/3.

So the equation of this line is y = 4/3 x. So the instantaneous rate of change of f(x) at x = 0..5 is

given by the tangent line to f(x) at x = .5. Ehich we can find by evaluating the value of the line at ,5.

We have f^'(x) = 4/3 (.5) = .667