Suppose your friend gives you the following problem: There is some function p(x) that represents profit p (in dollars) as a function of the number of units sold x. Your friend doesn't know what the function p(x) is explicitly, but does know that p'(x)=2 for all x. Your friend also knows that p(200)=3175.24. Now, your friend has enough information to approximate p(203) using tangent line approximation, but your friend really wants to know how accurate that approximation will be.

Question

Suppose your friend gives you the following problem: There is some function p(x) that represents profit p (in dollars) as a function of the number of units sold x. Your friend doesn't know what the function p(x) is explicitly, but does know that p'(x)=2 for all x. Your friend also knows that p(200)=3175.24. Now, your friend has enough information to approximate p(203) using tangent line approximation, but your friend really wants to know how accurate that approximation will be.

The question to you then is: What can you tell your friend about the accuracy of the tangent line approximation for p(203), given the above information about the function p? (You may explain yourself mathematically, or by presenting graphs, or just by discussing in complete sentences.)

Answer 1

since the slope is constant, the tangent line will be perfectly accurate. It is, after all approximating a straight line, right?