In real life everything is messier than in calculus. You are used to getting “exact” answers in mathematics. You will soon find situations where the only way to get an answer is to approximate it.

Over the year, you will learn several techniques for approximating. In college, you may take a course called Numerical Approximations, learning approximation techniques. If you use calculators or computers to find a solution; these are often approximations requiring a lot of arithmetic.

Graphing calculators approximate difference quotients using the symmetric difference quotient with a very small value of .

Remember when you zoomed in on a function and found that close-up it looked like a line. Over small distances near the point of tangency the tangent line has approximately the same y-coordinates as the function. This is called local linearity – over very short intervals most functions appear linear.

One way to approximate a function’s value is to travel along the tangent line from a point you know to a nearby point that you don’t know. You do this by writing the equation of the tangent line at the point you know and then moving a short distance along it. The line’s y-coordinate is close to the function’s and may be used to approximate it. Soon, when you study differential equations, you will use this idea in a slightly different way. When you get to integration and later infinite series you will learn more approximation techniques.

With any approximation, it is useful to know how close the approximation is to the value you are approximating. The first shot at this is to look at the concavity near where you are working. From the concavity you can tell if the tangent line lies above or below the curve. With that, you can determine whether you have an overestimate or an underestimate.

Now, let’s see how close we can get.

Course and Exam Description Unit 4 Section 4.6, Unit 10 Sections 10.2 and 10.10