MATH 221 Derivatives Assignment

To work with derivatives you have to know what a limit is, but to motivate why we are going to study limits let’s first look at the two classical problems that gave rise to the notion of a derivative: the tangent to a curve, and the instantaneous velocity of a moving object.

1. The tangent to a curve

Suppose you have a function y = f(x) and you draw its graph. If you want to find the tangent to the graph of f at some given point on the graph of f, how would you do that?

Let P be the point on the graph at which want to draw the tangent. If you are making a real paper and ink drawing you would take a ruler, make sure it goes through P and then turn it until it doesn’t cross the graph anywhere else.

If you are using equations to describe the curve and lines, then you could pick a point Q on the graph and construct the line through P and Q (“construct” means “find an equation for”). This line is called a “secant,” and it is of course not the tangent that you’re looking for. But if you choose Q to be very close to P then the secant will be close to the tangent.

So this is our recipe for constructing the tangent through P: pick another point Q on the graph, find the line through P and Q, and see what happens to this line as you take Q closer and closer to P. The resulting secants will then get closer and closer to some line, and that line is the tangent.

We’ll write this in formulas in a moment, but first let’s worry about how close Q should be to P. We can’t set Q equal to P, because then P and Q don’t determine a line (you need two points to determine a line). If you choose Q different from P then you don’t get the tangent, but at best something that is “close” to it. Some people have suggested that one should take Q “infinitely close” to P, but it isn’t clear what that would mean. The concept of a limit is meant to solve this confusing problem.

Are you struggling to keep up with the demands of your academic journey? Don't worry, we've got your back! Exam Question Bank is your trusted partner in achieving academic excellence for all kind of technical and non-technical subjects.

Our comprehensive range of academic services is designed to cater to students at every level. Whether you're a high school student, a college undergraduate, or pursuing advanced studies, we have the expertise and resources to support you.

To connect with expert and ask your query click here Exam Question Bank