Key Concepts

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Limits

Limits quantify what happens to the values of a function as we approach a given point. This can be defined notationally as:

limx6f(x)=L\lim_{x \rightarrow 6} f(x) = L

We can read this in simple terms as “the limit as x goes to 6 of f(x) approaches some value L”.

Introduction to Differential Calculus
Lesson 1 of 1
  1. 1
    Data scientists often are asked to make data-driven recommendations on the best way to solve a problem. Along the way, they build and fit statistical models to support their claims, make predictio…
  2. 2
    The first concept we will look at is something called a limit. Limits quantify what happens to the values of a function as we approach a given point. This can be defined notationally as: \lim_{x …
  3. 3
    Suppose we wanted to measure a runner’s instantaneous speed using a stopwatch. By instantaneous speed, we mean their speed at an exact moment in time. Let’s define f(t) as the runner’s distance …
  4. 4
    In the previous exercise, we discussed the idea of the derivative at a point as the instantaneous rate of change at that point. The idea behind the definition is that by finding how the function c…
  5. 5
    Derivative functions contain important information. Let’s look at a couple of examples. Going back to our running example, f(t) describes a runner’s position as a function of time; therefore, the…
  6. 6
    We have talked at length about derivatives abstractly. But how do we actually find derivative functions? We will outline this in this exercise and the next one. The answer is through a series of “…
  7. 7
    So we now have an idea on where to start differentiating (that is, take the derivative of) things like: \begin{aligned} 4\log(x) \ x^2 + \log(x) \ x^2\log(x) \ \end{aligned} Our final step is…
  8. 8
    We’ve discussed how to compute derivatives by hand. However, oftentimes when working on a computer, we have functions given to us in array form rather than an explicit equation. In this exercise, w…
  9. 9
    Congrats! You have finished your exploration of differential calculus. This has been a jam-packed lesson, so pat yourself on the back for making it through the material. Let’s review what we have l…

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