Limits and continuity basics of investing
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Use implicit differentiation to solve related rates problems. Approximate values of functions using linearization. Relate the graph of a function to properties of its derivative. Solve optimization problems by using the derivative to find extrema of functions. Integration Evaluate Riemann sums and interpret them geometrically and contextually. Compute the integral of a function as the limit of a Riemann sum.
Relate integrals and antiderivatives through the fundamental theorem of calculus. Use integration to compute the area between a curve and an axis. The Idea of Limits of Functions When we talked about functions before, we payed attention at the values of functions at specific points. The idea behind limits is to analyze what the function is "approaching" when x "approaches" a specific value.
To start getting used to this idea, let's turn to this graph: When x approaches the value "a" in the x axis, the function f x approaches "L" in the y axis. In this graph I drawed a big pink hole at the point a,L. Let's focus on the point 1,1. We can see from the graph that when x approaches 1, the function f x approaches 1. When this happens, we say that: This is read "the limit as x approaches 1 of x squared equals 1". Why Limits are Useful You might ask what this is useful for. Very good question.
Why would you need to know what the function is approaching? You already know the function equals 1 when x equals 1, right? As an example of this, let's consider the following function: Don't let this notation intimidate you! This only means that this function equals x2 when x is anything other than 1, and equals 0 when x equals 1. Let's see what the graph looks like: What does the function approach when x approaches 1?
It also approaches 1, right? It doesn't matter that the function is other than 1 at that point!
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