Limits: The Foundation of Calculus
Limits describe the behavior of a function as it approaches a specific point, often used for finding the rate of change of a function.
Intuitive Understanding of Limits
A limit describes the value that a function approaches as the input (usually x) gets closer and closer to a particular value.
ExampleConsider the function $f(x) = \frac{x^2 - 1}{x - 1}$. As x approaches 1, this function gets closer and closer to 2, even though it's undefined at x = 1. We write this as:
$$\lim_{x \to 1} \frac{x^2 - 1}{x - 1} = 2$$
Common MistakeThe limit of a function as it approaches a point is not the same as the function at that point. We don't care any bit about what it is at that point—only its behavior while as it approaches that point.
Estimating Limits from Tables and Graphs
Students learn to estimate limits using tables of values and graphs. This approach helps build intuition about function behavior near critical points.
ExampleTo estimate $\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$, we can create a table:
From this, we can estimate that the limit as x approaches 2 is 4.
TipWhen estimating limits from graphs, look at the y-values the function approaches as x gets very close to the point of interest from both sides.
The Derivative: Gradient Function and Rate of Change
The derivative is a fundamental concept in calculus, representing both the gradient function of a curve and the rate of change of a function.
Derivative as Gradient Function
The derivative at a point gives the slope of the tangent line to the curve at that point, giving us the rate of change of the function at that point. This interpretation allows us to understand how steep a curve is at any given point.
ExampleIf s(t) represents the position of an object at time t, then s'(t) gives the instantaneous velocity of the object at time t.
NoteThe dual interpretation of the derivative as both a gradient and a rate of change is crucial for a deep understanding of calculus and its applications.
Common MistakeOnly continuous functions can be differentiated at every point. Discontinuous functions do not have a derivative.
A function is discontinuous if the limit as it approaches a point from the left is not the same as if it approaches the same point from the right.
Gradient of a Curve as a Limit
The derivative is formally defined as a limit, connecting the concepts of limits and derivatives. Students in IB Math AA SL are expected to have an informal understanding of this relationship.
$$f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$
This limit represents the slope of the secant line as the two points on the curve get infinitely close to each other, eventually becoming the tangent line. Try it yourself by finding the gradient of the straight line between the points $(x,f(x))$ and $(x+h, f(x+h))$
Common MistakeStudents often confuse the limit definition of the derivative with the process of finding derivatives using rules. Remember, this limit definition is the foundation, while derivative rules are shortcuts built on this foundation.
Notations for Derivatives
Several notations are used to represent derivatives, and students should be familiar with all of them:
- Leibniz Notation: $\frac{dy}{dx}$ This notation emphasizes the idea of the derivative as a ratio of infinitesimal changes. This can also be done with different variables, such as $\frac{dV}{dr}$, $\frac{ds}{dt}$
- Lagrange Notation: $f'(x)$ This is often the most convenient notation for simple functions.
If y = x^2, we can write the derivative as:
- $\frac{dy}{dx} = 2x$ (Leibniz notation)
- $f'(x) = 2x$ (Lagrange notation)
- If y represents volume V and x represents radius r, we might write $\frac{dV}{dr} = 2\pi r h$ for a cylinder of height h.
