First Principles and Higher Derivatives in Mathematics
Continuity and Differentiability
Continuity and differentiability are fundamental concepts in calculus that describe the behavior of functions.
Continuity
A function is considered continuous at a point if there are no breaks, jumps, or holes in its graph at that point. Informally, we can think of a continuous function as one that can be drawn without lifting the pencil from the paper.
ExampleThe function $f(x) = x^2$ is continuous for all real numbers. On the other hand, the function $g(x) = 1/x$ is continuous everywhere except at $x = 0$, where it has a vertical asymptote.
Differentiability
A function is differentiable at a point if it has a well-defined derivative at that point. Geometrically, this means the function has a unique tangent line at that point.
NoteAll differentiable functions are continuous, but not all continuous functions are differentiable.
ExampleThe function $f(x) = |x|$ (absolute value of x) is continuous everywhere but not differentiable at $x = 0$ because it has a sharp corner at this point.
NoteContinuity and differentiability is not tested in the exam
Limits and Their Importance
Limits are a crucial concept in calculus, forming the foundation for understanding continuity, derivatives, and integrals.
Convergence and Divergence
When we talk about limits, we're often interested in what happens to a function as its input approaches a particular value. This behavior can be classified as either convergent or divergent.
- Convergence: The limit exists and approaches a finite value.
- Divergence: The limit doesn't exist or approaches infinity.
Consider the limit:
$$\lim_{x \to 2} \frac{x^2 - 4}{x - 2}$$
This limit converges to 4, which we can see by factoring the numerator:
$$\lim_{x \to 2} \frac{(x+2)(x-2)}{x - 2} = \lim_{x \to 2} (x+2) = 4$$
On the other hand, the limit:
$$\lim_{x \to 0} \frac{1}{x}$$
diverges because the function approaches positive infinity as $x$ approaches 0 from the right, and negative infinity as $x$ approaches 0 from the left.
Definition of Derivative from First Principles
The derivative of a function at a point represents the rate of change of the function at that point. In other words, if you draw a tangent to a graph at some point, the slope of that tangent represents the rate of change of the graph at that point.
The Limit Definition
The above graph shows a function $f(x)$ with 2 points, $(a,f(a))$ and $(a+h_1,f(a+h_1))$. The slope of the tangent line between the two points on the graph can be calculated using rise over run.
$$ \frac{rise}{run} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{f(a+h_1) -f(a)}{a + h_1 - a} = \frac{f(a+h_1) -f(a)}{h_1} $$
Hence the above expression is the slope of the straight line between any two points on the graph of $f$.
Another point $(a+h_2,f(a+h_2))$ where $a+h_2$ is closer to $a$ than $a+h_1$ was. The slope at this point would thus be
$$\frac{f(a+h_2) -f(a)}{h_2}$$
