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.
The 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.
All differentiable functions are continuous, but not all continuous functions are differentiable.
The 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.
Continuity 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
$$\frac{f(a+h_2) -f(a)}{h_2}$$
As $a+h$ gets closer to $a$, or in other words $h$ approaches $0$, the line approaches to becoming a tangent at point $(a.f(a))$.
Hence, the slope approaches the instantaneous (at a singular point) rate of change as $h$ approach $0$,
Therefore the derivative of a function $f(x)$ at a point $x$ is defined as
$$f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$$
$h$ can be both negative and positive. Further, we can just use $x$ instead of $a$ for better notation.
This definition encapsulates the idea of finding the instantaneous rate of change by considering smaller and smaller intervals.
When using this definition, it's often helpful to factor the numerator to cancel out the $h$ in the denominator before taking the limit.
Let's find the derivative of $f(x) = x^2$ using the first principles definition:
$$\begin{align*} f'(x) &= \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} \\ &= \lim_{h \to 0} \frac{(x+h)^2 - x^2}{h} \\ &= \lim_{h \to 0} \frac{x^2 + 2xh + h^2 - x^2}{h} \\ &= \lim_{h \to 0} \frac{2xh + h^2}{h} \\ &= \lim_{h \to 0} (2x + h) \\ &= 2x \end{align*}$$
Thus, we've shown that the derivative of $x^2$ is $2x$ using the first principles definition.
Higher Derivatives
Higher derivatives are obtained by repeatedly differentiating a function. They provide information about the rate of change of the rate of change, and so on.
Notation
There are several notations used for higher derivatives:
- Leibniz notation: $\frac{d^2y}{dx^2}$, $\frac{d^3y}{dx^3},...,\frac{d^ny}{dx^n}$, etc.
- Lagrange notation: $f''(x)$, $f'''(x),...,f^{(n)}(x)$, etc.
These ultimately represent the $n^{th}$ derivative.
The 2nd derivative is obtained by differentiating the first derivative, the 3rd derivative is obtained by differentiating the 2nd derivative and so on.
Hence the $n^{th}$ derivative is obtained by differentiating the $(n-1)^{th}$ derivative
For induction questions based around derivation, when proving for the $n^{th}$ derivative, you assume true for $k$ and prove for $k+1$.
Thus the $(k+1)^{th}$ derivative is basically just the derivative of the $k^{th}$ derivative!
Let's find the first three derivatives of $f(x) = x^3$:
First derivative: $f'(x) = 3x^2$ Second derivative: $f''(x) = 6x$ Third derivative: $f'''(x) = 6$
We can see that the fourth derivative and beyond will be zero for this polynomial function.
Applications and Connections
Understanding derivatives from first principles and higher derivatives is crucial for various applications in calculus and beyond.
- In physics, higher derivatives are used to describe motion. For instance, if $s(t)$ represents position, then $s'(t)$ is velocity, $s''(t)$ is acceleration, and $s'''(t)$ is jerk.
- In optimization problems, the first derivative helps find critical points, while the second derivative determines the nature of these points (maxima, minima, or inflection points).
- The concept of limits ties into the study of infinite geometric sequences (SL1.8), where we consider the behavior of sequences as they progress infinitely.
- Proof by mathematical induction (AHL 1.15) can be used to prove formulas for higher derivatives of certain functions.
When studying this topic, try to make connections with other areas of mathematics and real-world applications. This will deepen your understanding and make the concepts more meaningful.
In conclusion, the study of derivatives from first principles and higher derivatives forms a crucial foundation for advanced calculus and its applications. By mastering these concepts, students gain powerful tools for analyzing rates of change and function behavior across various disciplines.
