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A1 --> Y11
B1 --> Y12
C1 --> Y13
Calculus
Functions
What is a Relation?
In mathematics, a relation describes a connection or association between elements of two sets. It is a fundamental concept that generalizes how elements from one set can correspond to elements of another set.
Formal Definition of a Relation
A relation between two sets \(X\) and \(Y\) is any subset of the Cartesian product \(X \times Y\). The Cartesian product \(X \times Y\) is the set of all ordered pairs \((x, y)\), where \(x \in X\) and \(y \in Y\). Formally, a relation \(R\) is defined as:
\[ R \subseteq X \times Y \]
If \((x, y) \in R\), we say that \(x\) is related to \(y\) by \(R\), and we write \(x \, R \, y\).
Properties of Relations
Relations can have various properties depending on the specific subset of \(X \times Y\):
Reflexive Relation: A relation \(R \subseteq X \times X\) is reflexive if every element in \(X\) is related to itself: \[ \forall x \in X, \, (x, x) \in R \]
Symmetric Relation: \(R\) is symmetric if: \[ \forall x, y \in X, \, (x, y) \in R \implies (y, x) \in R \]
Transitive Relation: \(R\) is transitive if: \[ \forall x, y, z \in X, \, (x, y) \in R \land (y, z) \in R \implies (x, z) \in R \]
Antisymmetric Relation: \(R\) is antisymmetric if: \[ \forall x, y \in X, \, (x, y) \in R \land (y, x) \in R \implies x = y \]
Examples of Relations
Equality Relation: The relation \(R = \{(x, x) \mid x \in X\}\) is reflexive, symmetric, and transitive.
Divisibility Relation: For \(X = Y = \mathbb{Z}\), the relation \(R = \{(x, y) \mid x \text{ divides } y\}\) is reflexive, transitive, but not symmetric.
Friendship Relation: Let \(X\) be a set of people. The relation \(R = \{(x, y) \mid x \text{ is friends with } y\}\) is symmetric but not necessarily reflexive or transitive.
Order Relation: For \(X = Y = \mathbb{R}\), the relation \(R = \{(x, y) \mid x \leq y\}\) is reflexive, antisymmetric, and transitive.
What is a Function?
Now that we understand the concept of relations, we can define a function as a special type of relation with additional constraints.
Formal Definition of a Function
A function \(f\) is a relation between two sets \(X\) and \(Y\) that satisfies the following condition: for every element \(x \in X\), there exists exactly one element \(y \in Y\) such that \((x, y) \in f\). Formally:
\[ f \subseteq X \times Y, \quad \text{and} \quad \forall x \in X, \exists! y \in Y \text{ such that } (x, y) \in f \]
This means that no element of \(X\) can be associated with more than one element of \(Y\).
Important Notes
- Every function is a relation, but not every relation is a function.
- Functions are deterministic: for a given input \(x\), the output \(y\) is uniquely determined.
Examples of Functions
Mapping in a Graph:
Suppose \(X = \{\text{A, B, C}\}\) and \(Y = \{1, 2, 3, 4\}\). Let consider sets \[ \begin{align*} f &= \{(\text{A}, 1), (\text{B}, 2), (\text{C}, 3)\}\\ g &= \{(\text{A}, 2), (\text{B}, 2), (\text{C}, 3)\}\\ h &= \{({\color{red}\text{A}, 1}),({\color{red}\text{A}, 2}), (\text{B}, 2), (\text{C}, 3)\} \end{align*} \]
First two are functions, but the last one is not a function because element A is associated with two different elements. Let draw these mappings in a graph.
Injective Function
Non-injective Function
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A2 --> Y22
B2 --> Y22
C2 --> Y23
Not a Function
graph LR
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A3 --> Y31
A3 --> Y32
B3 --> Y32
C3 --> Y33
Numerical Sequence:
Let’s construct a set of pairs where each natural number \(n\) is paired with the fraction \(\frac{1}{n}\).
\[ f=\{(1,1),(2,1/2),(3,1/3),...\}=\{(n,1/n): n \in \mathbb{N}\} \]
This is good example of a function because each natural number is associated with a unique fraction. This kind of function is called a sequence.
Let us visualize this sequence in Python:
Show the code
where \(\mathbb{N}\) denotes the set of natural numbers. This function assigns each \(n\) the value \(1/n\).
We often denote sequences using \(a_n=1/n\).
This is an example of numerical sequence, where each natural number is associated with a number. We can define other types of sequences, for example, a sequence of matrices, a sequence of functions, etc. Term sequence is often used to denote a function that assigns each natural number a value any kind!
Matrix Sequence:
\[ f=\{(1,M_1),(2,M_2),(3,M_3),...\}=\{(n,M_n): n \in \mathbb{N}\} \]
where \(M_n\) is a matrix.
Real Function:
The quadratic function often written as \(f(x)=x^2\) is a function that assigns each \(x\) the value \(x^2\). In our notation, we write this as
\[ f=\{(x,x^2): x \in \mathbb{R}\} \]
where \(\mathbb{R}\) denotes the set of real numbers.
Show the code
Function Sequence:
Let us consider a sequence of functions:
\[ f=\{(1,x^1),(2,x^2),(3,x^3),(4,x^4),...\}=\{(n,x^n): n \in \mathbb{N}, x\in \mathbb{R}\} \]
Let us plot the first three functions of this sequence:
Show the code
where each element of the sequence is a function.
Determinant
The determinant of a matrix is a function that assigns each matrix a number. In our notation, we write this as
\[ f=\{(M,\text{det}(M)): M \text{ is a square matrix}\} \]
Matrix
A matrix itself is also a function that assigns each pair of indices \((i,j)\) where \(i,j \in \{1,2,...,n\}\) the element of the matrix at position \((i,j)\). In our notation, we write this as
\[ M=\{((i,j),M_{ij}): i,j \in \{1,2,...,n\}\} \]
for convenience, we write this as a two-dimensional array:
\[ M=\begin{bmatrix} M_{11} & M_{12} & \cdots & M_{1n} \\ M_{21} & M_{22} & \cdots & M_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ M_{n1} & M_{n2} & \cdots & M_{nn} \end{bmatrix} \]
Matrix addition
Matrix addition is a function that assigns each pair of matrices their sum. In our notation, we write this as
\[ f=\{((A,B),A+B): A,B \text{ are matrices of the same dimensions}\} \]
where \(A+B\) is the matrix whose elements are the sums of the elements of matrices \(A\) and \(B\), that is, for each \(i,j\) we have \[ (A+B)_{ij}=A_{ij}+B_{ij} \]
Python dictionary
A dictionary in Python is also a function that assigns each key a value. In our notation, we write this as:
Geogebra examples
Check the following examples in Geogebra:
- ax^3 + bx^2 + cx + d
- sin(ax + b)
- e^( -x^2 / s)
- x^2 + y^3 = 1
- ImplicitCurve(xx+yyy-1) which is the same as the previous one
- ImplicitCurve(sin(x) + cos(y) - 1)
Composition of functions
The composition of two functions \(f\) and \(g\) is written as \(f\circ g\). It is a function that assigns each \(x\) the value \((f\circ g)(x)=f(g(x))\). It is not hard to undestand mechanically what this means. We first apply \(g\) to \(x\) and then apply \(f\) to the result:
\[ x\mapsto g(x) \mapsto f(g(x)) \]
Example 1
Let \(f(x)=x^2\) and \(g(x)=x+1\). Then the composition of these functions is
\[ (f\circ g)(x)=f(g(x))=f(x+1)=(x+1)^2 \]
Example 2
Let \(f(x)=\sin(x)\) and \(g(x)=x^2\). Then the composition \(f\circ g\) is
\[ (f\circ g)(x)=f(g(x))=f(x^2)=\sin(x^2) \]
but \(g\circ f\) is
\[ (g\circ f)(x)=g(f(x))=g(\sin(x))=\sin^2(x) \]
From these examples we see that the composition of functions is not commutative, that is, in general \(f\circ g \neq g\circ f\).
Below is a Python code that visualizes the composition of two functions \(f\) and \(g\).
Show the code
import numpy as np
import matplotlib.pyplot as plt
def gf(x):
return np.sin(x**2)
def fg(x):
return np.sin(x)**2
fig, ax = plt.subplots(figsize=(6,4))
x=np.linspace(-6,6,1000)
plt.plot(x,gf(x),label='g(f(x))')
plt.plot(x,fg(x),label='f(g(x))')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend()
plt.show()
Inverse functions
A function is essentially nothing more than a set of ordered pairs with additional properties that for each ‘\(x\)’ there is exactly one ‘\(y\)’. Given this, we might ask: if we reverse the elements in these pairs, will we still have a function? We will certainly end up with a relation, but it may not necessarily be a function again. For it to remain a function, the original function must be injective!
Below we will show few examples of inverse functions.
\[ \begin{align*} f &= \{(1,2), (2,3), (3,4)\} \longrightarrow f^{-1} = \{(2,1), (3,2), (4,3)\} \\ f &= \{(1,2), (2,2), (3,4)\} \longrightarrow f^{-1} = \{(2,1), (2,2), (4,3)\} \quad {\color{red}\text{not a function}} \\ f &= \left\{ \left( n, \frac{1}{n} \right); n \in \mathbb{N} \right\} \longrightarrow f^{-1} = \left\{ \left( \frac{1}{n}, n \right); n \in \mathbb{N} \right\} \end{align*} \]
Now we will consider very important case. We will try to reapeat above reasoning for the function \(f(x)=x^2\). \[ \begin{align*} f &= \{(x, x^2); x \in \mathbb{R}\} \longrightarrow f^{-1} = \{(x^2, x); x \in \mathbb{R}\} \quad {\color{red}\text{not a function}} \end{align*} \]
Now check how this works by writing few elements of the function \(f\) and its inverse function \(f^{-1}\).
\[ \begin{align*} \{{\color{red}(-2, 4)}, {\color{blue}(-1, 1)}, {\color{green}(0, 0)}, {\color{orange}(1, 1)}, {\color{purple}(2, 4)}\} \longrightarrow \{{\color{red}(4, -2)}, {\color{blue}(1, -1)}, {\color{green}(0, 0)}, {\color{orange}(1, 1)}, {\color{purple}(4, 2)}\} \end{align*} \]
We see that the element \(1\) is associated with two different elements in the inverse function. This is because the function \(f(x)=x^2\) is not injective. Therefore, we cannot define its inverse function. So this function is not invertible.
If we consider the formula \(f(x)=x^2\) on the domain \([0,\infty)\), then the function becomes injective and we can define its inverse function. \[ \begin{align*} f &= \{(x, x^2); x \in [0, \infty)\} \longrightarrow f^{-1} = \{(x^2, x); x \in [0, \infty)\}= \{(y, \sqrt{y}); y \in [0, \infty)\} \end{align*} \]
Let us use python to visualize this:
Show the code
import numpy as np
import matplotlib.pyplot as plt
def f(x):
return x**2
def f_inv(x):
return np.sqrt(x)
fig, ax = plt.subplots(figsize=(10,10))
x=np.linspace(-3,3,100)
x_plus=np.linspace(0,3,100)
plt.plot(x,f(x),label='f(x)=x^2')
plt.plot(x_plus,f(x_plus),label='f(x)=x^2 for x>=0')
plt.plot(x_plus,f_inv(x_plus),label='f^{-1}(x)=\sqrt{x}')
# add -- line y=x
plt.plot(x,x,label='y=x',linestyle='--')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.ylim(-1,4)
plt.grid()
plt.legend()
plt.gca().set_aspect('equal', adjustable='box')
plt.show()<>:15: SyntaxWarning:
invalid escape sequence '\s'
<>:15: SyntaxWarning:
invalid escape sequence '\s'
/tmp/ipykernel_53971/1499674410.py:15: SyntaxWarning:
invalid escape sequence '\s'
Additional resources
Limits of functions
Sequences (\(\mathbb{N}\rightarrow\mathbb{R}\))
Do not forget a sequence is a function that assigns each natural number a value, whcih we write as
\[\{(n,a_n): n \in \mathbb{N}\}=\{(0, a_0), (1, a_1), (2, a_2), ...\}\]
The argument is most often omitted, so we write this as
\[\{a_n\}=\{a_0, a_1, a_2, ...\}\]
Standard examples of sequences are:
Arithmetic sequence
Consider:
- \(a_0\)
- \(a_1=a_0+r\)
- \(a_2=a_1+r=a_0+2r\)
- \(a_3=a_2+r=a_0+3r\)
- \(a_4=a_3+r=a_0+4r\)
- …
- \(a_n=a_{n-1}+r = a_0+nr\)
- …
The n-th term for an arithmetic sequence is \[a_n=a_0+nr\]. Example: \(a_n=2+3n\) giving the sequence \(2,5,8,11,...\).
Geometric sequence
Consider:
- \(a_0\)
- \(a_1=a_0\cdot q\)
- \(a_2=a_1\cdot q=a_0\cdot q^2\)
- \(a_3=a_2\cdot q=a_0\cdot q^3\)
- \(a_4=a_3\cdot q=a_0\cdot q^4\)
- …
- \(a_n=a_{n-1}\cdot q = a_0\cdot q^n\)
- …
The n-th term for a geometric sequence is \[a_n=a_0\cdot q^n\]. Example: \(a_n=2\cdot 3^n\) giving the sequence \(2,6,18,54,...\).
Fibbonacci sequence
Consider:
- \(a_0=0\)
- \(a_1=1\)
- \(a_2=a_1+a_0=1\)
- \(a_3=a_2+a_1=2\)
- \(a_4=a_3+a_2=3\)
- \(a_5=a_4+a_3=5\)
- …
- \(a_n=a_{n-1}+a_{n-2}\)
- …
The n-th term for a Fibbonacci sequence is \[a_n=a_{n-1}+a_{n-2}\].
Example: \(a_n=a_{n-1}+a_{n-2}\) giving the sequence \(0,1,1,2,3,5,8,...\).
See link for more examples of sequences: Oeis.org
Limits of sequences
An idea of a limit of sequence is very simple. It is a number that the sequence approaches as we take more and more elements of the sequence. For example, the sequence \(a_n=\frac{1}{n}\) approaches 0 as we take more and more elements of the sequence. We write this as
\[ \lim_{n \to \infty} a_n=0 \]
Formal definition of a limit of a sequence is as follows:
Definition: Limit of a sequence
A number \(g\) is the limit of a sequence \(a_n\) if for every \(\varepsilon>0\) there exists \(N\) such that for every \(n>N\) we have \(|a_n-g|<\epsilon\). We write this as \[\lim_{n \to \infty} a_n=g\].
Show the code
import numpy as np
import matplotlib.pyplot as plt
# Function definition
def a(n):
return 3 * n**2 / (n**2 + 1)
# Parameters for visualization
epsilon = 0.1 # Epsilon value to represent the "margin" around the limit
N = 6 # Chosen N for the definition of limit
# Generate data points
n = np.arange(1, 20) # Range of n values for plotting
a_values = a(n) # Compute the function values
# Compute the limit for large n (L = 3)
limit = 3
# Create the plot
fig, ax = plt.subplots(figsize=(8, 5))
plt.scatter(n, a_values, label="Sequence $a(n) = \\frac{3n^2}{n^2 + 1}$", color="blue") # Sequence points
plt.axhline(y=limit, color="green", linestyle="-", label="Limit $L = 3$") # The limit line
plt.axhline(y=limit + epsilon, color="red", linestyle="--", label="$L + \\varepsilon$") # Upper epsilon bound
plt.axhline(y=limit - epsilon, color="red", linestyle="--", label="$L - \\varepsilon$") # Lower epsilon bound
plt.axvline(x=N, color="orange", linestyle="--", label="$N = 10$") # Vertical line for N
# Annotate the plot
plt.title("Visualization of Limit of $a(n)$ as $n \\to \\infty$")
plt.xlabel("$n$")
plt.ylabel("$a(n)$")
plt.legend(loc="lower right")
plt.grid(True)
# Display the plot
plt.show()
Some important points to note about limits of sequences are:
- The limit of a sequence is unique. This means that if a sequence has a limit, then this limit is unique.
- If a sequence has a limit, then the sequence is bounded. This means that there exists a number \(M\) such that \(|a_n| \leq M\) for all \(n\).
To compute the limit of a sequence, we often use the following rules:
- The limit of a sum of sequences is the sum of the limits of the sequences.
\[ \lim_{n \to \infty} (a_n\pm b_n)=\lim_{n \to \infty} a_n\pm\lim_{n \to \infty} b_n \]
- The limit of a product of sequences is the product of the limits of the sequences.
\[ \lim_{n \to \infty} (a_n \cdot b_n)=\lim_{n \to \infty} a_n \cdot \lim_{n \to \infty} b_n \]
- The limit of a quotient of sequences is the quotient of the limits of the sequences.
\[ \lim_{n \to \infty} \frac{a_n}{b_n}=\frac{\lim\limits_{n \to \infty} a_n}{\lim\limits_{n \to \infty} b_n} \]
- The limit of a sequence raised to a power is the limit of the sequence raised to the power.
\[ \lim_{n \to \infty} a_n^k=(\lim_{n \to \infty} a_n)^k \]
The Squeeze Theorem
If \(a_n \leq b_n \leq c_n\) for all \(n\) and \(\lim\limits_{n \to \infty} a_n=\lim\limits_{n \to \infty} c_n=g\), then \(\lim\limits_{n \to \infty} b_n=g\).
Interpretation of the Squeeze Theorem is that if a sequence is squeezed between two sequences that have the same limit, then the sequence itself has the same limit.
Example
Let:
\[ a_n=-\frac{1}{n}, \quad b_n=\frac{\sin(n)}{n} , \quad c_n=\frac{1}{n} \]
Then we have \(a_n \leq b_n \leq c_n\) for all \(n\). We also have
\[ \lim_{n \to \infty} a_n=\lim_{n \to \infty} c_n=0 \]
Therefore, by the Squeeze Theorem, we have
\[ \lim_{n \to \infty} \frac{\sin(n)}{n}=0 \]
Show the code
import numpy as np
import matplotlib.pyplot as plt
def a(n):
return -1/n
def b(n):
return np.sin(n)/n
def c(n):
return 1/n
fig, ax = plt.subplots(figsize=(10,6))
n=np.arange(1,40)
plt.scatter(n,a(n),label='a(n)')
plt.scatter(n,b(n),label='b(n)')
plt.scatter(n,c(n),label='c(n)')
plt.xlabel('n')
plt.ylabel('f(n)')
plt.legend()
plt.show()
Limits of real functions (\(\mathbb{R}\rightarrow\mathbb{R}\))
Formal Definition
Definition: Let \(f(x)\) be a function defined on some open interval containing \(c\), except possibly at \(c\) itself. We say that
\[ \lim_{x \to c} f(x) = L \]
if for every \(\varepsilon > 0\) there exists a \(\delta > 0\) such that whenever \(0 < |x - c| < \delta\), we have
\[ |f(x) - L| < \varepsilon. \]
Geogebra:
Definition
Show the code
# Import necessary libraries
import numpy as np
import matplotlib.pyplot as plt
# Function Definitions
def func_1(x):
return np.sin(x)
def func_2(x):
return np.sin(1 / x)
# Parameters for f(x) = sin(x) at x = π
epsilon_1 = 0.2 # Chosen epsilon
delta_1 = 0.15 # Chosen delta
target_x_1 = np.pi
target_y_1 = func_1(target_x_1)
# Parameters for f(x) = sin(1/x) near x = 0
epsilon_2 = 0.2 # Chosen epsilon
delta_2 = 0.1 # Chosen delta
target_x_2 = 0
# Generate x-values
x_values_1 = np.linspace(target_x_1 - 2, target_x_1 + 2, 500)
delta_range_1 = np.linspace(target_x_1 - delta_1, target_x_1 + delta_1, 500)
x_values_2 = np.linspace(-0.5, 0.5, 1000)
x_values_2 = x_values_2[x_values_2 != 0] # Exclude zero for sin(1/x)
delta_range_2 = np.linspace(-delta_2, delta_2, 500)
delta_range_2 = delta_range_2[delta_range_2 != 0] # Exclude zero
# Calculate function values
func_values_1 = func_1(x_values_1)
delta_func_values_1 = func_1(delta_range_1)
func_values_2 = func_2(x_values_2)
delta_func_values_2 = func_2(delta_range_2)
# Epsilon bands
y_upper_1 = target_y_1 + epsilon_1
y_lower_1 = target_y_1 - epsilon_1
y_upper_2 = epsilon_2
y_lower_2 = -epsilon_2
# Create side-by-side plots
fig, axs = plt.subplots(2,1, figsize=(10, 8))
# First plot: f(x) = sin(x), epsilon-delta
axs[0].plot(x_values_1, func_values_1, label="f(x) = sin(x)", color="blue")
axs[0].axvline(target_x_1, color="red", linestyle="--", label=f"x = π (Target)")
axs[0].axvline(target_x_1 - delta_1, color="green", linestyle="--", label=f"x = π - δ")
axs[0].axvline(target_x_1 + delta_1, color="green", linestyle="--", label=f"x = π + δ")
axs[0].hlines([y_upper_1, y_lower_1], target_x_1 - 2, target_x_1 + 2, colors="orange", linestyles="--", label="f(x) ± ε")
axs[0].scatter([target_x_1], [target_y_1], color="red", s=100, label=f"Target Point (π, {round(target_y_1, 2)})")
axs[0].set_title("Epsilon-Delta for f(x) = sin(x) at x = π")
axs[0].set_xlabel("x")
axs[0].set_ylabel("f(x)")
axs[0].grid(alpha=0.4)
axs[0].legend()
# Second plot: f(x) = sin(1/x), epsilon-delta near zero
axs[1].plot(x_values_2, func_values_2, label="f(x) = sin(1/x)", color="blue")
axs[1].axvline(-delta_2, color="green", linestyle="--", label=f"x = -δ")
axs[1].axvline(delta_2, color="green", linestyle="--", label=f"x = δ")
axs[1].hlines([y_upper_2, y_lower_2], -0.5, 0.5, colors="orange", linestyles="--", label="f(x) ± ε")
axs[1].scatter([0], [0], color="red", s=100, label=f"Target Point (0, 0)")
axs[1].set_title("Epsilon-Delta for f(x) = sin(1/x) near x = 0")
axs[1].set_xlabel("x")
axs[1].set_ylabel("f(x)")
axs[1].grid(alpha=0.4)
axs[1].legend()
# Adjust layout and show
plt.tight_layout()
plt.show()
The limit of a function describes the behavior of the function as its input approaches a certain value. We denote it as:
\[ \lim_{x \to c} f(x) = L \]
This means that as \(x\) approaches \(c\), the values of \(f(x)\) get arbitrarily close to \(L\).
One-Sided Limits
Limits can also be evaluated as \(x\) approaches \(c\) from one side:
Left-hand limit: If \(x\) approaches \(c\) from the left (\(x \to c^-\)), then: \[ \lim_{x \to c^-} f(x) = L \]
Right-hand limit: If \(x\) approaches \(c\) from the right (\(x \to c^+\)), then: \[ \lim_{x \to c^+} f(x) = L \]
Show the code
The two-sided limit exists if and only if: \[ \lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L. \]
Properties of Limits
For functions \(f(x)\) and \(g(x)\) with limits \(\lim_{x \to c} f(x) = L_1\) and \(\lim_{x \to c} g(x) = L_2\):
Sum: \[ \lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x) = L_1 + L_2 \]
Product: \[ \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) = L_1 \cdot L_2 \]
Quotient (if \(\lim_{x \to c} g(x) \neq 0\)): \[ \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim\limits_{x \to c} f(x)}{\lim\limits_{x \to c} g(x)} = \frac{L_1}{L_2} \]
Power: \[ \lim_{x \to c} [f(x)]^k = \left(\lim_{x \to c} f(x)\right)^k = L_1^k, \quad k \in \mathbb{R}. \]
Limits at Infinity
If there is a number \(L\) such that for every \(\varepsilon > 0\) there exists a number \(M\) such that for all \(x > M\) we have \(|f(x) - L| < \varepsilon\), then we write: \[ \lim_{x \to \infty} f(x) = L \]
Similarly, if there is a number \(L\) such that for every \(\varepsilon > 0\) there exists a number \(M\) such that for all \(x < M\) we have \(|f(x) - L| < \varepsilon\), then we write:
\[ \lim_{x \to -\infty} f(x) = L \]
Asymptotes
In mathematics, functions often exhibit interesting behavior as their inputs grow very large or very small. One such phenomenon is when a function approaches a straight line, called an asymptote, as \(x\) becomes large.
If a function \(f(x)\) approaches a line \(y = ax + b\) as \(x\) grows, we say the function has an oblique asymptote defined by the equation:
\[ f(x) \sim ax + b \qquad \text{for large } x, \]
which formally means:
\[ \lim_{x \to \infty} \big(f(x) - (ax + b)\big) = 0. \]
To find the slope \(a\) of the asymptote:
\[ a = \lim_{x \to \infty} \frac{f(x)}{x}. \]
To find the intercept \(b\):
\[ b = \lim_{x \to \infty} \big(f(x) - ax\big). \]
Example: Find the asymptotes of the function \(f(x)=\frac{x^3+3x^2+1}{3x^2}\)
First, we find the slope of the asymptote:
\[ a=\lim_{x \to \infty} \frac{x^3+3x^2+1}{3x^2\cdot x} = \lim_{x \to \infty} \frac{1}{3} \]
Next, we find the intercept of the asymptote:
\[ \begin{align*} b &=\lim_{x \to \infty} \left(\frac{x^3+3x^2+1}{3x^2}-\frac{1}{3}x\right)\\ &=\lim_{x \to \infty} \left(\frac{x^3+3x^2+1-x^3}{3x^2}\right)\\ &=\lim_{x \to \infty} \left(\frac{3x^2+1}{3x^2}\right)\\ &=\lim_{x \to \infty} \left(1+\frac{1}{3x^2}\right)\\ &=1 \end{align*} \]
Therefore, the asymptote of the function \(f(x)=\frac{x^3+3x^2+1}{3x^2}\) is \(y=\frac{1}{3}x+1\).
Show the code
import numpy as np
import matplotlib.pyplot as plt
def f(x):
return (x**3+3*x**2+1)/(3*x**2)
def line(x):
return 1/3*x+1
fig, ax = plt.subplots(figsize=(6,4))
x=np.linspace(-10,10,100)
plt.plot(x,f(x),label='f(x)')
plt.plot(x,line(x),label='asymptote')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend()
plt.ylim(-10,10)
plt.show()
Countinuity
Definition
A function \(f(x)\) is continuous at a point \(x = a\) if the following three conditions are satisfied:
\(f(a)\) is defined.
This means that the function must have a value at \(x = a\). If \(f(a)\) is not defined, the function cannot be continuous at that point.\(\lim\limits_{x \to a} f(x)\) exists.
The limit of \(f(x)\) as \(x\) approaches \(a\) must exist. This ensures that the function does not have a “jump” or oscillation around \(x = a\).\(\lim\limits_{x \to a} f(x) = f(a)\).
The value of the limit as \(x\) approaches \(a\) must equal the actual value of the function at \(x = a\). This condition guarantees that there is no “hole” or discontinuity at the point.
If any of these conditions fail, the function is not continuous at \(x = a\).
Intuition
Continuity intuitively means that the graph of a function can be drawn without lifting the pen from the paper. For instance:
- Polynomials like \(f(x) = x^2 + 3x + 2\) are continuous everywhere.
- Piecewise functions may fail to be continuous if there is a jump, hole, or break in their definition.
Types of Discontinuities
When a function is not continuous at \(x = a\), we classify the discontinuity into three types:
Removable Discontinuity:
Occurs when \(\lim\limits_{x \to a} f(x)\) exists, but \(f(a)\) is not defined or differs from the limit.Example:
\[f(x) = \frac{x^2 - 1}{x - 1}\qquad x\in\mathbb{R}\setminus\{1\}\]
By computing the limit, we find that \(\lim\limits_{x \to 1} f(x) = 2\), so the discontinuity at \(x = 1\) is removable by setting \(f(1) = 2\).
\[ f(x) = \begin{cases} \frac{x^2 - 1}{x - 1}, & x \neq 1, \\ 2, & x = 1. \end{cases} \]
Jump Discontinuity:
Occurs when the left-hand limit \(\lim\limits_{x \to a^-} f(x)\) and right-hand limit \(\lim\limits_{x \to a^+} f(x)\) exist but are not equal.
Example: A piecewise function with different definitions on either side of \(x = a\).Infinite Discontinuity:
Occurs when \(f(x)\) approaches infinity or negative infinity as \(x\) approaches \(0\).
Example: \(f(x) = \frac{1}{x}\) has an infinite discontinuity at \(x = 0\).
Derivatives
Definition
The derivative of a function \(f(x)\) at a point \(x=a\) is defined as:
\[ f'({\color{red}a}) = \lim_{{\color{blue}h} \to 0} \frac{f({\color{red}a}+{\color{blue}h}) - f({\color{red}a})}{{\color{blue}h}} \] This limit represents the slope of the tangent line \(y\) to the curve \(f(x)\) at the point \((a, f(a))\).
\[ y=f'(a)(x-a)+f(a) \]
Show the code
import numpy as np
import matplotlib.pyplot as plt
def f(x):
return x**2
def f_prime(x):
return 2*x
def tangent_line(x, a):
return f_prime(a)*(x-a) + f(a)
fig, ax = plt.subplots(figsize=(6,4))
x=np.linspace(-10,10,100)
plt.plot(x,f(x),label='f(x)')
plt.plot(x,tangent_line(x,2),label='tangent line at x=2')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend()
plt.scatter(2,f(2),color='red')
plt.show()
Geogebra implementation
In input bar write the following commands:
- f(x)=x^2
- a=2
- P=(a,f(a))
- h=Slider[0, 1, 0.01]
- Q=(a+h, f(a+h))
- T=Line(P,Q)
Interpretation
Slope of the Tangent Line:
The derivative is the slope of the line that just “touches” the curve at a single point without crossing it (locally). This tangent line provides a linear approximation to the function near that point.Instantaneous Rate of Change:
If \(f(x)\) represents position, then \(f'(x)\) represents velocity—how quickly the position is changing at a specific moment.Geometric Insight:
At points where the derivative is:- Positive (\(f'(x) > 0\)): The function is increasing.
- Negative (\(f'(x) < 0\)): The function is decreasing.
- Zero (\(f'(x) = 0\)): The function has a horizontal tangent line, potentially a local maximum, minimum, or inflection point.
- Positive (\(f'(x) > 0\)): The function is increasing.
Practical Example:
Consider the function \(f(x) = x^2\). The derivative \(f'(x) = 2x\) tells us that the slope of the curve increases as \(x\) increases, meaning the curve becomes steeper as you move away from the origin.
Examplea
Sinus
Let \(f(x)=\sin(x)\). Then the derivative of \(f(x)\) at \(x=a\) is \(f'(a)=\cos(a)\). Now we will prove this.
Proof: From trigonometry, we know that \(\sin(a+h)=\sin(a)\cos(h)+\cos(a)\sin(h)\). Therefore, we have
\[ \begin{align*} f'({\color{red}a}) &=\lim_{{\color{blue}h} \to 0} \frac{\sin({\color{red}a}+{\color{blue}h})-\sin({\color{red}a})}{{\color{blue}h}}\\ &=\lim_{{\color{blue}h} \to 0} \frac{\sin({\color{red}a})\cos({\color{blue}h})+\cos({\color{red}a})\sin({\color{blue}h})-\sin({\color{red}a})}{{\color{blue}h}}\\ &=\lim_{{\color{blue}h} \to 0} \frac{\sin({\color{red}a})(\cos({\color{blue}h})-1)+\cos({\color{red}a})\sin({\color{blue}h})}{{\color{blue}h}}\\ &=\cos({\color{red}a})\lim_{{\color{blue}h} \to 0} \frac{\sin({\color{blue}h})}{{\color{blue}h}}\\ &=\cos({\color{red}a}) \end{align*} \]
Polyonomial
Let \(f(x)=x^3\). Then the derivative of \(f(x)\) at \(x=a\) is \(f'(a)=3a^2\). Now we will prove this.
Proof: We have
\[ \begin{align*} f'({\color{red}a})&= \lim_{{\color{blue}h} \to 0} \frac{({\color{red}a}+{\color{blue}h})^3-{\color{red}a}^3}{{\color{blue}h}}\\ &=\lim_{{\color{blue}h} \to 0} \frac{{\color{red}a}^3+3{\color{red}a}^2{\color{blue}h}+3{\color{red}a}{\color{blue}h}^2+{\color{blue}h}^3-{\color{red}a}^3}{{\color{blue}h}}\\ &=\lim_{{\color{blue}h} \to 0} 3{\color{red}a}^2+3{\color{red}a}{\color{blue}h}+{\color{blue}h}^2\\ &=3{\color{red}a}^2 \end{align*} \]
General Polynomial
Let \(f(x)=x^n\). Then the derivative of \(f(x)\) at \(x=a\) is \(f'(a)=na^{n-1}\). Now we will prove this.
Proof: We have
\[ \begin{align*} f'({\color{red}a})&= \lim_{{\color{blue}h} \to 0} \frac{({\color{red}a}+{\color{blue}h})^n-{\color{red}a}^n}{{\color{blue}h}}\\ &=\lim_{{\color{blue}h} \to 0} \frac{{\color{red}a}^n+n{\color{red}a}^{n-1}{\color{blue}h}+\binom{n}{2}{\color{red}a}^{n-2}{\color{blue}h}^2+\ldots+{\color{blue}h}^n-{\color{red}a}^n}{{\color{blue}h}}\\ &=\lim_{{\color{blue}h} \to 0} n{\color{red}a}^{n-1}+\binom{n}{2}{\color{red}a}^{n-2}{\color{blue}h}+\ldots+{\color{blue}h}^{n-1}\\ &=n{\color{red}a}^{n-1} \end{align*} \]
Check the link: Binomial theorem
Basic Derivatives
| Function \(f(x)\) | Derivative \(f'(x)\) |
|---|---|
| \(c\) | \(0\) |
| \(x\) | \(1\) |
| \(x^n\) | \(nx^{n-1}\) |
| \(e^x\) | \(e^x\) |
| \(a^x\) | \(a^x \ln(a)\) |
| \(\ln(x)\) | \(\frac{1}{x}\) |
| \(\log_a(x)\) | \(\frac{1}{x \ln(a)}\) |
| \(\sin(x)\) | \(\cos(x)\) |
| \(\cos(x)\) | \(-\sin(x)\) |
Basic Theorems
Sum or Difference of Derivatives:
\[ ({\color{red}f} \pm {\color{blue}g})'(x) = {\color{red}f}'(x) \pm {\color{blue}g}'(x) \]Product of Derivatives (Product Rule):
\[ ({\color{red}f} \cdot {\color{blue}g})'(x) = {\color{red}f}'(x) {\color{blue}g}(x) + {\color{red}f}(x) {\color{blue}g}'(x) \]Quotient of Derivatives (Quotient Rule):
\[ \left(\frac{{\color{red}f}}{{\color{blue}g}}\right)'(x) = \frac{{\color{red}f}'(x) {\color{blue}g}(x) - {\color{red}f}(x) {\color{blue}g}'(x)}{{\color{blue}g}^2(x)} \]Derivative of a Power Function (Power Rule):
\[ ({\color{red}f}^n)'(x) = n {\color{red}f}'(x) {\color{red}f}^{n-1}(x) \]
Examples
Second example
\[ \begin{align*} ({\color{blue}{\cos(x)}}{\color{red}{e^x}})' &= ({\color{blue}{\cos(x)}})' {\color{red}{e^x}} + {\color{blue}{\cos(x)}} ({\color{red}{e^x}})' \\[5pt] &= {\color{blue}{-\sin(x)}} {\color{red}{e^x}} + {\color{blue}{\cos(x)}} {\color{red}{e^x}} \\[5pt] &= ({\color{blue}{\cos(x)}} - {\color{blue}{\sin(x)}}) {\color{red}{e^x}} \end{align*} \]
Third example
\[ \begin{align*} \tan(x) &= \left(\frac{\color{blue}{\sin(x)}}{\color{red}{\cos(x)}}\right)' \\[5pt] &= \frac{(\color{blue}{\sin(x)})'(\color{red}{\cos(x)}) - (\color{red}{\cos(x)})'(\color{blue}{\sin(x)})}{(\color{red}{\cos(x)})^2} \\[5pt] &= \frac{\color{blue}{\cos(x)}\color{red}{\cos(x)} - (\color{red}{-\sin(x)})(\color{blue}{\sin(x)})}{(\color{red}{\cos(x)})^2} \\[5pt] &= \frac{\color{red}{\cos^2(x)} + \color{blue}{\sin^2(x)}}{\color{red}{\cos^2(x)}} \\[5pt] &= \frac{1}{\color{red}{\cos^2(x)}} \end{align*} \]
Chain Rule
This is a very important rule that allows us to differentiate composite functions. If \({\color{red}f}(x)={\color{blue}g}({\color{green}h}(x))\), then \[ {\color{red}f}'(x)={\color{blue}g}'({\color{green}h}(x)){\color{green}h}'(x) \]
First example
Let \({\color{red}f}(x)={\color{blue}\sin}({\color{green}x^2})\). Then
- \({\color{blue}g}(y)=\sin(y) \Rightarrow {\color{blue}g}'(y)=\cos(y)\)
- \({\color{green}h}(x)=x^2 \Rightarrow {\color{green}h}'(x)=2x\)
so
\[ \begin{align*} {\color{red}f}'(x)&={\color{blue}g}'({\color{green}h}(x)){\color{green}h}'(x)=\cos(x^2)\cdot 2x \end{align*} \]
Second example
Let \(f(x)=e^{x^2+\cos(x)}\). Then
- \(g(y)=e^y \Rightarrow g'(y)=e^y\)
- \(h(x)=x^2+\cos(x) \Rightarrow h'(x)=2x-\sin(x)\)
so
\[ \begin{align*} f'(x)&=g'(h(x))h'(x)=e^{x^2+\cos(x)}\cdot (2x-\sin(x)) \end{align*} \]
Higher Order Derivatives
Definition of derivative of a function \(f(x)\) at a point \(x=a\) is a number \(f'(a)\). We can define new function \(f'(x)\) by
\[ f'=\{(x,f'(x))\} \]
for \(x\) in the domain of \(f\) for which \(f'(x)\) is defined. This function is called the first derivative of \(f\). We can then define the second derivative of \(f\) as the first derivative of \(f'\), and so on.
- Second derivative: \((f')'(x)=f''(x)\)
- Third derivative: \((f'')'(x)=f'''(x)\)
- …
Geogebra implementation
In input bar write the following commands:
- f(x)=x3+2x2-4 x
- f1(x)=Derivative[f(x),x]
- f2(x)=Derivative[f1(x),x]
- f3(x)=Derivative[f2(x),x]
Finding Extrema
Extrema of a function are points where the function reaches its maximum or minimum values, either locally (in a neighborhood) or globally (on the entire domain). To find extrema, we use the following process:
Steps to Find Extrema
Compute the Derivative:
Find the first derivative \(f'(x)\) of the function \(f(x)\). The derivative represents the slope of the tangent line.Set the Derivative to Zero:
Solve the equation \(f'(x) = 0\). These points are called critical points, where the function’s slope is horizontal.Check for Undefined Points:
Identify points where \(f'(x)\) does not exist but \(f(x)\) is still defined. These can also be critical points.Use the Second Derivative Test (Optional):
Compute the second derivative \(f''(x)\) to determine the nature of each critical point:- If \(f''(x) > 0\), the function has a local minimum at that point.
- If \(f''(x) < 0\), the function has a local maximum at that point.
- If \(f''(x) = 0\), the test is inconclusive. Use other methods, such as analyzing the function’s behavior.
Boundary Points (if the domain is restricted):
For a function defined on a closed interval \([a, b]\), evaluate \(f(a)\) and \(f(b)\), as extrema can occur at the boundaries.
Necessary Conditions for Extrema
- A point \(x=c\) is a candidate for an extremum if:
- \(f'(c) = 0\) (critical point), or
- \(f'(c)\) is undefined but \(f(c)\) is defined.
- A point is a local extremum only if it satisfies the conditions above and meets additional criteria (e.g., second derivative test or sign change in \(f'(x)\)).
Example
Consider the function \(f(x) = x^3 - 3x^2 + 4\):
Compute the first derivative:
\(f'(x) = 3x^2 - 6x\).Solve \(f'(x) = 0\):
\(3x^2 - 6x = 0\)
\(x(x-2) = 0\)
Critical points are \(x = 0\) and \(x = 2\).Compute the second derivative:
\(f''(x) = 6x - 6\).- At \(x = 0\): \(f''(0) = -6\) (local maximum).
- At \(x = 2\): \(f''(2) = 6\) (local minimum).
- At \(x = 0\): \(f''(0) = -6\) (local maximum).
Evaluate boundaries (if applicable):
Not relevant here, as the domain is not restricted.
Conclusion
Extrema occur at \(x = 0\) (local maximum) and \(x = 2\) (local minimum). Use this structured approach to identify extrema systematically.
Applications in Computer Science
Finding extrema plays a critical role in computer science, particularly in solving optimization problems. Optimization involves determining the maximum or minimum value of a function, often under specific constraints. Here are some key applications:
1. Machine Learning
In machine learning, training models often involves minimizing a loss function, which measures the error between the predicted and actual values. For example:
- Gradient Descent: An optimization algorithm that uses derivatives to find the minimum of a function (e.g., loss function).
- Convergence to a Global Minimum: Finding extrema ensures the model learns the best possible parameters.
2. Data Analysis and Fitting
In data analysis, functions are fit to data points (e.g., regression models) by minimizing error metrics. The derivative helps identify the optimal parameters for the function.
3. Algorithm Design
Many algorithms in computer science involve optimization:
- Pathfinding Algorithms: Algorithms like Dijkstra’s and A* aim to minimize the cost of reaching a destination.
- Scheduling Problems: Finding the minimum time or resources required to complete tasks.
4. Operations Research
In operations research, optimization is used to solve problems involving resource allocation, scheduling, and logistics. For example:
- Linear Programming: Derivatives are used in the simplex method to maximize or minimize objective functions.
- Nonlinear Optimization: Extrema are crucial in finding optimal solutions to more complex problems.
5. Artificial Intelligence
In AI, optimization problems arise in areas like:
- Game Theory: Finding optimal strategies for agents by maximizing rewards or minimizing losses.
- Reinforcement Learning: Optimizing policies to maximize cumulative rewards over time.
Conclusion
The ability to find extrema is fundamental to numerous optimization problems in computer science, enabling efficient algorithms and solutions across various domains, from machine learning to operations research.
Taylor Series
Definition
The Taylor series of a function \(f(x)\) at a point \(x=a\) is an infinite sum of terms that are calculated from the derivatives of \(f(x)\) at \(x=a\). It is given by:
\[ \begin{align*} f(x) &= f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \cdots+ \frac{f^{n-1}(a)}{(n-1)!}(x-a)^{n-1}+\ldots\\ &=\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n \end{align*} \]
How to read this
First approximation
\[ \begin{align*} f(x) &\approx f(a) + f'(a)(x-a) \end{align*} \]
Second approximation
\[ \begin{align*} f(x) &\approx f(a) + f'(a)(x-a)+\frac{f''(a)}{2!}(x-a)^2 \end{align*} \]
etc.
Example: Let \(f(x)=\sin(x)\). Then the Taylor series of \(f(x)\) at \(x=0\) is
\[ \begin{align*} f(x)&=f(0)+f'(0)x+\frac{f''(0)}{2!}x^2+\frac{f'''(0)}{3!}x^3+\cdots\\ &=0+x-\frac{1}{2}x^2-\frac{1}{6}x^3+\cdots\\ &=\sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}s \end{align*} \]
Now we will plot first 5 aproximations of the function \(f(x)=\sin(x)\) using Taylor series.
Show the code
import numpy as np
from scipy.special import factorial
import matplotlib.pyplot as plt
def f(x):
return np.sin(x)
def taylor_series(x, n):
result = 0
for i in range(n):
result += (-1)**i * x**(2*i+1) / factorial(2*i+1)
return result
fig, ax = plt.subplots(figsize=(10,6))
x = np.linspace(-2*np.pi, 2*np.pi, 1000)
plt.plot(x, f(x), label='sin(x)', color='blue', linestyle='-', linewidth=2)
for i in range(1, 10):
plt.plot(x, taylor_series(x, i), label=f'(n={i})', linestyle='--')
plt.ylim(-2, 2)
plt.xlabel('x')
plt.ylabel('f(x)')
plt.legend()
plt.show()
Do this in Geogebra!!!
Examples
\[ \begin{align*} e^x&=1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots = \sum_{n=0}^{\infty} \frac{x^n}{n!}\\ \sin(x)&=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\cdots= \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}\\ \cos(x)&=1-\frac{x^2}{2!}+\frac{x^4}{4!}-\cdots= \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}\\ \ln(1+x)&=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots= \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}\\ \frac{1}{1+x}&=1-x+x^2-x^3+\cdots= \sum_{n=0}^{\infty} (-1)^n x^n \end{align*} \]
Integrals
Rieman Integral
The Riemann integral of a function \(f(x)\) over the interval \([a,b]\) is defined as:
\[ \int_{a}^{b} f(x) \, dx = \lim_{n \to \infty} \sum_{i=1}^{n} f(x_i) \Delta x \]
where \(\Delta x = \frac{b-a}{n}\) and \(x_i = a + i \Delta x\).
Show the code
import numpy as np
import matplotlib.pyplot as plt
# Define the function to integrate
def f(x):
return x**2
# Define the interval [a, b]
a, b = 0, 2
# Number of rectangles
n = 10
# Calculate the width of each rectangle
dx = (b - a) / n
# Define the x values for the left endpoints
x = np.linspace(a, b, n, endpoint=False)
# Calculate the y values (height of rectangles)
y = f(x)
# Plot the function
x_fine = np.linspace(a, b, 500)
plt.plot(x_fine, f(x_fine), label='f(x) = x^2', color='blue')
# Draw the rectangles
for i in range(n):
plt.bar(x[i], y[i], width=dx, align='edge', edgecolor='black', color='orange', alpha=0.5)
# Add labels and title
plt.title('Riemann Sum Approximation')
plt.xlabel('x')
plt.ylabel('f(x)')
plt.axhline(0, color='black', linewidth=0.8)
plt.axvline(0, color='black', linewidth=0.8)
plt.legend()
plt.grid(alpha=0.3)
# Show the plot
plt.show()
GO TO GEOGEBRA!!!
Use Integral, LowerSum, UpperSum to show the Riemann sum!
Properties of Integrals
Linearity
\[ \int_{a}^{b} [f(x) \pm g(x)] \, dx = \int_{a}^{b} f(x) \, dx \pm \int_{a}^{b} g(x) \, dx \]
\[ \int_{a}^{b} c f(x) \, dx = c \int_{a}^{b} f(x) \, dx \]
\[ \int_{a}^{c} f(x) \, dx = \int_{a}^{b} f(x) \, dx+\int_{b}^{c} f(x) \, dx \quad \text{for } [a,c]=[a,b]\cup[b,c] \]
Change of Variables
If \(x=g(t)\) is a differentiable function on \([a,b]\) and \(f\) is continuous on the range of \(g\), then
\[ \int_{a}^{b} f(g(t))g'(t) \, dt = \int_{g(a)}^{g(b)} f(x) \, dx \]
Integration by Parts
\[ \int u \, dv = uv - \int v \, du \]
Wolfram Alpha
Fundamental Theorem of Calculus
Let \(f(x)\) be a continuous function on \([a,b]\). Then the function \(F(x)\) defined by
\[ F(x) = \{(x, \int_{a}^{x} f(t) \, dt)\} \]
where in books we write
\[ F(x)=\int_{a}^{x} f(t) \, dt \]
is continuous on \([a,b]\), differentiable on \((a,b)\), and
\[ F'(x)=f(x) \]
Proof
Let \(x_0 \in (a,b)\). Then
\[ \begin{align*} F'(x_0)&=\lim_{h \to 0} \frac{\color{red}{F(x_0+h)}-\color{blue}{F(x_0)}}{h}\\ &=\lim_{h \to 0} \frac{\color{red}{\int_{a}^{x_0+h} f(t) \, dt}- \color{blue}{\int_{a}^{x_0} f(t) \, dt}}{h}\\ &=\lim_{h \to 0} \frac{\color{red}{\int_{a}^{x_0} f(t) \, dt} +\int_{x_0}^{x_0+h} f(t) \, dt -\color{blue}{\int_{a}^{x_0} f(t) \, dt}}{h}\\ &=\lim_{h \to 0} \frac{\int_{x_0}^{x_0+h} f(t) \, dt}{h}\\ &=\lim_{h \to 0} \frac{f(x_0+\theta h)h}{h}\\ &=f(x_0) \end{align*} \]
Antiderivative (indeffinite integral)
In calculus, an antiderivative, primitive function, primitive integral, or indefinite integral of a function \(f\) is a differentiable function \(F\) whose derivative is equal to the original function \(f\). This can be stated symbolically as \(F'(x) = f(x)\). The process of solving for antiderivatives is called antidifferentiation (or indefinite integration), and its opposite operation is called differentiation, which is the process of finding a derivative.
| \(f(x)\) | \(F(x)\) |
|---|---|
| \(x^n\) | \(\frac{x^{n+1}}{n+1} + C\) |
| \(\frac{1}{x}\) | \(\ln|x| + C\) |
| \(e^x\) | \(e^x + C\) |
| \(\sin(x)\) | \(-\cos(x) + C\) |
| \(\cos(x)\) | \(\sin(x) + C\) |
Using Antiderivatives in Definite Integrals
The definite integral of a function \(f(x)\) over an interval \([a, b]\) is defined as:
\[ \int_a^b f(x) \, dx \]
According to the Fundamental Theorem of Calculus, if \(F(x)\) is an antiderivative of \(f(x)\) (i.e., \(F'(x) = f(x)\)), the value of the definite integral is given by:
\[ \int_a^b f(x) \, dx = F(b) - F(a) \]
This means that the definite integral can be computed using the antiderivative \(F(x)\) evaluated at the endpoints \(a\) and \(b\).
Step-by-Step Process
To calculate a definite integral using an antiderivative:
Find the Antiderivative:
Determine \(F(x)\) such that \(F'(x) = f(x)\).Evaluate at the Bounds:
Compute \(F(b)\) and \(F(a)\).Subtract:
The result of the integral is \(F(b) - F(a)\).
Example: Compute a Definite Integral
Let \(f(x) = 2x\), and compute the integral over \([1, 3]\):
\[ \int_1^3 2x \, dx \]
Find the Antiderivative:
\[ \int 2x \, dx = x^2 + C \]Evaluate at the Bounds:
- \(F(3) = 3^2 = 9\)
- \(F(1) = 1^2 = 1\)
Subtract:
\[ \int_1^3 2x \, dx = F(3) - F(1) = 9 - 1 = 8 \]
Thus, the total signed area under \(f(x)\) from \(x=1\) to \(x=3\) is \(8\).
Applications of Definite Integrals
Definite integrals are widely used to compute quantities such as area, volume, arc length, surface area, and volumes of solids of revolution. Below is a summary of the key formulas and methods for these applications:
1. Area Under a Curve
The area between a curve \(y = f(x)\) and the \(x\)-axis over an interval \([a, b]\) is given by:
\[ A = \int_a^b f(x) \, dx \]
- For areas below the \(x\)-axis, take the absolute value of the integral or split the integral into positive and negative parts.
- For areas between two curves \(y = f(x)\) and \(y = g(x)\): \[ A = \int_a^b \left[ f(x) - g(x) \right] \, dx \]
2. Volume of a Solid
To compute the volume of a solid with a cross-sectional area \(A(x)\) perpendicular to the \(x\)-axis:
\[ V = \int_a^b A(x) \, dx \]
- For solids of revolution (rotating around the \(x\)-axis): \[ V = \pi \int_a^b \left[ f(x) \right]^2 \, dx \]
3. Arc Length of a Curve
The length of a curve \(y = f(x)\) from \(x = a\) to \(x = b\) is:
\[ L = \int_a^b \sqrt{1 + \left[ f'(x) \right]^2} \, dx \]
- For parametric curves \((x(t), y(t))\): \[ L = \int_a^b \sqrt{\left[ \frac{dx}{dt} \right]^2 + \left[ \frac{dy}{dt} \right]^2} \, dt \]
4. Surface Area of a Solid of Revolution
The surface area of a solid generated by rotating a curve \(y = f(x)\) about the \(x\)-axis is:
\[ S = 2\pi \int_a^b f(x) \sqrt{1 + \left[ f'(x) \right]^2} \, dx \]
Practical Applications
- Physics:
Displacement from velocity:
If \(v(t)\) is the velocity of an object, the displacement over \([t_1, t_2]\) is:\[ \int_{t_1}^{t_2} v(t) \, dt \]
Work done by a force:
If \(F(x)\) is the force applied over a distance, the total work is:
\[ \int_a^b F(x) \, dx \]
Differential Equations
A differential equation is a mathematical equation that relates a function with its derivatives. Differential equations are used to model various phenomena in physics, engineering, biology, economics, and more.
General Form
A differential equation can be written as:
\[ F\left(x, y, y', y'', \dots, y^{(n)}\right) = 0 \]
Here:
- \(y = y(x)\) is the unknown function (solution).
- \(y', y'', \dots, y^{(n)}\) are its derivatives.
Here \(F\) is a function that relates \(y\) with its derivatives and possibly the independent variable \(x\).
Note that in physics, \(x\) often represents time, and \(y\) represents a physical quantity (e.g., position, temperature, concentration).
For position \(x(t)\) function we can write the following differential equation:
\[ F(t, x(t), \frac{dx}{dt}, \frac{d^2x}{dt^2}, \dots, \frac{d^nx}{dt^n}) = 0 \]
where \(x(t)\) is the position of the object at time \(t\), and \(x'(t), x''(t), \dots\) are its derivatives with respect to time.
Order of a Differential Equation
The order of a differential equation is the highest order of the derivative present in the equation. For example:
\(y'' + 2y' + y = 0\) is a second-order ODE.
\(y' + y = x\) is a first-order ODE.
\((y'')^3 + y' = 0\) is a second-order ODE (nonlinear).
Depending on the order \(n\) in the equation, we need \(n\) initial conditions to solve the ODE. Otherwise, the solution will have \(n\) arbitrary constants.
Key Differences: Algebraic vs. Differential Equations
- In algebraic equations, we solve for a number of \(x\) that satisfies the equation.
- In differential equations, we solve for a function \(y(x)\) that satisfies the given relationship involving its derivatives.
Approaches to Solving Differential Equations
Symbolic Solvers
Tools like Wolfram Alpha, Mathematica, or the sympy library in Python can provide analytical (exact) solutions to many differential equations.Numerical Methods
Techniques such as Euler’s method, Runge-Kutta methods, and others are used to approximate solutions when an analytical solution is not feasible. These methods are implemented in software like Python, MATLAB, and similar platforms, allowing us to compute the numerical behavior of the solution.
Types of Differential Equations
- Ordinary Differential Equations (ODEs):
- Involves derivatives of a function with respect to a single variable.
- Example: \[ \frac{dy}{dx} + 2y = e^x \]
- Partial Differential Equations (PDEs):
- Involves partial derivatives of a function with respect to multiple variables.
- Example: \[ \frac{\partial u}{\partial t} = c^2 \frac{\partial^2 u}{\partial x^2} \]
- Linear Differential Equations:
- A differential equation is linear if it can be written as: \[ a_n(x)y^{(n)} + a_{n-1}(x)y^{(n-1)} + \cdots + a_0(x)y = g(x) \]
- Example: \[ y'' - 3y' + 2y = \sin(x) \]
- Nonlinear Differential Equations:
- A differential equation is nonlinear if it does not satisfy the linearity property.
- Example: \[ y'' + y^2 = 0 \]
Initial and Boundary Value Problems
- Initial Value Problem (IVP):
- Solve a differential equation with given initial conditions.
- Example: \[ \frac{dy}{dx} = 2x, \quad y(0) = 1 \]
- Boundary Value Problem (BVP):
- Solve a differential equation with specified values at the boundaries of the domain.
- Example: \[ y'' + y = 0, \quad y(0) = 1, \quad y(\pi) = 0 \]
Applications of Differential Equations
- Physics:
- Modeling motion (Newton’s laws).
- Heat conduction (Fourier’s law).
- Wave equations.
- Biology:
- Population growth models (Logistic equation).
- Spread of diseases (SIR model).
- Economics:
- Modeling capital accumulation.
- Growth models (Solow model).
- Engineering:
- Circuit analysis (RLC circuits).
- Control systems.
Links:
- List of linear ordinary differential equations
- List of nonlinear ordinary differential equations
- List of nonlinear partial differential equations
Solving Differential Equations
- Separable Equations:
- An ODE is separable if it can be written as: \[ \frac{dy}{dx} = g(x)h(y) \]
- Solution: \[ \int \frac{1}{h(y)} \, dy = \int g(x) \, dx \]
- Linear First-Order Equations:
- General form: \[ \frac{dy}{dx} + P(x)y = Q(x) \]
- Solution: \[ y(x) = \frac{1}{\mu(x)} \left( \int \mu(x) Q(x) \, dx + C \right) \] where \(\mu(x) = e^{\int P(x) \, dx}\) is the integrating factor.
- Second-Order Linear Equations:
- General form: \[ y'' + p(x)y' + q(x)y = g(x) \]
- Solution depends on whether the equation is homogeneous or non-homogeneous.
- Numerical Methods:
- For equations that cannot be solved analytically, numerical methods such as Euler’s method or Runge-Kutta methods are used.
Example of simple ODE
Solve the equation: \[ \frac{dy}{dx} = 2x, \quad y(0) = 1 \]
Separate variables: \[ dy = 2x \, dx \]
Integrate both sides:
\[ \int dy = \int 2x \, dx \] \[ y = x^2 + C \]
Apply the initial condition \(y(0) = 1\): \[ 1 = 0^2 + C \implies C = 1 \]
Final solution: \[ y = x^2 + 1 \]
Newton’s Second Law of Motion
Newton’s second law of motion states that the rate of change of momentum of an object is directly proportional to the net force acting on it. This law can be expressed as a differential equation:
\[ m \frac{d^2x}{dt^2} = F(x, v, t) \]
where: - \(m\) is the mass of the object, - \(\frac{d^2x}{dt^2}\) is the acceleration, - \(F(x, v, t)\) is the net force acting on the object, which may depend on position \(x\), velocity \(v\), and time \(t\).
This second-order ODE describes the motion of the object under the influence of external forces.
Solving this equation provides the position \(x(t)\) of the object as a function of time. The solution can be obtained analytically or numerically, depending on the complexity of the force function \(F(x, v, t)\).
- Let’s consider case with zero force \(F=0\) and \(m=1\). Then the equation becomes:
\[ \frac{d^2x}{dt^2} = 0 \]
This is a second-order linear homogeneous ODE with constant coefficients. The general solution is:
\[ x(t) = c_1 t + c_2 \]
where \(c_1\) and \(c_2\) are constants determined by initial conditions.
Consider that we know that \(x(0) = 1\) and \(v(0) = 2\). Then we have:
\[ x(0) = c_2 = 1 \]
and
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt} (c_1 t + c_2)= c_1 \]
so \(c_1 = 2\). Therefore the solution is \(x(t) = 2t + 1\).
We can check this solution by plotting the position \(x(t)\) and velocity \(v(t)\), as well as acceleration \(a(t)\) over time.
This agrees with the fact that the object moves with constant velocity and zero acceleration when no external force is acting on it.
Show the code
import numpy as np
import matplotlib.pyplot as plt
# Time points
t = np.linspace(0, 2, 100)
# Position x(t) = 2t + 1
x = 2 * t + 1
# Velocity v(t) = 2
v = 2
# Acceleration a(t) = 0
a = 0
# Plotting
plt.figure(figsize=(12, 6))
plt.subplot(1, 3, 1)
plt.plot(t, x, label='Position x(t) = 2t + 1', color='blue')
plt.xlabel('Time t')
plt.subplot(1, 3, 2)
plt.axhline(v, color='red', linestyle='-', label='Velocity v(t) = 2')
plt.xlabel('Time t')
plt.subplot(1, 3, 3)
plt.axhline(a, color='green', linestyle='-', label='Acceleration a(t) = 0')
plt.xlabel('Time t')
plt.suptitle("Motion of Object with F = 0 (position, velocity, acceleration)")
plt.tight_layout()
plt.show()
- Let’s consider a case with constant (gravitational) force $F = - mg $ and \(m = 1\). Then the equation becomes:
\[ \frac{d^2x}{dt^2} = -g \]
This is a second-order linear non-homogeneous ODE with constant coefficients.
The general solution obtained by double integration is: \[ x(t) = c_1 t + c_2 - \frac{1}{2}gt^2 \]
where \(c_1\) and \(c_2\) are constants determined by initial conditions.
Consider that we know that \(x(0) = 20\) and \(v(0) = 3\). Then we have:
\[ x(0) = c_1 \cdot 0 + c_2 - \frac{1}{2}g \cdot 0^2 = c_2 = 20 \]
and
\[ v(t) = \frac{dx}{dt} = \frac{d}{dt} (c_1 t + c_2 - \frac{1}{2}gt^2) = c_1 - gt \]
so \(c_1 = 3\).
Therefore the solution is \(x(t) = 3t + 20 - \frac{1}{2}gt^2\).
We can check this solution by plotting the position \(x(t)\) and velocity \(v(t)\), as well as acceleration \(a(t)\) over time.
Show the code
import numpy as np
import matplotlib.pyplot as plt
# Time points
t = np.linspace(0, 2.5, 100)
# Constants
g = 9.81 # acceleration due to gravity (m/s^2)
# Position x(t) = - 0.5 * g * t^2 + 3t + 20
x = - 0.5 * g * t**2 + 3 * t + 20
# Velocity v(t) = 3 - g * t
v = 3 - g * t
# Acceleration a(t) = -g
a = -g
# Plotting
plt.figure(figsize=(12, 6))
plt.subplot(1, 3, 1)
plt.plot(t, x, label='Position x(t) = 3t + 2 - 0.5gt^2', color='blue')
plt.xlabel('Time t')
plt.subplot(1, 3, 2)
plt.plot(t, v, label='Velocity v(t) = 3 - gt', color='red')
plt.xlabel('Time t')
plt.subplot(1, 3, 3)
plt.axhline(a, color='green', linestyle='-', label='Acceleration a(t) = -g')
plt.xlabel('Time t')
plt.suptitle("Motion of Object with F = -mg (position, velocity, acceleration)")
plt.tight_layout()
plt.show()