Comprehensive Calculus: Limits, Derivatives, Integrals, and Differential Equations Study Notes

The Exponential Function

• The exponential function is defined as $y = a^x$, where $a > 0$ and $x$ is a variable. This function exists for integral, rational, and irrational exponents.
• The Natural Exponential Function is defined as $y = e^x$, where $e = 2.7183$ is the Euler number.
• The Euler number is defined by the infinite series:     $e^x = \sum_{k=0}^{\infty} \frac{x^k}{k!}$

Properties and Graph of $e^x$

• The graph of $y = e^x$ for $x > 0$ is asymptotic to the $x$-axis and increases for all real values of $x$.
• Key mathematical properties include:
  1. $e^0 = 1$
  2. $\lim_{x \to -\infty} e^x = 0$
  3. $\lim_{x \to \infty} e^x = \infty$
  4. $e^x e^y = e^{x+y}$
  5. $\frac{e^x}{e^y} = e^{x-y}$
  6. $(e^x)^r = e^{xr}$

Calculus of Exponential Functions

• Derivative of $y = e^{f(x)}$: For a differentiable function $f$, the derivative is given by:     $y'(x) = f'(x) e^{f(x)}$
• Integral of $y = e^{f(x)}$: If $f$ is a differentiable function of $x$ such that $f'(x)$ is a constant (not a function of $x$), then:     $\int e^{f(x)} \, dx = \frac{1}{f'(x)} e^{f(x)} + c$

Examples: Derivatives

1. $y = e^{x^2}$
2. $y = e^{3x^2}$
3. $y = e^{3x+1}$
4. $y = e^{5/x}$
5. $y = e^{\sqrt{x}}$
6. $y = e^{x \sin(x)}$

Examples: Integrals

1. $\int e^{2x+1} \, dx$
2. $\int e^{x/3} \, dx$
3. $\int x^2 e^{-x^3} \, dx$
4. $\int 3e^{x/2} \, dx$
5. $\int x^2 e^{x^2} \, dx$
6. $\int x^3 e^{-x^4} \, dx$

The Logarithmic Function

• The logarithm of a number $b$ to the base $a$ ($\log_a(b)$) represents the power to which the base must be raised to equal $b$:     $\log_a(b) = k \implies b = a^k$
• Natural Logarithm: When the base is the Euler number $e$, it is denoted as $\ln(x)$.
• Remark: Since $\ln(x)$ is not defined for $x < 0$, the notation $\ln|x|$ is preferred in general contexts.

Properties and Graph of $\ln(x)$

• The graph of $y = \ln(x)$ for $x > 0$ is asymptotic to the $y$-axis.
• Logarithmic characteristics based on domain:
  1. $\ln(x) < 0$ for all $x \in \mathbb{R}$ such that $0 < x < 1$.
  2. $\ln(x) = 0$ for $x = 1$.
  3. $\ln(x) > 0$ for all $x \ge 1$.
• Mathematical properties:
  1. $\ln(xy) = \ln(x) + \ln(y)$
  2. $\ln(\frac{x}{y}) = \ln(x) - \ln(y)$
  3. $\ln(x^a) = a \ln(x)$, where $a$ is a constant.
  4. $\ln(b) = \frac{\log_a(b)}{\log_a(e)}$
  5. $\ln(e) = 1$
  6. $\lim_{x \to \infty} \ln(x) = \infty$
  7. $\lim_{x \to 0^+} \ln(x) = -\infty$
  8. $\ln(x)$ is a strictly increasing function.

Calculus of Logarithmic Functions

• Derivative of $\ln(f(x))$: Given $y = \ln(f(x))$, then:     $y' = \frac{f'(x)}{f(x)}$
• Integration resulting in Logarithms: The rule $\int x^n \, dx = \frac{x^{n+1}}{n+1} + c$ fails for $n = -1$. Since $\frac{d}{dx}(\ln(x)) = \frac{1}{x}$, it follows that:     $\int \frac{1}{x} \, dx = \ln|x| + c$     Alternatively, $\int \frac{1}{x} \, dx = \ln|kx|$ where $c = \ln|k|$.

Examples: Logarithmic Operations

1. Express as sum or difference:     (a) $\ln(\cot(x))$     (b) $\ln(\tan^2(x))$     (c) $\ln(x^2 - 4)$     (d) $\ln\left(\sqrt{\frac{x+1}{x-1}}\right)$     (e) $\ln(3x^2 \sin^4(x))$

2. Express as single logarithm:     (a) $\ln(x) + \ln(y) - 1$     (b) $\ln(\frac{e^3}{x}) + \ln(\frac{x}{e})$

General Exponential Function $a^x$

• Derivative of $a^x$: To find $\frac{d}{dx}(a^x)$, let $y = a^x$.     Taking natural logs: $\ln(y) = x \ln(a)$.     Implicit differentiation: $\frac{1}{y} \frac{dy}{dx} = \ln(a)$.     The result is:     $\frac{d}{dx}(a^x) = a^x \ln(a)$
• Integral of $a^x$: Based on the derivative result:     $\int a^x \, dx = \frac{a^x}{\ln(a)} + c$

Definite Integrals with Negative Bounds

• When evaluating $\int_a^b \frac{1}{x} \, dx$ for $a < 0$ and $b < 0$, the standard primitive $\ln(x)$ is undefined.
• However, for $x < 0$, the function $\ln(-x)$ exists and its derivative is $\frac{-1}{-x} = \frac{1}{x}$.
• Thus, the generalized form is:     $\int_a^b \frac{1}{x} \, dx = [\ln|x|]_a^b$
• Condition for Existence: This integral is undefined if the interval $[a, b]$ contains $x = 0$, as $y = \frac{1}{x}$ is discontinuous at that point.

Integration by Parts

• This technique is derived from the product rule of differentiation: $\frac{d}{dx}(uv) = v \frac{du}{dx} + u \frac{dv}{dx}$.

• By integrating both sides and rearranging, the formula is:     $\int u \, dv = uv - \int v \, du$

• Applications: This method applies to combinations including:

  • Polynomial and trigonometric
  • Polynomial and exponential
  • Polynomial and logarithm
  • Inverse trigonometric functions
  • Logarithmic functions
  • Exponential and trigonometric
  • Exponential and logarithm

• Integral of $\ln(x)$: Using integration by parts with $u = \ln(x)$, $dv = dx$, $du = \frac{dx}{x}$, and $v = x$:     $\int \ln(x) \, dx = x \ln(x) - \int x \frac{dx}{x} = x \ln(x) - x + c$

Trigonometric Functions: Derivatives and Integrals

Basic Trigonometric Identities

• $\csc(x) = \frac{1}{\sin(x)}$
• $\sec(x) = \frac{1}{\cos(x)}$
• $\cot(x) = \frac{1}{\tan(x)}$

Derivatives

1. $\frac{d}{dx}(\sin(x)) = \cos(x)$
2. $\frac{d}{dx}(\cos(x)) = -\sin(x)$
3. $\frac{d}{dx}(\tan(x)) = \sec^2(x)$
4. $\frac{d}{dx}(\sec(x)) = \sec(x) \tan(x)$
5. $\frac{d}{dx}(\csc(x)) = -\csc(x) \cot(x)$
6. $\frac{d}{dx}(\cot(x)) = -\csc^2(x)$

• General Power Rule: If $y = \sin^n(x)$, then $\frac{dy}{dx} = n \sin^{n-1}(x) \cos(x)$. If $y = \cos^n(x)$, then $\frac{dy}{dx} = n \cos^{n-1}(x) (-\sin(x))$.

Integrals

1. $\int \sin(x) \, dx = -\cos(x) + c$
2. $\int \cos(x) \, dx = \sin(x) + c$
3. $\int \tan(x) \, dx = -\ln|\cos(x)| + c = \ln|\sec(x)| + c$
4. $\int \sec^2(x) \, dx = \tan(x) + c$
5. $\int \csc^2(x) \, dx = -\cot(x) + c$

• Linear Arguments:     $\int \cos(ax+b) \, dx = \frac{1}{a} \sin(ax+b) + c$$\int \sin(ax+b) \, dx = -\frac{1}{a} \cos(ax+b) + c$

Integration of Trigonometric Powers

Even Powers of Sine and Cosine

Uses identities:

• $\cos^2(x) = \frac{1}{2}(1 + \cos(2x))$
• $\sin^2(x) = \frac{1}{2}(1 - \cos(2x))$

Odd Powers of Sine and Cosine

Uses the identity $\sin^2(x) + \cos^2(x) = 1$. The power is split (e.g., $\sin^5(x) = \sin^4(x) \sin(x)$) and solved via substitution.

Powers of Tangent

Uses the identity $\tan^2(x) = \sec^2(x) - 1$.

Product Form Inspections

1. $\int \sin(x) \cos^n(x) \, dx = -\frac{1}{n+1} \cos^{n+1}(x) + c$
2. $\int \cos(x) \sin^n(x) \, dx = \frac{1}{n+1} \sin^{n+1}(x) + c$

Inverse Trigonometric Functions

• Derivatives:

  1. $\frac{d}{dx}(\sin^{-1}(x)) = \frac{1}{\sqrt{1-x^2}}$
  2. $\frac{d}{dx}(\cos^{-1}(x)) = -\frac{1}{\sqrt{1-x^2}}$
  3. $\frac{d}{dx}(\tan^{-1}(x)) = \frac{1}{1+x^2}$

• Integration via Trigonometric Substitution:     To evaluate $\int \frac{1}{\sqrt{a^2 - b^2 x^2}} \, dx$, use $x = \frac{a}{b} \sin(\theta)$. Result: $\frac{1}{b} \sin^{-1}(\frac{bx}{a}) + c$.     To evaluate $\int \frac{1}{a^2 + b^2 x^2} \, dx$, use $x = \frac{a}{b} \tan(\theta)$. Result: $\frac{1}{ab} \tan^{-1}(\frac{bx}{a}) + c$.

Trigonometric Product Identities for Integration

For integrals of products with different linear arguments ($\alpha x$ and $\beta x$):

1. $\sin(\alpha) \sin(\beta) = \frac{1}{2} \cos(\alpha - \beta) - \frac{1}{2} \cos(\alpha + \beta)$
2. $\cos(\alpha) \cos(\beta) = \frac{1}{2} \cos(\alpha - \beta) + \frac{1}{2} \cos(\alpha + \beta)$
3. $\sin(\alpha) \cos(\beta) = \frac{1}{2} \sin(\alpha - \beta) + \frac{1}{2} \sin(\alpha + \beta)$
4. $\cos(\alpha) \sin(\beta) = \frac{1}{2} \sin(\alpha + \beta) - \frac{1}{2} \sin(\alpha - \beta)$

Hyperbolic Functions

Defined in terms of exponential functions:

1. $\sinh(x) = \frac{e^x - e^{-x}}{2}$
2. $\cosh(x) = \frac{e^x + e^{-x}}{2}$
3. $\tanh(x) = \frac{\sinh(x)}{\cosh(x)} = \frac{e^x - e^{-x}}{e^x + e^{-x}}$

Osborn's Rule

A rule for converting trigonometric identities into hyperbolic identities: replace sine and cosine with hyperbolic counterparts. If a term contains a product of two sines (explicitly or implicitly via tangent/cotangent), the sign of that product must be negated.

• Example: $\cos^2(x) + \sin^2(x) = 1 \implies \cosh^2(x) - \sinh^2(x) = 1$.

Derivatives of Hyperbolic Functions

1. $\frac{d}{dx}(\cosh(x)) = \sinh(x)$
2. $\frac{d}{dx}(\sinh(x)) = \cosh(x)$

Inverse Hyperbolic Functions

Expressed in logarithmic form:

1. $\sinh^{-1}(x) = \ln(x + \sqrt{1 + x^2})$
2. $\cosh^{-1}(x) = \ln(x + \sqrt{x^2 - 1})$
3. $\tanh^{-1}(x) = \frac{1}{2} \ln\left(\frac{1+x}{1-x}\right)$

Even and Odd Functions

• Even Functions: Satisfy $f(-x) = f(x)$. Symmetric about the $y$-axis. Examples: $|x|$, $x^2$, $\cos(x)$.     $\int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx$
• Odd Functions: Satisfy $f(-x) = -f(x)$. Symmetric about the origin (rotational symmetry). Examples: $x^3$, $\sin(x)$, $\tan(x)$.     $\int_{-a}^a f(x) \, dx = 0$

Ordinary Differential Equations (ODEs)

• An equation containing differential coefficients like $\frac{dy}{dx}$.
• Order: Determined by the highest derivative present.
• General Solution: Defines a family of curves.
• Particular Solution: Obtained when initial conditions are used to find specific constants.

First-Order ODEs

1. Separable Variables: Equations where terms can be grouped by variable (e.g., $\frac{dy}{dx} = xy \implies \int \frac{1}{y} \, dy = \int x \, dx$).
2. Exact Equations: Equations where one side is a direct derivative of a function. If not exact, an Integrating Factor ($I$) is used.
3. Linear Equations: Formed as $\frac{dy}{dx} + ay = b$, where $a$ and $b$ are functions of $x$ or constants.
   • Integrating Factor: $I(x) = e^{\int a \, dx}$.
   • General Solution: $\frac{d}{dx}(Iy) = bI \implies Iy = \int bI \, dx$.

Second-Order Linear ODEs with Constant Coefficients

Form: $a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = 0$. We assume valid solution $y = e^{\lambda x}$, leading to the Auxiliary Quadratic Equation: $a\lambda^2 + b\lambda + c = 0$.

Solutions based on Discriminate ($b^2 - 4ac$):

1. Identical Real Roots ($\lambda_1$): $y = e^{\lambda_1 x}(Ax + B)$
2. Distinct Real Roots ($\lambda_1, \lambda_2$): $y = A e^{\lambda_1 x} + B e^{\lambda_2 x}$
3. Complex Roots ($\alpha \pm \beta i$): $y = e^{\alpha x}(A \cos(\beta x) + B \sin(\beta x))$

Non-Homogeneous Equations

Form: $ay'' + by' + cy = f(x)$. The general solution is $y_n = y_c + y_p$, where:

• $y_c$ (Complementary Solution) solves the homogeneous equation $f(x) = 0$.
• $y_p$ (Particular Integral) is a specific solution suited to the type of $f(x)$ (Polynomial, Exponential, or Trigonometric).

Geometrical Applications of Calculus

• Area of a Sector (Polar Coordinates):     For a sector bounded by radius vectors at angles $\alpha$ and $\beta$:     $A = \int_{\alpha}^{\beta} \frac{1}{2} r^2 \, d\theta$

• Length of a Curve (Arc Length):

  1. Cartesian: $L = \int_a^b \sqrt{1 + \left(\frac{dy}{dx}\right)^2} \, dx$
  2. Parametric ($x=f(t), y=g(t)$): $L = \int_{t_1}^{t_2} \sqrt{\left(\frac{dx}{dt}\right)^2 + \left(\frac{dy}{dt}\right)^2} \, dt$
  3. Polar: $L = \int_{\alpha}^{\beta} \sqrt{r^2 + \left(\frac{dr}{d\theta}\right)^2} \, d\theta$