Physics and Mathematics

The relationship between the angle θ and the derivative is:

$\frac{dy}{dx} = \tan θ$

Where:

• θ is the angle the tangent line makes with the positive x-axis
• dy/dx is the value of the derivative at that point

This means:

• When dy/dx > 0 (positive slope): 0° < θ < 90° (function is increasing)
• When dy/dx = 0 (zero slope): θ = 0° (function has a horizontal tangent, possibly at extreme points)
• When dy/dx < 0 (negative slope): 90° < θ < 180° (function is decreasing)
• When dy/dx approaches ±∞: θ approaches 90° (function has a vertical tangent)

Example: At a point where the derivative equals 1, the tangent line makes a 45° angle with the x-axis because tan(45°) = 1.

The derivative of $e^x$ is:

$\frac{d}{dx}(e^x) = e^x$

This property makes $e^x$ special among all exponential functions because it is the only exponential function that equals its own derivative.

For general exponential functions:

• $\frac{d}{dx}(a^x) = a^x \ln(a)$
• Only when $a = e$ does the derivative equal the original function

This self-derivative property is why $e$ appears throughout calculus, differential equations, and mathematical modeling of natural growth and decay processes.

Example: When modeling population growth where the rate of change is proportional to the population, the solution involves $e^x$.

A local minimum of a function f(x) occurs at x = x₀ when:

1. The first derivative equals zero: f'(x₀) = 0
2. The second derivative is positive: f''(x₀) > 0

This means:

• The slope of the tangent line at x₀ is zero (tangent is horizontal)
• The function is concave upward at that point
• The function value f(x₀) is less than values at nearby points

Example: For f(x) = x² - 6x + 10, the minimum occurs at x = 3 where f'(3) = 0 and f''(3) = 2 > 0

Geometric interpretation of derivatives at critical points:

1. First derivative (f'(x)):

   • Represents the slope of the tangent line
   • At critical points (maxima/minima): f'(x) = 0, meaning the tangent is horizontal
   • Before a maximum: f'(x) > 0 (slope is positive, function increasing)
   • After a maximum: f'(x) < 0 (slope is negative, function decreasing)
   • Before a minimum: f'(x) < 0 (slope is negative, function decreasing)
   • After a minimum: f'(x) > 0 (slope is positive, function increasing)

2. Second derivative (f''(x)):

   • Represents the rate of change of the slope
   • Determines the concavity of the curve
   • At a maximum: f''(x) < 0 (concave downward)
   • At a minimum: f''(x) > 0 (concave upward)
   • At an inflection point: f''(x) = 0 (concavity changes)

Example: In projectile motion, the position function has a maximum at the peak where velocity (first derivative) is zero and acceleration (second derivative) is negative due to gravity.

Key integration rules:

Trigonometric functions:

• $\int \sin x\,dx = -\cos x + C$
• $\int \cos x\,dx = \sin x + C$
• $\int \tan x\,dx = -\ln|\cos x| + C = \ln|\sec x| + C$
• $\int \cot x\,dx = \ln|\sin x| + C$
• $\int \sec x\,dx = \ln|\sec x + \tan x| + C$
• $\int \csc x\,dx = \ln|\csc x - \cot x| + C$

Other common forms:

• $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$ (where $n \neq -1$)
• $\int \frac{1}{x}\,dx = \ln|x| + C$
• $\int \frac{1}{x^2+a^2}\,dx = \frac{1}{a}\tan^{-1}\frac{x}{a} + C$
• $\int \frac{1}{\sqrt{a^2-x^2}}\,dx = \sin^{-1}\frac{x}{a} + C$
• $\int \sec^2 x\,dx = \tan x + C$
• $\int \csc^2 x\,dx = -\cot x + C$

Significant digits (or figures) include all certain digits plus one uncertain/estimated digit:

Identification rules:

• All non-zero digits are significant
• Zeros between non-zero digits are significant (e.g., 1.05 has 3)
• Leading zeros are never significant (e.g., 0.00234 has 3)
• Trailing zeros after decimal are significant (e.g., 2.00 has 3)
• Trailing zeros in whole numbers are ambiguous (e.g., 1200 has 2-4, use scientific notation to clarify)

Rounding rules:

• If next digit > 5: round up
• If next digit < 5: leave unchanged
• If next digit = 5: round up if previous digit is odd, leave unchanged if even

Calculation rules:

1. Multiplication/Division: Result should have same number of significant digits as the measurement with fewest significant digits
2. Addition/Subtraction: Result should have same number of decimal places as the measurement with fewest decimal places

Example: 25.2 × 1374 ÷ 33.3 = 1040 (not 1039.7838...) Since 25.2 and 33.3 have 3 significant digits each, the answer has 3 significant digits.

Rules for significant figures in calculations:

For multiplication/division:

• The result has the same number of significant figures as the measurement with the FEWEST significant figures
• Example: 12.0 kg ÷ 7.0 kg = 1.7 (not 1.714285...)
  • Both measurements have 2 significant figures, so result has 2 significant figures

For addition/subtraction:

• The result has the same number of DECIMAL PLACES as the measurement with the fewest decimal places
• Example: 24.36 + 0.0623 + 256.2 = 280.7
  • 256.2 has only 1 decimal place, so result has 1 decimal place

Process for addition/subtraction:

1. Identify the measurement with fewest decimal places
2. Align all numbers by decimal point
3. Round all values to match the measurement with fewest decimal places
4. Perform the calculation
5. Round the final answer to match the fewest decimal places

Note: Remember that digits before the decimal point don't matter for addition/subtraction rules; only the number of decimal places matters.

Systematic approach to complex integration problems:

1. Analyze the integrand

   • Identify function type (polynomial, trigonometric, exponential, rational, etc.)
   • Look for patterns that match standard integration formulas
   • Check for products of functions that suggest specific techniques

2. Select appropriate technique based on integrand structure:

   • Simple function: Use direct formulas from integration table
   • Product of functions: Consider u-substitution or integration by parts
   • Rational functions: Try partial fraction decomposition
   • Trigonometric functions: Check for trigonometric identities
   • Functions with square roots: Consider trigonometric substitution

3. Apply the selected technique:

   • U-substitution: Let u = g(x) and find du = g'(x)dx
   • Integration by parts: Use $\int u\,dv = uv - \int v\,du$
   • Partial fractions: Decompose $\frac{P(x)}{Q(x)}$ into simpler fractions
   • Trigonometric substitutions:
     • For $\sqrt{a^2-x^2}$, use $x = a\sin\theta$
     • For $\sqrt{a^2+x^2}$, use $x = a\tan\theta$
     • For $\sqrt{x^2-a^2}$, use $x = a\sec\theta$

4. Verify your answer by differentiation

   • Take the derivative of your answer to confirm it equals the original integrand

5. For definite integrals:

   • Substitute the limits of integration into the antiderivative
   • Compute the difference: F(b) - F(a)

Key decision point: If one technique doesn't work, systematically try alternatives until a solution emerges.

The Triangle Rule of Vector Addition is a graphical method for finding the sum of two vectors:

Procedure:

1. Draw the first vector ($\vec{A}$) with its proper magnitude and direction
2. From the head of the first vector, draw the second vector ($\vec{B}$) with its proper magnitude and direction
3. The resultant vector ($\vec{R} = \vec{A} + \vec{B}$) is drawn from the tail of the first vector to the head of the second vector

Why it works:

• Vectors follow the principle of vector addition where both magnitude and direction matter
• The triangle formed represents the actual path traveled if both vectors were applied sequentially
• Mathematically demonstrates the commutative property of vector addition

Example Application: When a ball moves at 3 m/s inside a tube while the tube moves at 4 m/s perpendicular to its length:

• Draw the 3 m/s vector (tube motion)
• From its head, draw the 4 m/s vector (perpendicular motion)
• The resultant is 5 m/s at an angle of 53° from the tube's direction

Note: The resultant can be calculated using the Pythagorean theorem ($R = \sqrt{A^2 + B^2}$) for perpendicular vectors, or the law of cosines for non-perpendicular vectors.

Vector Operations with Equal-Magnitude Vectors

For two vectors $\vec{A}$ and $\vec{B}$ of equal magnitude (5 units) at angle $\theta = 60°$:

1. Resultant Vector Magnitude ($|\vec{A} + \vec{B}|$):

• Formula: $|\vec{A} + \vec{B}| = \sqrt{A^2 + B^2 + 2AB\cos\theta}$
• Calculation:
  • $|\vec{A} + \vec{B}| = \sqrt{5^2 + 5^2 + 2(5)(5)\cos60°}$
  • $= \sqrt{25 + 25 + 50(0.5)}$
  • $= \sqrt{75} = 5\sqrt{3} \approx 8.66$ units

2. Difference Vector Magnitude ($|\vec{A} - \vec{B}|$):

• Formula: $|\vec{A} - \vec{B}| = \sqrt{A^2 + B^2 - 2AB\cos\theta}$
• Calculation:
  • $|\vec{A} - \vec{B}| = \sqrt{5^2 + 5^2 - 2(5)(5)\cos60°}$
  • $= \sqrt{25 + 25 - 25}$
  • $= \sqrt{25} = 5$ units

Key Insight: For equal-magnitude vectors at angle $\theta$:

• When $\theta = 0°$ (parallel): Maximum resultant ($2A$), minimum difference (0)
• When $\theta = 180°$ (antiparallel): Minimum resultant (0), maximum difference ($2A$)
• When $\theta = 60°$: The difference vector has the same magnitude as either original vector

Alternative Approach: The difference vector can also be calculated by considering $\vec{A} + (-\vec{B})$, where the angle between them is $180° - 60° = 120°$.