Calculus Study Notes: Limits and Derivatives

Calculus Study Notes

Definitions

• Calculus: A branch of mathematics that studies continuous change, primarily through derivatives and integrals.
• Limits: The value that a function approaches as the input approaches some value.
• Derivative: Measures the rate of change of a function; represented as f'(x) or dy/dx.
• Integral: Represents the accumulation of quantities, such as areas under curves; denoted as ∫f(x)dx.

Key Concepts

1. Limits

   • Notation: ( \lim_{x \to c} f(x) )
   • One-sided limits: ( \lim_{x \to c^-} f(x) ) (left), ( \lim_{x \to c^+} f(x) ) (right)
   • Properties:
     • Limit laws (sum, product, quotient)
     • Squeeze theorem

2. Derivatives

   • Definition:
     • ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} )
   • Interpretation: Slope of the tangent line to the graph of the function at a point.
   • Rules:
     • Power Rule: ( d/dx [x^n] = nx^{n-1} )
     • Product Rule: ( d/dx [uv] = u'v + uv' )
     • Quotient Rule: ( d/dx [u/v] = (u'v - uv')/v^2 )
     • Chain Rule: ( d/dx [f(g(x))] = f'(g(x))g'(x) )

3. Applications of Derivatives

   • Finding tangents and normals.
   • Identifying local maxima and minima (Critical points).
   • Analyzing function behavior (increasing/decreasing intervals).

4. Integrals

   • Definite Integral: Represents the area under a curve from a to b:
     • ( \int_a^b f(x)dx )
   • Indefinite Integral: General formula for antiderivatives:
     • ( \int f(x)dx = F(x) + C )
   • Fundamental Theorem of Calculus: Connects derivatives and integrals:
     • If F is an antiderivative of f, then:
       • ( \int_a^b f(x)dx = F(b) - F(a) )

5. Techniques of Integration

   • Substitution
   • Integration by parts
   • Partial fractions
   • Numerical integration (Trapezoidal rule, Simpson’s rule)

Important Applications

• Modeling physical systems with rates of change (velocity, acceleration).
• Area and volume calculations for geometrical figures.
• Solving differential equations that describe dynamic systems.

Notation Summary

• ( f'(x) ): Derivative of f with respect to x.
• ( \int f(x)dx ): Indefinite integral of f.
• ( \int_a^b f(x)dx ): Definite integral of f from a to b.
• ( e^x ), ( \ln(x) ): Exponential and logarithmic functions often used in calculus.

Important Limits and Derivatives

• ( \lim_{x \to 0} \frac{\sin x}{x} = 1 )
• ( \frac{d}{dx} e^x = e^x )
• ( \frac{d}{dx} \ln x = \frac{1}{x} )

Understanding these concepts is crucial for advanced study in mathematics and its applications in science, engineering, and economics.

Calculus

• Calculus studies continuous change, primarily using derivatives and integrals.
• Derivatives measure rates of change, while integrals represent accumulation.

Limits

• A limit is the value a function approaches as its input approaches a specified value.
• Notation: ( \lim_{x \to c} f(x) ) represents the limit of f(x) as x approaches c.
• One-sided limits approach c from either the left (( \lim_{x \to c^-} f(x) )) or right (( \lim_{x \to c^+} f(x) )).
• Limits have properties like sum, product, and quotient laws, as well as the Squeeze Theorem.

Derivatives

• The derivative of a function measures the instantaneous rate of change at any point.
• Notation: ( f'(x) ) or ( dy/dx )
• The definition of the derivative is: ( f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h} )
• Derivatives graphically represent the slope of the tangent line to the function at a given point.
• Power Rule: ( d/dx [x^n] = nx^{n-1} )
• Product Rule: ( d/dx [uv] = u'v + uv' )
• Quotient Rule: ( d/dx [u/v] = (u'v - uv')/v^2 )
• Chain Rule: ( d/dx [f(g(x))] = f'(g(x))g'(x) )

Applications of Derivatives

• Derivatives are used to find tangents and normals to curves.
• They help identify local maxima and minima (critical points).
• Derivatives analyze function behavior, determining increasing or decreasing intervals.

Integrals

• The definite integral represents the area under the curve of a function between two specified points.
• Notation: ( \int_a^b f(x)dx ) represents the definite integral from a to b.
• The indefinite integral represents the general formula for all antiderivatives.
• Notation: ( \int f(x)dx = F(x) + C )
• The Fundamental Theorem of Calculus establishes a connection between derivatives and integrals:
  • If F(x) is an antiderivative of f(x), then ( \int_a^b f(x)dx = F(b) - F(a) ).

Techniques of Integration

• Substitution: Simplifies the integrand by substituting a new variable.
• Integration by parts: Uses the product rule in reverse to evaluate integrals.
• Partial fractions: Decomposes rational functions into simpler fractions for integration.
• Numerical integration: Uses methods like the Trapezoidal rule and Simpson’s rule to approximate definite integrals.

Important Applications of Calculus

• Modeling: Calculus models physical systems with rates of change (velocity, acceleration).
• Geometry: Calculus calculates areas and volumes of geometric shapes.
• Solving Differential Equations: Calculus helps solve equations describing dynamic systems in various fields.

Important Notation

• ( f'(x) ): Derivative of f(x) with respect to x.
• ( \int f(x)dx ): Indefinite integral of f(x).
• ( \int_a^b f(x)dx ): Definite integral of f(x) from a to b.
• ( e^x ) and ( \ln(x) ): Exponential and logarithmic functions often used in calculus.

Important Limits and Derivatives

• ( \lim_{x \to 0} \frac{\sin x}{x} = 1 )
• ( \frac{d}{dx} e^x = e^x )
• ( \frac{d}{dx} \ln x = \frac{1}{x} )