Math 1550: Calculus I Class Notes

Contents

• Limits and Derivatives

• 2.1 The Tangent and Velocity Problems

• 2.2 The Limit of a Function

• 2.3 Calculating Limits Using the Limit Laws

• 2.5 Continuity

• 2.6 Limits at Infinity; Horizontal Asymptotes

• 2.7 Derivatives and Rates of Change

• 2.8 The Derivative as a Function

• Differentiation Rules

• Applications of Differentiation

• Integrals

• Further Applications of Integration

2.1 The Tangent and Velocity Problems

• Average Velocity:

  • The average velocity of an object is calculated as:

    • v_avg = (s(t1) - s(t0)) / (t1 - t0)

  • Example: For a rock thrown on Mars:

    • Height function: y(t) = 10t - 1.86t²

    • Find v_avg over [1, 4]:

      • v_avg = (y(4) - y(1)) / (4 - 1) = 0.7 m/s.

• Instantaneous Velocity:

  • The instantaneous velocity is found by:

    • v(t) = lim (h -> 0)[s(t + h) - s(t)] / h

  • Example: Calculate at t = 1 gives about 6.28 m/s.

2.2 The Limit of a Function

• Definition: lim x→a f(x) = L if for all x close to a, f(x) is arbitrarily close to L.

• Finding Limits Graphically:

  • Use graphical methods and values near a.

  • Example: Calculate limits for the piecewise function by inspecting values near the points.

2.3 Calculating Limits Using the Limit Laws

• Limit Laws: Useful for simplifying limit calculations

  • Sum, Difference, Product, and Quotient Rules defined for limits.

• Direct Substitution: If p and q are polynomials:

  • lim x→a p(x) = p(a)

• Example:

  • Find lim x→−3 (2x³ + 6x² - 9) using direct substitution.

2.5 Continuity

• Definition: f is continuous at x = a if:

  1. f(a) is defined,

  3. lim x→a f(x) = f(a).

• Discontinuities:

  • Removable, jump, and infinite discontinuities mentioned.

2.6 Limits at Infinity; Horizontal Asymptotes

• Definitions:

  • lim x→∞ f(x) = L or lim x→−∞ f(x) = L denotes horizontal asymptotes.

  • Example: Study the behavior of rational functions as x → ±∞.

2.7 Derivatives and Rates of Change

• Definition of Derivative:

  • f’(x) = lim h→0 [f(x + h) - f(x)] / h.

• Physical Interpretation: Slope of tangent line or instantaneous rate of change.

2.8 The Derivative as a Function

• Derivative can be treated as a function of x.

• Notation variations: f’(x), dy/dx.

Differentiation Rules

• Constant, Power, Sum, and Product Rules:

  • Basic rules outline.

• Example: Differentiate polynomial functions and apply correctly.

Applications of Differentiation

• Finding Extrema:

  • Local vs Absolute Max/Min: Definitions provided and implications discussed.

• Mean Value Theorem: Discussed with conditions for function continuity and differentiability.

Integrals & Applications of Integration

• Concept of areas and net changes.

• Fundamental Theorem of Calculus.

Further Applications of Integration

• Application to physics and daily scenarios. Examples included.