Calculus Lecture Notes

Table and Limit Estimation

  • Complete the table: Round answers to four decimal places.

    • Expression: \lim_{x \to 0} (\sqrt{x+1} - 1)
    • Values in table:
    • x = -0.1: f(x) = 0.5132
    • x = -0.01: f(x) = 0.5013
    • x = -0.001: f(x) = 0.4988
    • x = 0: f(x) = 0.5000
    • x = 0.001: f(x) = 0.5001
    • x = 0.01: f(x) = 0.5015
    • x = 0.1: f(x) = 0.5132

  • Estimate the limit: Use values from the table to estimate the limit.

  • Graphing Utility: Graph the function to confirm the limit is 0.5000.

Existence of Limits

  • Use the graph of function f to determine if each limit exists: (If the limit does not exist, enter DNE.)
    • \lim_{x \to -2} f(x)
    • (a) f(-2) = DNE
    • (b) \lim_{x \to -\infty} f(x) = DNE
    • (c) f(0) = 4
    • (d) \lim_{x \to 0} f(x) = DNE
    • (e) f(2) = DNE
    • (f) \lim_{x \to 2} f(x) = 0.5
    • (g) f(4) = DNE
    • (h) \lim_{x \to 4} f(x) = 6

One-Sided Limits

  • Discussion of one-sided limits and their values when approaching -2:

    • From the right, approaches +∞
    • From the left, approaches -∞
    • Vertical Asymptote exists at x = -2, hence f(-2) = DNE.

  • At x = 2: Left limit approaches 0.5, right limit confirms 0.5, meaning limit exists.

  • At x = 4: Solid circle indicates defined limit, and right limit calculation supports this.

Function Continuity

  • Three conditions for continuity at a point c:
    1. f(c) is defined.
    2. The limit \lim_{x \to c} f(x) exists.
    3. The limit equals the function value: \lim_{x \to c} f(x) = f(c).

Discontinuities

  • Assessment of discontinuities for the function:
    • Function of the form f(x) = \frac{x + 3}{x^2 - 2x - 15}.
    • Set denominator = 0 to find discontinuities. Factorizing:
      • x^2 - 2x - 15 = (x - 5)(x + 3)
      • Roots: x = 5, -3.
      • Type of Discontinuity: Removable if limits are finite; nonremovable if not.
      • At x = -3: Limit exists, so removable discontinuity.
      • At x = 5: Function undefined, limit approaches ∞, hence nonremovable.

Derivatives Using Limit Process

  • Find the derivative of function:
    • For f(x) = x^2 + x - 9:
    • f'(x) = 2x + 1.
    • Using limit definition:
      • f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}.
    • Compute:
      • f(x+h) = (x+h)^2 + (x+h) - 9
      • Expand and simplify to reach derivative.

Trigonometric Function Derivative

  • Find the derivative of: y = \cos(8x):
    • y' = -8 \sin(8x) using the chain rule of differentiation.

Implicit Differentiation

  • Differentiate the equation x^4 = xy + y^2 = 4 with respect to x. Use product rule where necessary.
  • Rearranging gives formula for dy/dx:
    • dy/dx = \frac{4 - 4x^3}{x - 2y}.

Relative and Absolute Maxima

  • Difference between relative and absolute maximum:
    • Relative Maximum: Peak of the graph within proximity - the local maximum.
    • Absolute Maximum: The greatest value of the function on a defined interval.

Critical Numbers and Intervals of Increase/Decrease

  • Consider the function f(x) = -5x^3 + 15x.
    • Critical numbers: x = -1, 1.
    • Increasing on: (-\infty, -1) U (1, \infty).
    • Decreasing on: (-1, 1).

Concavity and Inflection Points

  • For function f(x) = x^2 - 5x^3, determine intervals of concavity:
    • Find second derivative and set to zero to solve for inflection points.
    • Check the sign distribution on intervals divided by those points to determine concavity.

Antiderivatives and Integrals

  • General antiderivative notation and process discussed:
    • \int f(x)dx = F(x) + C where C is the constant of integration.

Evaluating Definite Integrals

  • Process of evaluating definite integral using fundamental theorem demonstrated.
    • Use limits of the antiderivatives evaluated at endpoints.

Limit of Function at Non-Defined Points

  • Limits approaching infinity and use of L'Hôpital's rule to evaluate limits of 0/0 or ∞/∞ forms are demonstrated.