math 21

• University of the Philippines Diliman
• MATHEMATICS 21
• Elementary Analysis I
• Course Module
• Institute of Mathematics
• UP Institute of Mathematics
• c⃝2018 by the Institute of Mathematics, University of the Philippines Diliman.
• All rights reserved.
• No part of this document may be distributed in any way, shape, or form, without prior written permission from the Institute of Mathematics, University of the Philippines Diliman.
• Mathematics 21 Module Writers and Editors

  • Carlo Francisco Adajar
  • Michael Baysauli
  • Katrina Burdeos
  • Lawrence Fabrero
  • Alip Oropeza
• Limits


  The limiting value that a function approaches as the input approaches a particular value
• Continuity


  A function is continuous at a point if the function's value at that point is equal to the limit of the function as the input approaches that point
• Evaluating limits


  1. Substituting the given value

  2. Using algebraic manipulation

  3. Using other techniques
• One-sided limits exist when a function approaches a value from the left or right
• Limits involving infinity include infinite limits and limits at infinity
• Formal definition of limits


  Using epsilon-delta notation to precisely define the limit of a function
• Intermediate Value Theorem


  If a continuous function takes on two different values, it must also take on all intermediate values
• Squeeze Theorem


  If a function is bounded above and below by two functions that have the same limit, then the original function also has that limit
• New classes of functions

  • Inverse functions
  • Exponential and logarithmic functions
  • Inverse circular functions
  • Hyperbolic functions
  • Inverse hyperbolic functions
• Derivative


  The rate of change of a function at a point
• Differentiability


  A function is differentiable at a point if it has a derivative at that point
• Techniques of differentiation


  1. Basic differentiation rules

  2. Chain rule

  3. Implicit differentiation

  4. Logarithmic differentiation
• L'Hôpital's Rule


  A method for evaluating limits of the form 0/0 or ∞/∞ by taking the limit of the ratio of the derivatives of the numerator and denominator
• Mean Value Theorem


  If a function is continuous on a closed interval, then there exists at least one point in the interval where the derivative is equal to the average rate of change over the interval
• Relative extrema


  Local maximum or minimum values of a function
• Concavity


  The curvature of a function, determined by the sign of the second derivative
• Graph sketching


  Identifying key features like critical points, asymptotes, and using derivatives to determine the shape of the graph
• Rectilinear motion


  Motion along a straight line, described by position, velocity, and acceleration functions
• Related rates


  Solving problems involving the rates of change of related quantities
• Local linear approximation


  Using the derivative to approximate a function near a point
• Absolute extrema


  Global maximum or minimum values of a function on an interval
• Limit of a function


  The number to which the function value gets closer and closer as x approaches a certain number a
• If the limit of f(x) as x approaches a exists, it is unique
• Limit of a constant c as x approaches a

  The constant c
• Limit of x as x approaches a

  The number a
• Evaluating limits using limit theorems

  If lim x→a f(x) = L1 and lim x→a g(x) = L2, then:

  1. lim x→a [f(x) ± g(x)] = L1 ± L2

  2. lim x→a [cf(x)] = cL1

  3. lim x→a [f(x)g(x)] = L1L2

  4. lim x→a f(x)/g(x) = L1/L2, provided g(a) ≠ 0
• If f(x) is not defined at x=a, the limit of f(x) as x approaches a may still exist
• If the limit of f(x) as x approaches a does not approach a real number, then the limit does not exist
• One-sided limit


  The limit of a function f(x) as x approaches a from one side (either left or right)
• One-sided limit from the left

  • The limit of f(x) as x approaches a from values less than a