Exam Notes: Risk Management and Investment Strategies

Families of Functions and Curves

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Fundamental Concepts

Absolute Value Graphs

  • Graph of y=f(x)y = |f(x)| :
    • Reflects the portion of the graph of f(x)f(x) that lies below the x-axis over the x-axis.
    • Example: Obtain the graph of y=2x3y = |2x - 3| from y=2x3y = 2x - 3.

Sketching Graphs

  • Problems:
    • Sketch y=1x2y = |1 - x^2|
    • Sketch:
      • f(x)=cosxf(x) = |\cos x|
      • f(x)=cosx+cosxf(x) = \cos x + |\cos x|

Greatest Integer Function

  • Definition: The greatest integer function x\lfloor x \rfloor gives the greatest integer less than or equal to xx.
    • Examples:
      • 2.7=2\lfloor 2.7 \rfloor = 2
      • 2.3=3\lfloor -2.3 \rfloor = -3
      • 4=4\lfloor 4 \rfloor = 4

Function Composition and Commutativity

  • Commutativity: Check if fg=gff \circ g = g \circ f for non-constant functions.
    • If yes, find examples.
    • If no, demonstrate why not.

Quick Check Answers

  1. Function Operations:

    • (f+g)(x)=x32+x(f + g)(x) = \sqrt[3]{x} - 2 + x; x0x \geq 0
    • (fg)(x)=x32x(f - g)(x) = \sqrt[3]{x} - 2 - x; x0x \geq 0
    • (fg)(x)=3x3/22x(fg)(x) = 3x^{3/2} - 2x; x0x \geq 0
    • (f/g)(x)=x32x(f/g)(x) = \dfrac{\sqrt[3]{x} - 2}{x}; x>0x \gt 0
  2. Composition:

    • (fg)(x)=2x(f \circ g)(x) = 2 - x; x0x \geq 0
    • (gf)(x)=2x2(g \circ f)(x) = \sqrt{2 - x^2}; 2x2-\sqrt{2} \leq x \leq \sqrt{2}
  3. Vectors Movements: Right; 2; Up; 1.

  4. True/False:

    • (a) W
    • (b) Yes

Families of Functions

Overview

  • Definition: Functions are grouped into families based on their defining formulas or common characteristics.

Families of Curves

  • Constant Function Graph: f(x)=cf(x) = c is represented as a horizontal line.

    • Example: y=cy = c forms horizontal lines when cc varies.
  • Parameters: Constants varied to create families of curves.

    • Example:
      • Linear equation: y=mx+by = mx + b
      • Keeping bb constant:
        • Family of Lines Slope mm
      • Keeping mm constant:
        • Family of Parallel Lines Intercept bb

Figures

  • Figure 0.3.1:

    • Horizontal lines with various values of cc.
  • Figure 0.3.2:

    • Curves with varying parameters mm and bb:
      • (a) y=mx+by = mx + b with fixed bb and varying mm.
      • (b) y=mx+by = mx + b with fixed mm and varying bb.

Extended readings:

www.youtube.com
Lesson 9 Graphing Absolute Value and Greatest Integer Functions
www.snow.edu
[PDF] 11.1 Additional Graphs of Functions
industri.fatek.unpatti.ac.id
[PDF] 035-Calculus-Howard-Anton-Irl-Bivens ... - Teknik Industri Unpatti

Power Functions: The Family ( y = x^n )

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Definition of Power Functions

  • A function of the form f(x)=xpf(x) = x^p, where pp is constant, is called a power function.
  • We consider cases where p=np = n is a positive integer.

Graphical Representation

The graphs for the curves y=xny = x^n for n=1,2,3,4,n = 1, 2, 3, 4, and 55 show distinct characteristics:

  • n=1n = 1: A straight line with slope 1 through the origin.
  • n=2n = 2: A parabola that opens upward with its vertex at the origin.

Shape of the Curve for n2n \geq 2

The shape of the curve y=xny = x^n depends on whether nn is even or odd:

Even Values of nn

  • Symmetry: Graphs are symmetric about the y-axis.
  • Shape: Resemble the graph of y=x2y = x^2.
  • Key Points: Pass through (1,1)(-1, 1), (0,0)(0, 0), and (1,1)(1, 1).
  • Behavior: As nn increases, graphs flatten over the interval 1<x<1-1 \lt x \lt 1 and become steeper for x>1x \gt 1 and x<1x \lt -1.

Odd Values of nn

  • Symmetry: Graphs are symmetric about the origin.
  • Shape: Resemble the graph of y=x3y = x^3.
  • Key Points: Pass through (1,1)(-1, -1), (0,0)(0, 0), and (1,1)(1, 1).
  • Behavior: Similar flattening and steepening as even nn, flattening over 1<x<1-1 \lt x \lt 1 and steepening for x>1x \gt 1 and x<1x \lt -1.

Remark on Flattening and Steepening

  • Interval 1<x<1-1 \lt x \lt 1:
    • As nn increases, the absolute value of xnx^n decreases, causing graphs to flatten.
  • Interval x>1x \gt 1 or x<1x \lt -1:
    • As nn increases, the absolute value of xnx^n increases, causing graphs to steepen.

This understanding of power functions is crucial for calculus, as it helps in analyzing the behavior of polynomial functions and their derivatives. The symmetry and transformation properties lay the foundation for more complex mathematical concepts.

Extended readings:

math.libretexts.org
3.3: Power Functions and Polynomial Functions - Math LibreTexts
math.libretexts.org
3.3: Power Functions and Polynomial Functions
www.khanacademy.org
End behavior of polynomials (article) - Khan Academy

Families of Functions

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The Family y=xny = x^{-n}

Graph Shape and Properties

  • Even nn:

    • Functions f(x)=1/xnf(x) = 1/x^n are even and symmetric about the yy-axis.
    • General shape resembles y=1/x2y = 1/x^2.
    • Pass through points (1,1)(-1, 1) and (1,1)(1, 1).
    • As nn increases, graphs become steeper for 1<x<0-1 \lt x \lt 0 and 0<x<10 \lt x \lt 1, but flatter for x>1x \gt 1 and x<1x \lt -1.
  • Odd nn:

    • Functions f(x)=1/xnf(x) = 1/x^n are odd and symmetric about the origin.
    • General shape resembles y=1/xy = 1/x.
    • Pass through points (1,1)(1, 1) and (1,1)(-1, -1).
    • As nn increases, graphs become steeper for 1<x<0-1 \lt x \lt 0 and 0<x<10 \lt x \lt 1, but flatter for x>1x \gt 1 and x<1x \lt -1.
  • Discontinuity:

    • Both even and odd nn exhibit a break at the origin because division by zero is undefined.

Insight

  • Symmetry helps predict the graph's behavior when altering nn.
  • Visualizing changes in steepness provides an understanding of function rates and limits.

Inverse Proportions

  • Definition:

    • A variable yy is inversely proportional to a variable xx if there is a positive constant kk such that y=kxy = \frac{k}{x}.
  • Behavior:

    • Graphs are similar in shape to y=1/xy = 1/x but can be compressed or stretched.
    • If xx doubles, yy becomes half; if xx triples, yy becomes one-third, etc.
  • Equation Relation:

    • Expressed as xy=kxy = k, indicating the product of the variables is a constant.

Insight

  • Understanding inverse relationships is vital in real-world applications, such as physics and economics.
  • The relationship xy=kxy = k simplifies identification and computation.

Example Analysis

Table 0.3.1

xxyy
0.86.25
2.52
41.25
6.251
100.8
0.510
  1. (a) Data Analysis:

    • The product xyxy remains constant, suggesting yy is inversely proportional to xx.
  2. (b) Function Expression:

    • y=kxy = \frac{k}{x}, where kk is a constant matching product values from the table.
  3. (c) Graphing:

    • Plot y=kxy = \frac{k}{x} and the given data for x>0x \gt 0 to visualize inverse proportionality.

Insight

  • Analyzing tables for inversely proportional data increases comprehension of how variables interact.
  • Graphs offer visual confirmation of mathematical relationships.

By understanding these mathematical concepts and relationships, we can interpret and predict various real-world phenomena.

Extended readings:

math.libretexts.org
2.4: Families of Functions - Mathematics LibreTexts
study.com
Family of Functions | Graphs & List - Lesson - Study.com
flexbooks.ck12.org
Function Families | CK-12 Foundation

Notes on Inverse Proportion and Power Functions

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Inverse Proportion

  • Definition: An inverse proportion between two variables xx and yy means xy=kxy = k for a constant kk. This implies that if one variable increases, the other decreases proportionally.

  • Example:

    • Equation: Given xy=5xy = 5, then y=5xy = \frac{5}{x}.
    • Graph: The graph is a hyperbola, shown in Figure 0.3.6, demonstrating inverse proportionality by the curve bending towards the axes as xx or yy increases.
  • Boyle's Law:

    • Statement: States that if a fixed amount of an ideal gas is held at a constant temperature, the product of the pressure PP and volume VV is constant, PV=kPV = k.
    • Graph: Illustrated in Figure 0.3.7, showing the inverse relationship, where doubling the pressure halves the volume.

Power Functions with Noninteger Exponents

  • General Form: If p=1np = \frac{1}{n}, where nn is a positive integer, then f(x)=xp=xnf(x) = x^p = \sqrt[n]{x}.

  • Special Cases:

    • f(x)=x1/2=xf(x) = x^{1/2} = \sqrt{x}
      • Domain: [0,+)[0, +\infty) because square roots of negative numbers are imaginary.
      • Graph: Extends only over non-negative xx.
    • f(x)=x1/3=x3f(x) = x^{1/3} = \sqrt[3]{x}
      • Domain: (,+)(-\infty, +\infty) since cube roots are defined for all real numbers.
      • Graph: Extends over the entire xx-axis.
  • Graph Behavior:

    • For y=xy = \sqrt{x} and y=xy = -\sqrt{x}, the graphs form the upper and lower halves of a parabola x=y2x = y^2 (Figure 0.3.8c).
    • Power functions with odd nn extend over the entire xx-axis, while those with even nn only extend over non-negative xx.
  • Examples with Fractional Exponents:

    • f(x)=x2/3f(x) = x^{2/3}
      • Graph: Figure 0.3.9 shows this function.
    • Other examples: f(x)=x35,f(x)=x7/8f(x) = \sqrt[5]{x^3}, f(x) = x^{-7/8}.

Technology Mastery

  • Graphing Utilities:

    • May omit portions due to complexity with fractional exponents.
    • Solutions:
      • If pp is even and qq is odd, use g(x)=xp/qg(x) = |x|^{p/q}.
      • If both pp and qq are odd, use g(x)=(xx)xp/qg(x) = \left(\frac{|x|}{x}\right)|x|^{p/q}.
  • Practice: Use graphing utilities to visualize functions like f(x)=x35f(x) = \sqrt[5]{x^3} and f(x)=x7/8f(x) = x^{-7/8}, capturing all significant features.

Extended readings:

math.libretexts.org
3.3: Power Functions and Polynomial Functions - Math LibreTexts
hhsmrgarvey.weebly.com
[PDF] 4.5-Power Functions and Equations
www.youtube.com
Power Functions with Rational (Fractional) Exponents Writing and ...

Polynomials and Rational Functions

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Polynomials

  • Definition: A polynomial in xx is a function expressible as a sum of terms cxnc \cdot x^n, where cc is a constant and nn is a nonnegative integer.
  • Examples:
    • 2x+12x + 1
    • 3x2+5x23x^2 + 5x - \sqrt{2}
    • x3x^3
    • 44 (can be represented as 4x04x^0)
    • 5x7x4+35x^7 - x^4 + 3

Key Concepts

  • Coefficients: Constants c_0,c_1,,c_nc\_0, c\_1, \ldots, c\_n in the expressions.
  • Degree: The highest power of xx with a nonzero coefficient. Polynomials can have the following degrees and corresponding names:
    • Linear: Degree 1, e.g., 3+5x3 + 5x
    • Quadratic: Degree 2, e.g., x23x+1x^2 - 3x + 1
    • Cubic: Degree 3, e.g., 2x372x^3 - 7
    • Quartic: Degree 4, e.g., 8x49x3+5x38x^4 - 9x^3 + 5x - 3
    • Quintic: Degree 5, e.g., 3+x3+x5\sqrt{3} + x^3 + x^5
  • Zero Polynomial: Considered to have an undefined degree.
  • Natural Domain: All polynomials have a natural domain of (,+)(-\infty, +\infty); however, their behavior varies by degree.

Polynomial Graphs

  • Characteristics: Smooth curves without discontinuities or sharp corners.
  • Behavior: Exhibits a wave-like pattern with peaks and valleys, degree nn has fewer than nn peaks/valleys.
  • Graph Shapes:
    • Degree 0 and 1: Line
    • Degree 2: Parabola
    • Higher degrees: More complex curves

Graph Examples

  • Degree 2, 3, 4, and 5 Graphs: Represent typical polynomial graphs of increasing complexity.

Rational Functions

  • Definition: A function expressed as the ratio of two polynomials, f(x)=P(x)Q(x)f(x) = \frac{P(x)}{Q(x)}.
  • Components:
    • P(x)P(x) and Q(x)Q(x) are polynomials.
  • Domain Note: Excludes values where Q(x)=0Q(x) = 0.

Additional Insights

  • Polynomial Expansion: Terms like (x24)3(x^2 - 4)^3 can be expanded using binomial formulas.
  • Roots and Behavior: Graphs of polynomials mirror underlying equations, reflecting root multiplicity and leading coefficient influence on end behavior.

By understanding fundamentals like polynomial degrees, coefficients, and graph behavior, complex functions like rational functions become more comprehensible.

Extended readings:

math.libretexts.org
1.6: Polynomials and Rational Functions - Mathematics LibreTexts
mathquest.carroll.edu
Powers, Polynomials, and Rational Functions
math.libretexts.org
1.4.3: Graphs of Polynomial Functions - Mathematics LibreTexts