Grade 11
  1. Algebra  1.1. Sequences and Series  1.1.1. Arithmetic Sequence  1.1.2. Arithmetic Series  1.1.3. Geometric Sequence  1.1.4. Geometric Series  1.1.5. Convergence and Divergence  1.1.6. Sum to Infinity  1.1.7. Sigma Notation  1.2. Complex Numbers  1.2.1. Imaginary and Complex Numbers  1.2.2. Operations with Complex Numbers  1.2.3. Polar and Cartesian Forms  1.2.4. Complex Conjugates  1.2.5. Modulus and Argument  1.2.6. De Moivre's Theorem  1.2.7. Powers and Roots of Complex Numbers  1.3. Binomial Theorem  1.3.1. Expansion of Binomials  1.3.2. General Term  1.3.3. Binomial Coefficients  1.3.4. Applications of Binomial Theorem  1.3.5. Pascal's Triangle
  2. Functions and Graphs  2.1. Types of Functions  2.1.1. Polynomial Functions  2.1.2. Rational Functions  2.1.3. Exponential Functions  2.1.4. Logarithmic Functions  2.1.5. Trigonometric Functions  2.1.6. Piecewise Functions  2.2. Transformations of Functions  2.2.1. Translations  2.2.2. Reflections  2.2.3. Dilations and Stretches  2.2.4. Rotations  2.3. Inverse Functions  2.3.1. Finding Inverses  2.3.2. Graphs of Inverse Functions  2.3.3. Properties of Inverse Functions  2.3.4. Restrictions for Inverses
  3. Trigonometry  3.1. Trigonometric Ratios and Identities  3.1.1. Basic Trigonometric Ratios  3.1.2. Pythagorean Identities  3.1.3. Sum and Difference Formulas  3.1.4. Double Angle Formulas  3.1.5. Half Angle Formulas  3.1.6. Product-to-Sum and Sum-to-Product Formulas  3.2. Trigonometric Equations  3.2.1. Solving Trigonometric Equations  3.2.2. General Solutions in Trigonometric Equations  3.3. Graphs of Trigonometric Functions  3.3.1. Sine Graph  3.3.2. Cosine Graph  3.3.3. Tangent Graph  3.3.4. Amplitude and Period Adjustments  3.3.5. Phase Shifts in Graphs of Trigonometric Functions  3.4. Applications of Trigonometry  3.4.1. Laws of Sines and Cosines  3.4.2. Area of Triangles  3.4.3. Solving Triangles  3.4.4. Bearings and Navigation
  4. Calculus  4.1. Limits and Continuity  4.1.1. Concept of a Limit  4.1.2. One-Sided Limits  4.1.3. Limits at Infinity  4.1.4. Continuity of a Function  4.1.5. Types of Discontinuities  4.2. Differentiation  4.2.1. Definition of Derivative  4.2.2. Rules of Differentiation  4.2.3. Higher Order Derivatives  4.2.4. Chain Rule  4.2.5. Product Rule  4.2.6. Quotient Rule  4.3. Applications of Differentiation  4.3.1. Rate of Change  4.3.2. Tangents and Normals  4.3.3. Maxima and Minima  4.3.4. Increasing and Decreasing Functions  4.3.5. Optimization Problems  4.3.6. Curve Sketching  4.3.7. Related Rates  4.4. Integration  4.4.1. Antiderivatives  4.4.2. Indefinite Integrals  4.4.3. Definite Integrals  4.4.4. Fundamental Theorem of Calculus  4.4.5. Techniques of Integration  4.4.6. Area Under a Curve  4.5. Applications of Integration  4.5.1. Area Between Curves  4.5.2. Volumes of Solids of Revolution  4.5.3. Average Value of a Function
  5. Vectors and Matrices  5.1. Vectors  5.1.1. Representation of Vectors  5.1.2. Vector Operations  5.1.3. Dot Product  5.1.4. Cross Product  5.1.5. Applications in Geometry  5.1.6. Projection of Vectors  5.2. Matrices  5.2.1. Types of Matrices  5.2.2. Matrix Operations  5.2.3. Determinants  5.2.4. Inverse of a Matrix  5.2.5. Solving Systems of Linear Equations  5.2.6. Cramer's Rule
  6. Probability and Statistics  6.1. Probability  6.1.1. Basic Probability Concepts  6.1.2. Conditional Probability  6.1.3. Independent and Dependent Events  6.1.4. Bayes' Theorem  6.1.5. Combinatorial Probability  6.2. Random Variables  6.2.1. Discrete Random Variables  6.2.2. Continuous Random Variables  6.2.3. Probability Distributions  6.3. Probability Distributions  6.3.1. Binomial Distribution  6.3.2. Normal Distribution  6.3.3. Poisson Distribution  6.3.4. Uniform Distribution  6.4. Statistics  6.4.1. Measures of Central Tendency  6.4.2. Measures of Dispersion  6.4.3. Data Collection and Representation  6.4.4. Correlation and Regression  6.4.5. Sampling Techniques
  7. Coordinate Geometry  7.1. Straight Lines  7.1.1. Slope and Intercept Form  7.1.2. Distance Formula  7.1.3. Midpoint Formula  7.1.4. Angle Between Lines  7.1.5. Equation of Lines  7.2. Conic Sections  7.2.1. Circle  7.2.2. Parabola  7.2.3. Ellipse  7.2.4. Hyperbola  7.2.5. Properties of Conics  7.2.6. Parametric Equations of Conics  7.2.7. Polar Coordinates of Conics
  8. Mathematical Reasoning  8.1. Logic  8.1.1. Statements and Logical Connectives  8.1.2. Truth Tables  8.1.3. Logical Equivalence  8.1.4. Quantifiers  8.1.5. Proof Techniques  8.2. Proofs  8.2.1. Direct Proof  8.2.2. Indirect Proof  8.2.3. Proof by Contradiction  8.2.4. Proof by Mathematical Induction

Grade 11 → Calculus → Applications of Differentiation ↓


Curve Sketching


Curve sketching is a method used in calculus to create a rough graph of a function based on its derivatives. Understanding the important features of a graph can help in understanding the behavior of various functions in mathematics. This skill is particularly useful for visualizing the shape and position of a function without using a graphing technique. Curve sketching involves analyzing the function through differentiation, providing information about its increasing and decreasing behavior, bend, concavity, and asymptotic behavior.

Basic concepts

Differentiation is the mathematical process of finding the derivative of a function, which expresses how the output of a function changes as its input changes. To sketch a curve effectively, one must understand the first and second derivatives of a function:

  1. First derivative (f'(x)): It provides information about the slope of the tangent to the curve at any given point. When f'(x) > 0, the function is increasing, and when f'(x) < 0, it is decreasing.
  2. Second derivative (f''(x)): This provides information about the curvature of the function. A positive second derivative suggests that the function is concave upward, while a negative second derivative indicates that it is concave downward.

Steps of curve sketching

The following steps outline a structured approach to graphing the curve of a function:

1. Identify the domain

The domain of a function is the set of all possible inputs (or x values) for which the function is defined. Understanding the domain helps avoid evaluating the function at points where it is not defined, such as division by zero or a negative value under a square root for a real function.

2. Set the blocking

Find the x-intercept and y-intercept of the function. x intercept is found by setting f(x) = 0 and solving for x. y intercept is found by evaluating f(0).

3. Analyze symmetry (if applicable)

Determine if the function has any symmetry, since symmetric functions have repeating shapes that make graphing simpler. - Even functions are symmetric about y axis. - Odd functions are symmetric about the origin.

4. Find the derivative

Calculate the first derivative f'(x) and the second derivative f''(x) to study the behavior of the function.

5. Determine the critical points

Critical points occur where the first derivative is zero or undefined. These points are possible locations for local maximums, minimums, or inflection points.

6. Test interval

Use the intervals determined by the critical points to understand where the function is increasing or decreasing.

7. Analyze the concavity

Using the second derivative, determine the intervals where the function is concave up or concave down. The points where the concavity changes are the points of inflection.

8. Evaluate asymptotic behavior

Identify any horizontal or vertical asymptotes that the curve reaches as x goes to infinity or to certain fixed values, respectively.

9. Draw curves

Compile all the information you have gathered to create a rough outline of the graph of the function, and mark the intercepts, critical points, and asymptotes.

Examples of curve graphing

Example 1: Quadratic function

Consider the function f(x) = x^2 - 4x + 3.

Step-by-step process:

  1. Set the blocking:
    • x -intercept: Solve x^2 - 4x + 3 = 0 Factoring gives (x - 1)(x - 3) = 0 Thus, x = 1 and x = 3.
    • y intercept: f(0) = 0^2 - 4*0 + 3 = 3.
  2. Find the first derivative: 
    f'(x) = 2x - 4
  3. Find the critical points by setting f'(x) = 0: 
    2x – 4 = 0
    On solving, we get x = 2.
  4. Use the second derivative to analyze the concavity: 
    f''(x) = 2
    Since f''(x) = 2 gt 0, the function is always concave upward.
  5. Sketch the curve using the information you gathered.
(1,0) (3,0) (0,3)

As shown in the example, the quadratic f(x) = x^2 - 4x + 3 passes through a minimum at x = 2, which is the vertex of the parabola.

Example 2: Rational function

Consider the function g(x) = (x^2 - 1)/(x - 2).

Step-by-step process:

  1. Set the blocking:
    • x-intercept: Set g(x) = 0 to get x^2 - 1 = 0 Therefore, x = ±1.
    • y intercept: g(0) = (0^2 - 1)/(0 - 2) = 1/2.
  2. Domain: The function is undefined at x = 2 (vertical asymptote).
  3. Find the first derivative using the quotient rule: 
    G'(x) = frac{(2x)(x - 2) - (x^2 - 1)(1)}{(x - 2)^2}
  4. Analyze asymmetric behavior:
    • Horizontal asymptote: Since x → ±∞, g(x) → x
  5. Sketch the curve.
(-1,0) (1,0) (x=2)

Graphing the rational function g(x) = frac{(x^2 - 1)}{x - 2} by completing the above steps reveals x and y-intercepts, behavior, and asymptotes.

Conclusion

Curve sketching is a graphical topic in mathematics that leverages the principles of calculus to interpret functions visually. By following a structured process, one can create an accurate representation of a function's behavior. Understanding how to calculate derivatives and how to use them to identify key features such as intercepts, critical points, concavities, and asymptotes is essential to mastering curve sketching. The value of this skill is evident in both mathematical theory and real-world applications, as it helps bring complex functions to life through visual representation.


Grade 11 → 4.3.6


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