Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Solve optimization problems using differential calculus.
• Apply the fundamental counting principles.
• Use geometric series for annuities and sinking funds.
• Apply trigonometric identities and equations.
• Interpret and analyze statistical data.

Lesson summary

This week's focus is on consolidating key concepts and refining problem-solving skills across the Grade 12 Mathematics curriculum. Final examination preparation is crucial, and this week's structured revision aims to boost your confidence and preparedness. Mastering these mathematical principles is essential not only for academic success but also for informed decision-making in various aspects of life. For example, understanding financial mathematics is vital for managing personal finances and understanding economic trends, crucial in the South African context where financial literacy is paramount.

Lesson notes

This week's revision covers the following key areas: Optimization Problems, Counting Principles, Financial Mathematics (Annuities and Sinking Funds), Trigonometry, and Statistics.

A. Optimization Problems (Calculus): Optimization involves finding the maximum or minimum value of a function subject to certain constraints. This often involves finding the critical points of a function (where the derivative is zero or undefined) and using the first or second derivative test to determine if these points correspond to maxima, minima, or points of inflection.

Steps for Solving Optimization Problems: Understand the problem: Read the problem carefully and identify the quantity to be optimized (maximized or minimized) and the constraints.

Draw a diagram: If possible, draw a diagram to visualize the problem.

Introduce variables: Assign variables to the relevant quantities.

Formulate a function: Express the quantity to be optimized as a function of the variables.

Use constraints to eliminate variables: If necessary, use the constraints to express the function in terms of a single variable.

Find critical points: Find the derivative of the function and set it equal to zero to find the critical points. Also, consider points where the derivative is undefined.

Determine maximum or minimum: Use the first or second derivative test to determine if the critical points correspond to maxima or minima.

Answer the question: State the maximum or minimum value and the values of the variables that achieve it.

Example: A farmer in the Free State wants to build a rectangular enclosure for his sheep. He has 100 meters of fencing. What dimensions should the enclosure have to maximize the area? Let the length of the enclosure be l and the width be w. The perimeter is 2l + 2w = 100, so l + w = 50, and w = 50 - l. The area is A = l w = l(50 - l) = 50l - l 2 . To maximize the area, find the derivative: dA/dl = 50 - 2l.

Set the derivative equal to zero: 50 - 2l = 0, so l =

2

5. Then w = 50 - 25 =

2

5. The second derivative is d 2 A/dl 2 = -2, which is negative, so l = 25 is a maximum.

Therefore, the enclosure should be a square with sides of 25 meters to maximize the area.

B. Counting Principles (Permutation and Combination): Fundamental Counting Principle: If there are m ways to do one thing and n ways to do another, then there are m n ways to do both.

Permutation: An arrangement of objects in a specific order. The number of permutations of n objects taken r at a time is given by: P(n, r) = n! / (n - r)!

Combination: A selection of objects where order does not matter. The number of combinations of n objects taken r at a time is given by: C(n, r) = n! / (r! (n - r)!)

Example: How many different 3-digit numbers can be formed using the digits 1, 2, 3, 4, and 5 if repetition is allowed? If repetition is not allowed?

Repetition Allowed: For each digit, there are 5 choices. So, the total number of 3-digit numbers is 5 5 * 5 =

1

2

5. Repetition Not Allowed: This is a permutation problem. We are choosing 3 digits from 5 without repetition and order matters. So, the number of 3-digit numbers is P(5, 3) = 5! / (5 - 3)! = 5! / 2! = (5 4 3 2 1) / (2 1) =

6

0. C. Financial Mathematics (Annuities and Sinking Funds): Annuity: A series of equal payments made at regular intervals.

Future Value of an Annuity: The total value of the payments and interest earned at the end of the annuity period. FV = P * [((1 + i) n - 1) / i] where P = payment, i = interest rate per period, n = number of periods.

Present Value of an Annuity: The amount of money needed today to fund a series of future payments. PV = P * [(1 - (1 + i) -n ) / i] where P = payment, i = interest rate per period, n = number of periods.

Sinking Fund: An account where regular payments are made to accumulate a specific amount of money in the future. The formula is derived from the future value of an annuity.

Example: A recent graduate in Gauteng wants to save R500 per month for 5 years for a deposit on a house. If the interest rate is 8% per annum, compounded monthly, how much will she have saved after 5 years? P = R500 i = 8% / 12 = 0.08/12 = 0.0066667 (approximately) n = 5 years 12 months/year = 60 months FV = 500 [((1 + 0.0066667) 60 - 1) / 0.0066667] FV ≈ R36,707.56

D. Trigonometry: Trigonometric Identities: Equations that are true for all values of the variables for which the expressions are defined (e.g., sin 2 θ + cos 2 θ = 1, tan θ = sin θ / cos θ).

Trigonometric Equations: Equations involving trigonometric functions (e.g., sin x = 0.5).

Compound Angle Formulae: Formulae for trigonometric functions of sums and differences of angles (e.g., sin(A + B) = sin A cos B + cos A sin B).

Double Angle Formulae: Formulae for trigonometric functions of double angles (e.g., sin 2A = 2 sin A cos A). Solving 2D and 3D Problems using Trigonometry: Applying trigonometric ratios (sine, cosine, tangent) to solve problems involving triangles and angles in two and three dimensions.

The Sine rule: a/sinA = b/sinB = c/sinC.