Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Apply differentiation rules to analyze functions.
• Solve optimization problems using calculus.
• Calculate definite and indefinite integrals.
• Solve probability problems using counting principles.
• Use tree and Venn diagrams for probability.

Lesson summary

This week focuses on consolidating your understanding of Calculus (Differentiation and Integration) and Probability. These are critical sections in the Grade 12 Mathematics curriculum and frequently feature prominently in the final examination. Mastery of these topics is not only crucial for your final grade but also provides a strong foundation for future studies in fields like engineering, economics, statistics, and data science. In South Africa, understanding calculus helps in analyzing economic trends, predicting population growth, and optimizing resource allocation.

Lesson notes

2.1 Calculus: Differentiation Differentiation is the process of finding the derivative of a function. The derivative represents the instantaneous rate of change of a function at a particular point.

Power Rule: If f(x) = x n , then f'(x) = nx n-1 Constant Multiple Rule: If f(x) = kg(x), where k is a constant, then f'(x) = k*g'(x)

Sum/Difference Rule: If f(x) = u(x) ± v(x), then f'(x) = u'(x) ± v'(x)

Product Rule: If f(x) = u(x) v(x), then f'(x) = u'(x)v(x) + u(x)v'(x)

Quotient Rule: If f(x) = u(x) / v(x), then f'(x) = [u'(x)v(x) - u(x)v'(x)] / [v(x)] 2 Chain Rule: If f(x) = g(h(x)), then f'(x) = g'(h(x)) h'(x)

Applications of Differentiation: Stationary Points: Points where f'(x) =

0. These points can be local maxima, local minima, or points of inflection.

Intervals of Increase/Decrease: If f'(x) > 0, the function is increasing. If f'(x) 0, the function is concave up. If f''(x) 4 - 2x 2 + 5x - 7 Solution: f'(x) = 3(4x 3 ) - 2(2x) + 5 - 0 (Applying power rule, constant multiple rule, and sum/difference rule) f'(x) = 12x 3 - 4x + 5 Example 2: Find the derivative of f(x) = (x 2 + 1)(2x - 3)

Solution: Let u(x) = x 2 + 1 and v(x) = 2x - 3 u'(x) = 2x and v'(x) = 2 f'(x) = (2x)(2x - 3) + (x 2 + 1)(2) (Applying product rule) f'(x) = 4x 2 - 6x + 2x 2 + 2 f'(x) = 6x 2 - 6x + 2 Example 3: A farmer wants to fence off a rectangular field along a straight river. No fence is needed along the river. If the farmer has 1000 meters of fencing, what are the dimensions of the field that maximize the area?

Solution: Let x be the width of the field and y be the length along the river.

Perimeter: 2x + y = 1000 => y = 1000 - 2x Area: A = x*y = x(1000 - 2x) = 1000x - 2x 2 To maximize the area, find the critical points: A'(x) = 1000 - 4x = 0 x = 250 y = 1000 - 2(250) = 500 Therefore, the dimensions are 250 meters by 500 meters. 2.2 Calculus: Integration Integration is the reverse process of differentiation. It involves finding a function whose derivative is a given function.

Power Rule: ∫x n dx = (x n+1 )/(n+1) + C, where n ≠ -1 and C is the constant of integration.

Constant Multiple Rule: ∫kf(x) dx = k∫f(x) dx, where k is a constant.

Sum/Difference Rule: ∫[f(x) ± g(x)] dx = ∫f(x) dx ± ∫g(x) dx Definite Integrals: Used to calculate the area under a curve between two points. The Fundamental Theorem of Calculus states that if F'(x) = f(x), then ∫ a b f(x) dx = F(b) - F(a).

Example 1: Find the indefinite integral of ∫(4x 3 - 6x 2 + 2x + 1) dx Solution: ∫(4x 3 - 6x 2 + 2x + 1) dx = 4∫x 3 dx - 6∫x 2 dx + 2∫x dx + ∫1 dx = 4(x 4 /4) - 6(x 3 /3) + 2(x 2 /2) + x + C = x 4 - 2x 3 + x 2 + x + C Example 2: Evaluate the definite integral ∫ 1 3 (2x + 3) dx Solution: ∫ 1 3 (2x + 3) dx = [x 2 + 3x] 1 3 (Applying the power rule and sum rule) = [(3 2 + 3(3)) - (1 2 + 3(1))] = (9 + 9) - (1 + 3) = 18 - 4 = 14 2.3 Probability Probability is the measure of the likelihood that an event will occur.

Basic Probability: P(A) = Number of favorable outcomes / Total number of possible outcomes Permutations: The number of ways to arrange n distinct objects in a specific order. nPr = n! / (n-r)! where r is the number of objects selected from n.

Combinations: The number of ways to choose r objects from n distinct objects without regard to order. nCr = n! / [r!(n-r)!] Conditional Probability: The probability of an event A occurring, given that event B has already occurred. P(A|B) = P(A ∩ B) / P(B)

Independent Events: Two events are independent if the occurrence of one does not affect the probability of the other. P(A ∩ B) = P(A)

P(B)

Tree Diagrams: Visual tools used to represent and calculate probabilities in sequential events.

Venn Diagrams: Visual tools used to represent relationships between sets and probabilities.

Example 1: A bag contains 5 red balls and 3 blue balls. What is the probability of drawing a red ball?

Solution: P(Red) = Number of red balls / Total number of balls = 5 / (5 + 3) = 5/8 Example 2: How many ways can you arrange 4 books on a shelf?

Solution: This is a permutation problem since the order matters. Number of ways = 4! = 4 3 2 * 1 = 24 Example 3: A committee of 3 people is to be chosen from a group of 6 people. How many different committees can be formed?

Solution: This is a combination problem since the order does not matter. Number of committees = 6C3 = 6! / (3! 3!) = (6 5 4) / (3 2 * 1) = 20 Example 4: A box contains 2 defective light bulbs and 18 good light bulbs. Two bulbs are selected at random without replacement. What is the probability that both bulbs are defective?