Lesson Notes By Weeks and Term v5 - Grade 12

Lesson Video

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Performance objectives

• Apply differentiation rules to various functions.
• Use derivatives to find stationary points and sketch cubic graphs.
• Solve optimization problems.
• Calculate probabilities for independent and dependent events.
• Use counting principles in probability problems.

Lesson summary

This week's focus is on consolidating and strengthening your understanding of Calculus and Probability, two crucial sections in the Grade 12 Mathematics syllabus. Calculus is the mathematical study of continuous change, vital for modeling real-world phenomena from population growth to the trajectory of a soccer ball. Probability allows us to quantify uncertainty, a skill indispensable in various fields, including finance, sports analytics, and even understanding disease outbreaks like the COVID-19 pandemic. In South Africa, understanding interest rates (calculus based models) and insurance (probability) are extremely important for managing your personal finances effectively.

Lesson notes

Calculus: Differentiation 2.1 Differentiation Rules: Differentiation is the process of finding the derivative of a function, which represents the instantaneous rate of change of the function.

The following are key rules: Power Rule: If \(f(x) = ax^n\), then \(f'(x) = nax^{n-1}\)

Constant Rule: If \(f(x) = c\), where \(c\) is a constant, then \(f'(x) = 0\)

Constant Multiple Rule: If \(f(x) = c \cdot g(x)\), then \(f'(x) = c \cdot g'(x)\)

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

Chain Rule: If \(f(x) = g(h(x))\), then \(f'(x) = g'(h(x)) \cdot h'(x)\)

Example 1: Find the derivative of \(f(x) = 3x^4 - 2x^2 + 5x - 7\).

Solution: Applying the power rule, constant multiple rule, and sum/difference rule: \(f'(x) = 3(4x^3) - 2(2x) + 5(1) - 0\) \(f'(x) = 12x^3 - 4x + 5\) 2.2 Stationary Points and Graph Sketching: Stationary Points: Points where the derivative is equal to zero, i.e., \(f'(x) = 0\). These points can be local maxima, local minima, or points of inflection.

Increasing/Decreasing Intervals: If \(f'(x) > 0\), the function is increasing. If \(f'(x) 0\), the function is concave up (like a cup). If \(f''(x) 0\) (Increasing) \(-1 3\): \(f'(4) = 3(4)^2 - 6(4) - 9 = 48 - 24 - 9 = 15 > 0\) (Increasing) Therefore, the function is increasing for \(x 3\), and decreasing for \(-1 \(l + w = 50\) => \(l = 50 - w\)

Area: \(A = lw = (50 - w)w = 50w - w^2\) To maximize the area, we find the derivative and set it to zero: \(A'(w) = 50 - 2w\) \(50 - 2w = 0\) \(2w = 50\) \(w = 25\) Therefore, \(l = 50 - 25 = 25\) The enclosure should be a square with sides of 25 meters to maximize the area. Probability 2.4 Independent and Dependent Events: Independent Events: The outcome of one event does not affect the outcome of the other. \(P(A \text{ and } B) = P(A) \cdot P(B)\)

Dependent Events: The outcome of one event does affect the outcome of the other. \(P(A \text{ and } B) = P(A) \cdot P(B|A)\), where \(P(B|A)\) is the probability of B given that A has occurred.

Example 4: A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random, one after the other, without replacement. What is the probability that both balls are red?

Solution: \(P(\text{First ball is red}) = \frac{5}{8}\) \(P(\text{Second ball is red | First ball is red}) = \frac{4}{7}\) (Since one red ball has been removed) Therefore, \(P(\text{Both balls are red}) = \frac{5}{8} \cdot \frac{4}{7} = \frac{20}{56} = \frac{5}{14}\) 2.5 Tree Diagrams and Venn Diagrams: These are visual tools to represent probabilities.

Tree Diagrams: Useful for sequential events.

Venn Diagrams: Useful for representing sets and their intersections.

Example 5: A survey of 100 students showed that 60 take Mathematics, 40 take Physics, and 20 take both. Draw a Venn diagram and find the probability that a randomly selected student takes either Mathematics or Physics.

Solution: Students taking only Mathematics: 60 - 20 = 40 Students taking only Physics: 40 - 20 = 20 Students taking neither: 100 - (40 + 20 + 20) = 20 \(P(\text{Mathematics or Physics}) = \frac{40 + 20 + 20}{100} = \frac{80}{100} = \frac{4}{5}\) 2.6 Counting Principles (Permutations and Combinations): Permutation: An arrangement of objects in a specific order. \(P(n, r) = \frac{n!}{(n-r)!}\) (where n is the total number of objects and r is the number of objects being arranged)

Combination: A selection of objects where order does not matter. \(C(n, r) = \binom{n}{r} = \frac{n!}{r!(n-r)!}\)

Example 6: How many different ways can you arrange the letters in the word "MATHEMATICS"?

Solution: There are 11 letters in the word "MATHEMATICS".

However, some letters are repeated: M (2 times), A (2 times), T (2 times).

The number of arrangements is: \(\frac{11!}{2!2!2!} = \frac{39916800}{8} = 4989600\) Guided Practice (With Solutions)

Question 1: Find the derivative of \(f(x) = \frac{2x^3 - 5x + 1}{x}\).

Solution: First, simplify the function: \(f(x) = 2x^2 - 5 + \frac{1}{x} = 2x^2 - 5 + x^{-1}\) Now, differentiate: \(f'(x) = 4x - 0 - x^{-2} = 4x - \frac{1}{x^2}\)

Commentary: The key here is to simplify the function before differentiating. This makes applying the power rule much easier.

Question 2: Determine the maximum area of a rectangle with a perimeter of 40 cm.

Solution: Let length = l, width = w.

Perimeter: \(2l + 2w = 40\) => \(l + w = 20\) => \(l = 20 - w\)

Area: \(A = lw = (20 - w)w = 20w - w^2\) \(A'(w) = 20 - 2w\) Set \(A'(w) = 0\): \(20 - 2w = 0\) => \(w = 10\) So, \(l = 20 - 10 = 10\) Maximum Area = \(10 \times 10 = 100\) cm²

Commentary: This is a classic optimization problem. Remember to express the area in terms of a single variable before differentiating.

Question 3: A box contains 4 green balls and 6 yellow balls. Two balls are drawn at random, with replacement. What is the probability that the first ball is green and the second ball is yellow?

Solution: Since the balls are drawn with replacement, the events are independent.