Geometry, Trig & Calculus

Introduction to Differential Calculus and Fundamental Probability

Grade 11 · Grade 12

✓By the end of this lesson students will be able to determine the derivative of a function from first principles.
✓By the end of this lesson students will be able to apply rules of differentiation to find the derivative of polynomial functions.
✓By the end of this lesson students will be able to calculate measures of central tendency (mean, median, mode) for ungrouped data.
✓By the end of this lesson students will be able to apply the addition and multiplication rules for probability, distinguishing between mutually exclusive and independent events.

Key concepts

The Derivative from First Principles

The derivative of a function f(x) with respect to x, denoted as f'(x) or \frac{dy}{dx}, represents the instantaneous rate of change of the function at any point x. It is defined as the limit of the average gradient as the change in x (denoted by h or \Delta x) approaches zero.

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

Rules of Differentiation for Polynomials

For polynomial functions, we can use specific rules to find the derivative more efficiently than using first principles. The most common rule is the Power Rule, along with rules for constants and sums/differences of functions. These rules are derived from first principles.

If f(x) = cx^n, then f'(x) = cnx^{n-1}. If f(x) = c (a constant), then f'(x) = 0. If f(x) = g(x) \pm h(x), then f'(x) = g'(x) \pm h'(x).

Measures of Central Tendency

Measures of central tendency describe the 'centre' or typical value of a data set. The mean is the arithmetic average, the median is the middle value when data is ordered, and the mode is the most frequently occurring value.

\text{Mean (}\bar{x}\text{)} = \frac{\sum x}{n}

Basic Probability and Events

Probability is the measure of the likelihood of an event occurring. An event is a specific outcome or set of outcomes from an experiment. We distinguish between mutually exclusive events (cannot happen at the same time) and independent events (occurrence of one does not affect the probability of the other).

P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B). \text{If A and B are mutually exclusive, } P(A \text{ and } B) = 0, \text{ so } P(A \text{ or } B) = P(A) + P(B). \text{If A and B are independent, } P(A \text{ and } B) = P(A) \times P(B).

Key facts to remember

1The derivative f'(x) gives the gradient of the tangent to the curve y = f(x) at any point x.
2The power rule for differentiation states that if f(x) = ax^n, then f'(x) = nax^{n-1}.
3The mean (\bar{x}) is the sum of all values divided by the number of values (\frac{\sum x}{n}).
4The median is the middle value of an ordered data set. If there are two middle values, the median is their average.
5The mode is the value that appears most frequently in a data set.
6Mutually exclusive events cannot occur at the same time, so P(A \text{ and } B) = 0.
7Independent events are when the occurrence of one event does not affect the probability of the other, so P(A \text{ and } B) = P(A) \times P(B).
8The Addition Rule for probability is P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B).

Worked examples

Example 1

Determine the derivative of f(x) = 3x^2 - 2x + 5 from first principles.

If(x+h) = 3(x+h)^2 - 2(x+h) + 5

IIf(x+h) = 3(x^2 + 2xh + h^2) - 2x - 2h + 5

IIIf(x+h) = 3x^2 + 6xh + 3h^2 - 2x - 2h + 5

IVf(x+h) - f(x) = (3x^2 + 6xh + 3h^2 - 2x - 2h + 5) - (3x^2 - 2x + 5)

Vf(x+h) - f(x) = 6xh + 3h^2 - 2h

VI\frac{f(x+h) - f(x)}{h} = \frac{h(6x + 3h - 2)}{h}

VII\frac{f(x+h) - f(x)}{h} = 6x + 3h - 2

VIIIf'(x) = \lim_{h \to 0} (6x + 3h - 2)

9f'(x) = 6x - 2

Answer

f'(x) = 6x - 2

Ensure you expand (x+h)^2 correctly and factorise 'h' from the numerator before taking the limit. Do not substitute h=0 until after cancelling 'h'.

Example 2

Differentiate the following functions with respect to x: \n a) g(x) = 4x^3 - \frac{3}{x} + \sqrt{x} - 10 \n b) y = (2x+1)(x-3)

Ia) Rewrite g(x) in the form cx^n: g(x) = 4x^3 - 3x^{-1} + x^{1/2} - 10x^0

IIApply the power rule to each term: g'(x) = 4(3)x^{3-1} - 3(-1)x^{-1-1} + \frac{1}{2}x^{1/2-1} - 0

IIIg'(x) = 12x^2 + 3x^{-2} + \frac{1}{2}x^{-1/2}

IVg'(x) = 12x^2 + \frac{3}{x^2} + \frac{1}{2\sqrt{x}}

Vb) Expand the expression first: y = 2x^2 - 6x + x - 3

VIy = 2x^2 - 5x - 3

VIIDifferentiate term by term: \frac{dy}{dx} = 2(2)x^{2-1} - 5(1)x^{1-1} - 0

VIII\frac{dy}{dx} = 4x - 5

Answer

a) g'(x) = 12x^2 + \frac{3}{x^2} + \frac{1}{2\sqrt{x}} \n b) \frac{dy}{dx} = 4x - 5

Always rewrite terms with x in the denominator or under a root sign as x^n before differentiating. Expand products or quotients before differentiating, unless using the product/quotient rule (which is Grade 12 advanced content).

Example 3

A survey of 100 students showed that 40 students like Mathematics (M), 30 like Physical Sciences (P), and 10 like both. \n a) What is the probability that a randomly chosen student likes Mathematics or Physical Sciences? \n b) Are liking Mathematics and liking Physical Sciences mutually exclusive events? \n c) Are liking Mathematics and liking Physical Sciences independent events?

IGiven: Total students = 100. P(M) = \frac{40}{100} = 0.40, P(P) = \frac{30}{100} = 0.30, P(M \text{ and } P) = \frac{10}{100} = 0.10

IIa) Using the Addition Rule: P(M \text{ or } P) = P(M) + P(P) - P(M \text{ and } P)

IIIP(M \text{ or } P) = 0.40 + 0.30 - 0.10

IVP(M \text{ or } P) = 0.70 - 0.10 = 0.60

Vb) For mutually exclusive events, P(M \text{ and } P) must be 0. Since P(M \text{ and } P) = 0.10 \neq 0, liking Mathematics and Physical Sciences are NOT mutually exclusive events.

VIc) For independent events, P(M \text{ and } P) must be equal to P(M) \times P(P).

VIICalculate P(M) \times P(P) = 0.40 \times 0.30 = 0.12.

VIIISince P(M \text{ and } P) = 0.10 \neq 0.12, liking Mathematics and Physical Sciences are NOT independent events.

Answer

a) P(M or P) = 0.60 \n b) No, they are not mutually exclusive. \n c) No, they are not independent.

A Venn diagram can be very helpful to visualise the probabilities and relationships between events, especially for 'and' and 'or' scenarios.

Common mistakes

✗Algebraic errors when expanding (x+h)^2 or (x+h)^3 in first principles, leading to incorrect simplification.
✗Forgetting to rewrite terms like \frac{1}{x^n} or \sqrt{x} as x^{-n} or x^{1/2} before applying the power rule.
✗Confusing mutually exclusive events with independent events; they are distinct concepts with different conditions.
✗Incorrectly applying the Addition Rule for probability, especially forgetting to subtract P(A \text{ and } B) when events are not mutually exclusive.
✗Not ordering data before attempting to find the median.

Exam tips

★For first principles questions, show all steps clearly, including the factorisation of 'h' from the numerator and the limit notation (\lim_{h \to 0}).
★Always simplify the function as much as possible (e.g., expand brackets, rewrite surds/fractions with exponents) before differentiating using the rules.
★When solving probability problems, carefully read the question to identify whether events are mutually exclusive or independent, as this determines which formula to use.
★Draw Venn diagrams or tree diagrams for probability problems to help visualise the situation and verify your calculations, especially for complex scenarios.

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