Geometry, Trig & Calculus

Introductory Calculus, Statistics & Probability

SSS 1 · SSS 2 · SSS 3

✓By the end of this lesson students will be able to differentiate simple polynomial functions from first principles and using the power rule.
✓By the end of this lesson students will be able to integrate simple polynomial functions.
✓By the end of this lesson students will be able to calculate measures of central tendency (mean, median, mode) for grouped and ungrouped data.
✓By the end of this lesson students will be able to calculate measures of dispersion (range, variance, standard deviation) for grouped and ungrouped data.
✓By the end of this lesson students will be able to solve basic probability problems involving mutually exclusive and independent events.

Key concepts

Differentiation from First Principles

Differentiation is the process of finding the rate of change of a function with respect to its variable. From first principles, the derivative of a function y = f(x) is given by considering a small change in x, denoted as δx, which causes a small change in y, denoted as δy. The derivative, dy/dx, is the limit of the ratio δy/δx as δx approaches zero.

dy/dx = lim (δx→0) [f(x+δx) - f(x)] / δx

Differentiation (Power Rule)

The power rule is a fundamental rule for differentiating polynomial functions. If a function is of the form y = ax^n, where 'a' is a constant and 'n' is a real number, its derivative can be found by multiplying the coefficient by the power and then reducing the power by one.

If y = ax^n, then dy/dx = nax^(n-1)

Integration (Power Rule)

Integration is the reverse process of differentiation. It is used to find the original function when its derivative is known. For polynomial functions, the power rule for integration involves increasing the power by one and dividing the term by the new power. For indefinite integrals, a constant of integration 'C' must always be added.

∫ax^n dx = [a/(n+1)]x^(n+1) + C (for n ≠ -1)

Measures of Central Tendency

These are statistical measures that describe the central position of a set of data. They include the Mean (average), Median (middle value when data is ordered), and Mode (most frequent value). For grouped data, specific formulas are used to estimate these values.

Mean (grouped data): Σfx / Σf\nMedian (grouped data): L + [(N/2 - cf_b) / f_m] * c\nMode (grouped data): L + [ (f_m - f_b) / (2f_m - f_b - f_a) ] * c\n(where L=lower class boundary, N=total frequency, cf_b=cumulative frequency before median class, f_m=frequency of median/modal class, c=class width, f_b=frequency before modal class, f_a=frequency after modal class)

Measures of Dispersion

These measures describe how spread out the data points are. They include the Range (difference between highest and lowest values), Variance (average of the squared differences from the mean), and Standard Deviation (square root of the variance).

Variance (grouped data): [Σfx² / Σf] - (Σfx / Σf)²\nStandard Deviation: √Variance

Basic Probability

Probability is the measure of the likelihood that an event will occur. It is expressed as a number between 0 and 1. An event is a subset of the sample space (all possible outcomes). Mutually exclusive events cannot happen at the same time. Independent events do not affect each other's occurrence.

P(E) = (Number of favourable outcomes) / (Total number of possible outcomes)\nP(A or B) = P(A) + P(B) (for mutually exclusive events)\nP(A and B) = P(A) * P(B) (for independent events)

Key facts to remember

1The derivative dy/dx represents the instantaneous rate of change of y with respect to x, or the gradient of the tangent to the curve at a point.
2The derivative of a constant is zero.
3Integration is the reverse process of differentiation; indefinite integrals always require a constant of integration, 'C'.
4For grouped data, the mean is Σfx / Σf, where x is the midpoint of each class interval.
5The median is the middle value when data is arranged in order; for grouped data, it is found using interpolation.
6The mode is the most frequent value; for grouped data, it is the class with the highest frequency (modal class), and the exact mode is found by interpolation.
7Variance measures the average squared deviation from the mean, while standard deviation is its square root, providing a measure of spread in the original units.
8Probability of an event E, P(E), is between 0 and 1, inclusive. P(E) = 0 means impossible, P(E) = 1 means certain.

Worked examples

Example 1

1. (a) Differentiate y = 3x² - 5x + 2 with respect to x from first principles.\n(b) Find the gradient of the curve y = 2x³ - 4x + 1 at the point where x = 2.

I(a) Given y = f(x) = 3x² - 5x + 2

IIf(x+δx) = 3(x+δx)² - 5(x+δx) + 2

III= 3(x² + 2xδx + (δx)²) - 5x - 5δx + 2

IV= 3x² + 6xδx + 3(δx)² - 5x - 5δx + 2

Vf(x+δx) - f(x) = (3x² + 6xδx + 3(δx)² - 5x - 5δx + 2) - (3x² - 5x + 2)

VI= 6xδx + 3(δx)² - 5δx

VII[f(x+δx) - f(x)] / δx = (6xδx + 3(δx)² - 5δx) / δx

VIII= 6x + 3δx - 5

9dy/dx = lim (δx→0) (6x + 3δx - 5)

10= 6x - 5

11(b) Given y = 2x³ - 4x + 1

12Differentiating using the power rule:

13dy/dx = d/dx (2x³) - d/dx (4x) + d/dx (1)

14= 2 * 3x^(3-1) - 4 * 1x^(1-1) + 0

15= 6x² - 4x⁰

16= 6x² - 4

17To find the gradient at x = 2, substitute x = 2 into dy/dx:

18dy/dx |_(x=2) = 6(2)² - 4

19= 6(4) - 4

20= 24 - 4

21= 20

Answer

(a) dy/dx = 6x - 5\n(b) Gradient = 20

Remember to show all steps clearly, especially for differentiation from first principles, as it carries significant marks in exams.

Example 2

2. The table below shows the distribution of marks scored by 50 students in a Mathematics test.\n\nMarks (x) | Frequency (f)\n----------|--------------\n1-10 | 5\n11-20 | 8\n21-30 | 12\n31-40 | 15\n41-50 | 7\n51-60 | 3\n\nCalculate, correct to one decimal place:\n(a) The mean mark.\n(b) The median mark.\n(c) The standard deviation.

IFirst, create a frequency distribution table with class boundaries, midpoints (x), fx, x², and fx².

IIClass | Class Boundaries | f | Midpoint (x) | fx | Cumulative Frequency (cf) | x² | fx²

III------|------------------|---|--------------|----|-------------------------|----|------

IV1-10 | 0.5-10.5 | 5 | 5.5 | 27.5 | 5 | 30.25 | 151.25

V11-20 | 10.5-20.5 | 8 | 15.5 | 124.0 | 13 | 240.25 | 1922.00

VI21-30 | 20.5-30.5 | 12 | 25.5 | 306.0 | 25 | 650.25 | 7803.00

VII31-40 | 30.5-40.5 | 15 | 35.5 | 532.5 | 40 | 1260.25 | 18903.75

VIII41-50 | 40.5-50.5 | 7 | 45.5 | 318.5 | 47 | 2070.25 | 14491.75

951-60 | 50.5-60.5 | 3 | 55.5 | 166.5 | 50 | 3080.25 | 9240.75

10Totals| | Σf=50 | | Σfx=1475 | | | Σfx²=52512.5

11(a) Mean mark (x̄) = Σfx / Σf

12= 1475 / 50

13= 29.5

14(b) Median mark: N = 50, so N/2 = 25th position. The 25th value falls in the 21-30 class (cf=25).

15Median class = 21-30

16Lower class boundary (L) = 20.5

17Cumulative frequency before median class (cf_b) = 13

18Frequency of median class (f_m) = 12

19Class width (c) = 10.5 - 0.5 = 10

20Median = L + [(N/2 - cf_b) / f_m] * c

21= 20.5 + [(25 - 13) / 12] * 10

22= 20.5 + [12 / 12] * 10

23= 20.5 + 1 * 10

24= 20.5 + 10

25= 30.5

26(c) Standard Deviation (σ) = √[ (Σfx² / Σf) - (Σfx / Σf)² ]

27= √[ (52512.5 / 50) - (1475 / 50)² ]

28= √[ 1050.25 - (29.5)² ]

29= √[ 1050.25 - 870.25 ]

30= √180

31= 13.4164...

32Rounding to one decimal place: 13.4

Answer

(a) Mean mark = 29.5\n(b) Median mark = 30.5\n(c) Standard deviation = 13.4

Ensure to use correct class boundaries (0.5-10.5, etc.) for continuous data and the correct formulas for grouped data calculations. Rounding should only be done at the final step.

Example 3

3. A bag contains 5 red balls and 3 blue balls. Two balls are drawn at random, one after the other, without replacement.\n(a) Draw a tree diagram to represent the possible outcomes.\n(b) Find the probability that:\n (i) Both balls are red.\n (ii) The balls are of different colours.

I(a) Total number of balls = 5 (Red) + 3 (Blue) = 8 balls.

IIFirst Draw:

IIIP(Red 1st) = 5/8

IVP(Blue 1st) = 3/8

VSecond Draw (without replacement):

VIIf 1st is Red (4 Red, 3 Blue left, total 7):

VIIP(Red 2nd | Red 1st) = 4/7

VIIIP(Blue 2nd | Red 1st) = 3/7

9If 1st is Blue (5 Red, 2 Blue left, total 7):

10P(Red 2nd | Blue 1st) = 5/7

11P(Blue 2nd | Blue 1st) = 2/7

12Tree Diagram (text representation):

13Start

14├── Red (5/8)

15│ ├── Red (4/7) --> RR (5/8 * 4/7 = 20/56)

16│ └── Blue (3/7) --> RB (5/8 * 3/7 = 15/56)

17└── Blue (3/8)

18 ├── Red (5/7) --> BR (3/8 * 5/7 = 15/56)

19 └── Blue (2/7) --> BB (3/8 * 2/7 = 6/56)

20(b) (i) Probability that both balls are red (RR):

21P(RR) = P(Red 1st) * P(Red 2nd | Red 1st)

22= (5/8) * (4/7)

23= 20/56

24= 5/14

25(b) (ii) Probability that the balls are of different colours (RB or BR):

26P(Different colours) = P(RB) + P(BR)

27P(RB) = P(Red 1st) * P(Blue 2nd | Red 1st)

28= (5/8) * (3/7)

29= 15/56

30P(BR) = P(Blue 1st) * P(Red 2nd | Blue 1st)

31= (3/8) * (5/7)

32= 15/56

33P(Different colours) = 15/56 + 15/56

34= 30/56

35= 15/28

Answer

(a) (Tree diagram as described in steps)\n(b) (i) P(Both Red) = 5/14\n(b) (ii) P(Different colours) = 15/28

When drawing without replacement, the total number of items and the number of specific items change for subsequent draws. Tree diagrams are excellent for visualising sequential events.

Common mistakes

✗Forgetting to add the constant of integration 'C' for indefinite integrals.
✗Incorrectly applying the power rule for differentiation or integration (e.g., adding 1 to power for differentiation).
✗Using class limits instead of class boundaries for grouped data calculations (especially for median and mode).
✗Confusing mutually exclusive events with independent events (e.g., adding probabilities for independent events when they should be multiplied).
✗Calculation errors, especially with negative signs or squaring numbers when finding variance/standard deviation.

Exam tips

★Always show your working steps clearly, especially for calculus problems and grouped data calculations, as method marks are crucial.
★For differentiation from first principles, ensure you correctly expand (x+δx)^n and simplify before taking the limit.
★When dealing with grouped data, construct a clear and accurate frequency distribution table with all necessary columns (midpoints, fx, cf, fx², etc.).
★Read probability questions carefully to determine if events are independent or mutually exclusive, and if drawing is with or without replacement.

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