Maths

Integrate \csc^2 x

-\cot x + C

Integrate \sec x \tan x

\sec x + C

What is \csc^2 x equal to?

\cot^2 x + 1

What is \sec^2 x equal to?

\tan^2 x + 1

Express \sin^2 x using \cos^2 x

1 - \cos^2 x

Express \sin^2 x in terms of \cos(2x)

\frac{1}{2}(1-\cos(2x))

Integrate \sin x

-\cos x + C

Integrate \cos x

\sin x + C

Integrate \sec^2 x

\tan x + C

Integrate \csc x \cot x

-\csc x + C

Express \cos^2 x in terms of \cos(2x)

\frac{1}{2}(1+\cos(2x))

Integrate \frac{1}{x}

\ln |x| + C

Differentiate \sin x

\cos x

Differentiate \tan x

\sec^2 x

Differentiate \cot x

-\csc^2 x

Differentiate \sec x

\sec x \tan x

Differentiate \cos x

-\sin x

Differentiate \csc x

-\csc x \cot x

Differentiate \sin(2x)

2\cos(2x)

Differentiate \cot(3x)

-3\csc^2(3x)

Differentiate \sin^2 x

2\sin x \cos x

Differentiate \cos(0.5x)

-0.5\sin(0.5x)

Differentiate \sin x \cos x

\cos(2x)

Differentiate 2\cos(0.5x)

-\sin(0.5x)

Double angle formula for \sin(2\theta)

2\sin\theta\cos\theta

Double angle formula for \cos(2\theta)

\cos^2\theta-\sin^2\theta or 2\cos^2\theta-1 or 1-2\sin^2\theta

Double angle formula for \tan(2\theta)

\frac{2\tan\theta}{1 - \tan^2\theta}

Key considerations for tangent addition formula

Signs: The numerator has the same sign as in \tan(A \pm B). The denominator has the opposite sign.

Representation of a rational number in a proof

\frac{a}{b} , where a and b are integers and b \neq 0

Effect of the transformation y=|f(x)| on a graph

Reflects any part of the graph below the x-axis in the x-axis.

Effect of the transformation y=f(|x|) on a graph

Reflects the part of the graph for x \geq 0 in the y-axis, discarding the original part for x < 0.

Formula for the n-th term of an arithmetic sequence

an = a1 + (n-1)d

Formula for the n-th term of a geometric sequence

u_n = ar^{n-1}

Condition for a geometric series to be convergent

Convergent if |r|<1, where r is the common ratio.

Definition of a periodic sequence

A sequence is periodic if its terms repeat in a cycle, i.e., u{n+k} = un for some fixed integer k (period).

Angle a vector makes with the coordinate axes

If vector \mathbf{a} = x \mathbf{i} + y \mathbf{j} + z \mathbf{k} makes an angle \thetax with the positive x-axis, then \cos(\thetax) = \frac{x}{|\mathbf{a}|}, and similarly for \thetay and \thetaz.

Radian to degree conversion factor

1 \text{ radian} = \frac{180^\circ}{\pi} or \pi \text{ radians} = 180^\circ

Formula for arc length in a circle (radians)

l=r\theta

Formula for the area of a circle sector (radians)

A = \frac{1}{2} r^2\theta

What is the graph of y = \arcsin(x)?

A visual representation of the inverse sine function.

Domain of y = \arcsin x

[-1, 1] or -1 \leq x \leq 1

Range of y = \arcsin x

[-\pi/2, \pi/2] or -\frac{\pi}{2} \leq y \leq \frac{\pi}{2}

What is the graph of y = \arccos(x)?

A visual representation of the inverse cosine function.

Domain of y = \arccos x

[-1, 1] or -1 \leq x \leq 1

Range of y = \arccos x

[0, \pi] or 0 \leq y \leq \pi

What is the graph of y = \arctan(x)?

A visual representation of the inverse tangent function.

Domain of y = \arctan x

(-\infty, \infty) or -\infty < x < \infty

Range of y = \arctan x

(-\pi/2, \pi/2) or -\frac{\pi}{2} < y < \frac{\pi}{2}

Differentiate y = e^{kx}

\frac{dy}{dx} = ke^{kx}

Differentiate y = \ln(x)

\frac{dy}{dx} = \frac{1}{x}

Differentiate y = a^{kx}

\frac{dy}{dx} = (a^{kx})k\ln(a)

State the Chain Rule for differentiation

\frac{dy}{dx} = \frac{dy}{du} \times \frac{du}{dx}

State the Product Rule for differentiation

\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}

State the Quotient Rule for differentiation

\frac{d}{dx}\left(\frac{u}{v}\right) = \frac{v\frac{du}{dx} - u\frac{dv}{dx}}{v^2}

Formula for differentiating parametric equations

\frac{dy}{dx} = \frac{dy/dt}{dx/dt}

Purpose of implicit differentiation

A technique used to differentiate functions where y cannot be easily expressed as an explicit function of x.

Condition for a function f(x) to be concave

f(x) is concave if f''(x) \leq 0 for all values of x in that interval.

Condition for a function f(x) to be convex

f(x) is convex if f''(x) \geq 0 for all values of x in that interval.

Definition of a point of inflection

A point on a curve where the second derivative f''(x) changes sign (and is commonly zero).

Integrate e^x

e^x + C

Formula for integration by parts

\int u\frac{dv}{dx} dx = uv - \int v\frac{du}{dx} dx

Express \tan^2\theta as an identity

\sec^2\theta - 1

What trigonometric identity equals 1?

\sec^2\theta - \tan^2\theta

Express \cot^2\theta as an identity

\csc^2\theta - 1

What is \sin^2\theta + \cos^2\theta?

1

What is \sec^2x - 1 equal to?

\tan^2x

What is \sec^2x - \tan^2x equal to?

1

What is \cot^2x + 1 equal to?

\csc^2x

What is \csc^2x - \cot^2x equal to?

1

Limitations and applications of linear regression models

Do not extrapolate data beyond the observed range. Should not be used to find a value of x for a given y if the model is only valid for predicting y from x.

Characteristics of an experiment in probability

A repeatable process that results in a number of possible outcomes.

Definition of an event in probability

A collection of one or more outcomes from an experiment.

Definition of a sample space

The set of all possible outcomes of an experiment.

Properties of mutually exclusive events

They cannot occur at the same time (no overlap). P(A \text{ or } B) = P(A) + P(B)

Properties of independent events

The occurrence of one does not affect the other. P(A \text{ and } B) = P(A) \times P(B).

Probability mass function for a Binomial Distribution B(n,p)

P(X = r) = \binom{n}{r} p^r (1-p)^{n-r}

Conditions for modelling with a binomial distribution

• fixed number of trials (n)
• two possible outcomes (success/failure)
• fixed probability of success (p) in each trial
• Trials are independent of each other

Interpretation of the Product Moment Correlation Coefficient (r)

Describes the linear correlation between two variables, taking a value between -1 and 1.

• -1: strong negative linear correlation
• 0: no linear correlation
• 1: strong positive linear correlation

Justification for using a linear model

As the data points lie reasonably close to a straight line.

When is an estimate considered reliable in regression?

When interpolation (predicting within the range of observed data) is used.

Adjusting integral bounds for areas below the x-axis

If integrating to find area and the curve goes below the x-axis, evaluate the integral for that segment, take its absolute value, and add it. For a definite integral with negative result for that segment, you can "flip the numbers on the integral bracket" or multiply by -1 to get a positive area.

Importance of diagrams for velocity-time integration

When integrating a velocity-time graph to find displacement or distance, a diagram helps identify segments where velocity is negative (below the x-axis). For total distance, these negative areas must be taken as positive before summing.

Common result for differentiating \sin^2 x and the identity for \sin(2x)

Both are equal to 2\sin x \cos x. (Note: \frac{d}{dx}(\sin^2 x) = 2\sin x \cos x and \sin(2x) = 2\sin x \cos x)

Proof strategy for \sin x = \cos(90^\circ - x)

Consider a right-angled triangle. If one acute angle is x, the other is 90^\circ - x. The side opposite x is adjacent to 90^\circ - x. Thus, \sin x = \text{opposite/hypotenuse} = \cos(90^\circ - x).

Express \sin^2 x in terms of \cos(2x)

\sin^2 x = \frac{1}{2}(1-\cos(2x))

Express \cos^2 x in terms of \cos(2x)

\cos^2 x = \frac{1}{2}(1+\cos(2x))

Solving trigonometric equations of the form f(ax)

1. Multiply the given range for \theta by the coefficient a to get the range for a\theta.
2. Find all solutions for a\theta within this multiplied range (using periodicity of the function) BEFORE dividing by a to find the values of \theta.

Implication of a sign change for a continuous function

If f(a) and f(b) have opposite signs and f(x) is continuous on [a,b], then there is evidence to suggest that a root (where f(x)=0) lies between a and b.

Parametric integration limits and graph direction

When setting up a definite integral for parametric equations, if the integration is from x1 to x2 (where x1 = x(t1) and x2 = x(t2)), the value of t_1 (corresponding to the start of the interval) typically goes on the bottom limit of the integral. The user's note states 'Top', which might refer to a specific case or could be a misunderstanding of general integration limits.

Effect of multiplying an integral by -1

If you multiply a definite integral by -1, you should flip the bounds of integration, i.e., -\inta^b f(x)dx = \intb^a f(x)dx.

Property of integrals with constants

You can take any constants (multiplicative factors) out of the integral to simplify calculations (e.g. \int kf(x)dx = k \int f(x)dx).

Formula for integrating parametrics to find area

\int y\, \frac{dx}{dt}\, dt

Method to find R in R\cos(\theta-\alpha)=C type problems

Given R\cos\alpha = 4 and R\sin\alpha = \text{another value}, square both equations and add them: (R\cos\alpha)^2 + (R\sin\alpha)^2 = 4^2 + (\text{another value})^2 \Rightarrow R^2(\cos^2\alpha + \sin^2\alpha) = \dots \Rightarrow R^2 = \dots \Rightarrow R = \sqrt{\dots}. Alternatively, divide the equations to find \tan\alpha, then use a right-angled triangle where R is the hypotenuse, and adjacent/opposite sides are related to R\cos\alpha and R\sin\alpha respectively.

Summing a series that combines arithmetic and geometric sequences

You can split them up and sum them separately: find the sum of the arithmetic part and the sum of the geometric part, then add the results.

Essential elements for clear proof wording

Clearly state assumptions, define variables, show logical steps, and draw a final conclusion.

Crucial step when a proof meets in the middle

An indication that the proof is complete, typically by showing that the two sides or intermediate expressions are equivalent, e.g., "Therefore, [initial statement/equation] is true."

Chain Rule for a Composite Function f(g(x))

The derivative is f'(g(x))g'(x). For example, if f(x) = x^7 and g(x) = x^2+1, the derivative of f(g(x)) = (x^2+1)^7 would be 7(x^2+1)^6 \cdot 2x = 14x(x^2+1)^6.

Formula for the volume of a sphere

V = \frac{4}{3}\pi r^3

Phrasing for model limitations

Use phrases like: "The model does not consider…", "The model indicates that… (which is incorrect in reality)…", "The model is not valid for times after/before [specific point] (as it doesn't make physical sense)".

Simplify e^{3\ln 2} using logarithmic rules

e^{3\ln 2} = e^{\ln(2^3)} = 2^3 = 8