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If you're looking for the IB Math AI formula booklet, you've come to the right place! Below you'll find all relevant formulas, with explanations and notes.
IB Math AI Formula Booklet
Prior Learning – SL and HL
| Name | Formula | Notes/Annotations |
|---|---|---|
| Area of a parallelogram | A = bh where b is the base, h is the height | Remember: h is the perpendicular height, not the slanted side. |
| Area of a triangle | A = 1/2(bh) where b is the base, h is the height | Common mistake: don’t forget the ½ factor. |
| Area of a trapezoid | A = 1/2(a+b)h where a and b are the parallel sides, h is the height | Remember: h must be perpendicular to the parallel sides, not the slanted sides. |
| Area of a circle | A = πr2 where r is the radius | Remember: click the π sign on your calculator rather than providing an approximation. |
| Circumference of a circle | C = 2πr Where r is the radius | Don’t forget: you can also use C = πd if the diameter d is given instead of the radius. |
| Volume of a cuboid | V = lwh Where l is the length, w is the width, h is the height | Remember: mixing up length, width, and height doesn’t matter, just make sure you’re multiplying all three dimensions. |
| Volume of a cylinder | V = πr2h where r is the radius, h is the height | Common mistake: h is the vertical height, not the slant height of the curved surface. |
| Volume of a prism | V = Ah where A is the area of cross-section, h is the height | Don’t forget: the cross-section must be the same shape all the way through the prism. |
| Area of the curved surface of a cylinder | A = 2πrh where r is the radius, h is the height | Tip: imagine “unrolling” the curved surface into a rectangle with sides 2πr and h. |
| Distance between two points (x1, y1) and (x2, y2) | d = √(x1 - x2)2 + (y1 - y2)2 | Common mistake: don’t forget to square the differences before adding – subtracting directly will give the wrong result. |
| Coordinates of the midpoint of a line segment with endpoints (x1 , x2) (y1 , y2) | (x1 + x2/2 , y1 + y2/2) | Common mistake: average each coordinate separately. |
Prior Learning – HL Only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Solutions of a quadratic equation | The solutions of ax2 + bx + c = 0 are x = -b±√b2 - 4ac / 2a , a≠0 where a is the coefficient of x2 (must not be 0), b is the coefficient of x, and c is the constant term | Common mistake: Forgetting that a ≠ 0; if a = 0, it’s not a quadratic equation. |
Topic 1: Number and algebra – SL and HL
| Name | Formula | Notes/Annotations |
|---|---|---|
| The nth term of an arithmetic system | un = u1 + (n - 1)d where u1 is the first term, n is the term number, and d is the common difference | Common mistake: forget to add the first term u₁; some students only multiply (n−1) by d. |
| The sum of n terms of an arithmetic sequence | Sn = n/2 (2u1 + (n - 1)d) ; Sn = n/2 (u1 + un) where u1 is the first term, un is the nth term, d is the common difference, and n is the number of terms | Tip: both formulas work – use the one that’s easiest based on what you know. |
| The nth term of a geometric sequence | un = u1rn-1 where u₁ is the first term of the sequence, r is the common ratio, n is the position of the term in the sequence (term number), un is the value of the nth term. | Remember: r is the common ratio, not the difference; don’t confuse geometric and arithmetic sequences. |
| The sum of n terms of a finite geometric sequence | Sn = u1(rn - 1)/r - 1 = u1(1 - rn)/1 - r , r ≠ 1 where u1 is the first term, r is the common ratio, and n is the number of terms | Common mistake: confusing the finite sum formula with the infinite geometric series formula – don’t use this formula if n → ∞. |
| Compound interest | FV = PV × (1 + r/100k)kn where PV is the present value, n is the number of years, k is the number of compounding periods per year, r% is the nominal annual rate of interest | Tip: make sure to adjust r and n according to the compounding frequency k; using the annual rate directly without dividing by k is a common error. |
| Exponents and logarithms | ax = b ⇔ x = logab where a> 0, b>0, a≠1 where a > 0, a ≠ 1, x > 0 , y > 0, and m is a real number | Common mistake: remember the base a must be positive and not equal to 1; otherwise the logarithm is undefined. |
| Percentage error | ϵ = |VA - VE / VE| × 100% where VE is the exact value and VA is the approximate value of V. | Remember: Express the result as a percentage by multiplying by 100. |
Topic 1: Number and algebra – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Laws of logarithms | logaxy = logax + logay loga x/y = logax - logay logaxm = mlogax logax = logbx/logba where a is the base of the logarithm, b is another base and m is a real number | Remember: Change-of-base formula lets you convert logs to any base you can calculate. |
| The sum of an infinite geometric sequence | S∞ = u1/1 - r, |r| < 1 where u1 is the first term and r is the common ratio | Common mistake: this formula only works if |r| < 1; otherwise, the series does not converge. |
| Complex numbers | z = a + bi where a is the real part, b is the imaginary part, and i = - 1 is the imaginary unit. | Remember: when adding or subtracting complex numbers, combine real parts with real parts and imaginary parts with imaginary parts only. |
| Discriminant | ∆ = b2 - 4ac where Δ is the discriminant, and a,b, and c are constants of the quadratic equation ax2 + bx + c = 0 | Common mistake: don’t forget to calculate 4ac correctly; errors here often lead to wrong root classification. |
| The nth term of a geometric sequence | un = u1rn-1 u₁ is the first term of the sequence, r is the common ratio, n is the term number, un is the value of the nth term | Remember: r is the common ratio, not the difference; don’t confuse geometric and arithmetic sequences. |
| Modulus-argument (polar) and exponential (Euler) form | z = r (cosθ + isinθ) = reiθ = r cisθ where r is the modulus of z and θ is the argument of z
| Tip: r is the modulus (distance from origin) and θ is the argument (angle from positive x-axis); check your angle quadrant carefully. |
| Determinant of a 2x2 matrix | A = (a b c d) ⇒ det A = |A| = ad - bc where a, b, c, d are the four entries of the matrix | Remember: The determinant is zero if the rows or columns are linearly dependent. |
| Inverse of a 2x2 matrix | A = (a b c d) ⇒ A-1 = 1/det A (d -b -c a), ad ≠ bc where a, b, c, d are the entries of the matrix A | Tip: Swap a and d, then change the signs of b and c before dividing by the determinant. |
| Power formula for a matrix | Mn = PDn P-1 where P is the matrix of eigenvectors and D is the diagonal matrix of eigenvalues | Tip: Diagonalizing the matrix first makes raising it to a power much easier. |
Topic 2: Functions – SL and HL
| Name | Formula | Notes/Annotations |
|---|---|---|
| Equations of a straight line | y = mx + c; ax + by + d = 0; y - y1 = m(x - x1) where m is the gradient (slope), c is the y-intercept, a,b,d are real constants defining the line, and (x1, y1) is a point on the line | Tip: check c by setting x=0 and reading the y-intercept. |
| Gradient formula | m = y2 - y1/x2 - x1 where m is the gradient of the line, and (x1, y1) and (x2, y2) are two points on the line | Tip: always subtract in the same order for x and y; mixing the order will flip the sign of the gradient. |
Axis of symmetry of the graph of a quadratic function | f(x) = ax2 + bx + c = 0 ⇒ axis of symmetry is x = −b/2a where a, b, c are real constants defining the quadratic. The axis of symmetry is a vertical line through the vertex of the parabola. | Tip: The axis of symmetry gives the x-coordinate of the vertex. Substitute this value back into f(x)to find the y-coordinate. |
Topic 2: Functions – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Logistic function | f(x) = L/1+Ce-kx , L, k, C>0 where L is the maximum value of the function, k is the growth rate, C is a positive constant related to the initial value, and x is the independent variable | Quick check: For x = 0, f(0) = L / (1 + C), which gives the initial value. |
Topic 3: Geometry and trigonometry – SL and HL
| Name | Formula | Notes/Annotations |
|---|---|---|
| Distance between two points (x1, y1, z1) and (x2, y2, z2) | d = √(x1 - x2)2 +(y1 - y2)2 + (z1 - z2)2 where (x1 , x2);(y1 , y2);(z1 , z2) are two points in 3D space, and d is the distance between them | Remember: the distance is always non-negative – if you get a negative value, you likely forgot the square root. |
| Coordinates of the midpoint of a line segment with endpoints (x1, y1, z1) and (x2, y2, z2) | x1 + x2 /2 ; y1 + y2 /2 ; z1 + z2 / 2 where (x1 , x2);(y1 , y2);(z1 , z2) are two points in 3D space, and d is the distance between them | Tip: the midpoint is equidistant from both endpoints – use the distance formula to check your result if you’re unsure. |
| Volume of a right-pyramid | V = 1/3Ah where A is the area of the base, h is the height | Common mistake: h must be the perpendicular height from the apex to the base, not the slant height. |
| Volume of a right cone | V = 1/3πr2h where r is the radius, h is the height | Common mistake: use the vertical height h, not the slant height, when applying the formula. |
| Area of the curved surface of a cone | A = πrl where r is the radius, l is the slant height | Common mistake: make sure to use the slant height l, not the vertical height h, when finding the curved surface area. |
| Volume of a sphere | V = 4/3πr3 where r is the radius | Remember: the radius is measured from the center to the surface – using diameter instead without halving will double your result. |
| Surface area of a sphere | A = 4πr2 where r is the radius | Remember: the radius must be squared – mixing up r2 and r3 is a frequent error. |
| Sine rule | a/sinA = b/sinB = c/sinC where a, b, c are the lengths of the sides of a triangle, and A, B, C are the angles opposite those sides. | Tip: the angles must be in the same triangle as their opposite sides; don’t mix angles and sides from different triangles. |
| Cosine rule | c2 = a2 + b2 - 2abcosC cosC = a2 + b2 - c2 / 2ab where a, b, c are the sides of a triangle, and C is the angle opposite side c | Common mistake: make sure to square the sides correctly and use the angle opposite the side labeled c; mixing these up will give wrong results. |
| Area of a triangle | A = 1/2absinC where a and b are two sides of the triangle, and C is the angle between those sides | Tip: the angle C must be between the two given sides; using an angle not included will give an incorrect area. |
| Length of an arc | l = θ/360 × 2π where r is the radius, θ is the angle measured in degrees | Common mistake: Using degrees directly without conversion; it will give the wrong arc length. |
| Area of a sector | A = θ/360 × πr2 where r is the radius, θ is the angle measured in degrees | Remember: The area of a sector is proportional to the central angle and the square of the radius. |
Topic 3: Geometry and trigonometry – HL Only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Length of an arc | l = rθ where r is the radius, θ is the angle measured in radians | Common mistake: make sure the angle θ is in radians; using degrees without conversion will give the wrong arc length. |
| Area of a sector | A = 1/2r2θ where r is the radius, θ is the angle measured in radians | Tip: the angle θ must be in radians; using degrees without converting will result in an incorrect area. |
| Identities | cos2θ + sin2 θ = 1 tanθ = sinθ/cosθ where θ is the angle | Quick check: These identities are fundamental for simplifying trigonometric equations. |
| Transformation matrices | (cos2θ sin2θ sin2θ -cos2θ), reflection in the line y = (tanθ)x (k 0 0 1), horizontal stretch / stretch parallel to x-axis with a scale factor of k (1 0 0 k), vertical stretch / stretch parallel to y-axis with a scale factor of k (k 0 0 k), enlargement , with a scale of k, centre (0, 0) (cosθ sinθ -sinθ cosθ), clockwise rotation of angle θ about the origin (θ>0) where θ is the angle and k is the scale factor | Common mistake: Confusing horizontal and vertical stretches; the position of k matters. |
| Magnitude of a vector | |v| = √v12 + √v22 + √v32 , where v = (v1, v2, v3) where v1, v2, v3 are the components of the vector v | Tip: don’t forget to square each component before summing; a common mistake is taking the square root of each component separately instead of the sum of squares. |
| Vector equation of a line | r = a + λb where r is the position vector of a point on the line, a is a fixed point on the line, b is the direction vector of the line, and λ is a scalar parameter | Tip: Remember, if you’re asked for the parametric form, just break it into components x, y, z. |
| Parametric form of the equation of a line | x = x0 + λl, y = y0 + λm, z = z0 + λn where (x0, y0, z0) is a point on the line, (l, m, n) is the direction vector of the line, and λ is a scalar parameter | Common mistake: mixing up the direction vector with a point on the line – (l, m, n) must represent direction, not a location. |
| Scalar product | v⋅w = v1w1 + v2w2 + v3w3 where v = (v1, v2, v3) , w = (w1, w2, w3) v⋅w = |v||w| cosθ where θ is the angle between v and w | Tip: the scalar product is zero if vectors are perpendicular (θ = 90°); a common mistake is forgetting this when checking orthogonality. |
| Angle between two vectors | cosθ = v1w1 + v2w2 + v3w3 / |v||w| where v = (v1, v2, v3) and w = (w1, w2, w3) are two vectors, |v||w| are their magnitudes, and θ is the angle between them | Note: The plane is uniquely determined by its normal vector and any point lying on it. |
| Vector product | v × w = (v2w3 - v3w2 ,v3w1 - v1 - w3 , v1w2 - v2w1) where v = (v1, v2, v3) , w = (w1, w2, w3) |v × w| = |v||w| sinθ, where θ is the angle between v and w | Quick check: The resulting vector will be perpendicular to both v and w. |
| Area of a parallelogram | A = |v × w| where v and w form two adjacent sides of a parallelogram | Remember: The magnitude of the cross product gives the area directly. |
Topic 4: Statistics and probability – SL and HL
| Name | Formula | Notes/Annotations |
|---|---|---|
| Interquartile range | IQR = Q3 - Q1 where Q3 is the upper quartile and Q1 is the lower quartile
| Remember: For even-sized datasets, quartiles are often interpolated between data points – check your method (inclusive or exclusive). |
| Mean, x, of a set data | x = Σ(k, i = l) fixi/n where n = Σ(k, i = l)fi
where xi is the mean, xi is the i-th data value, fi is the frequency of xi , k is the number of distinct data values, and n is the total frequency | Remember: The mean is sensitive to outliers, so extreme values can distort it. |
| Probability of an event A | P(A) = n(A)/n(U) where n(A) is the number of favorable outcomes for event A, and n(U) is the total number of possible outcomes in the sample space
| Quick check: If multiple events are considered, confirm they’re mutually exclusive or not, as this affects calculations. |
| Complementary events | P(A) + P(A') = 1 where P(A) is the probability of event A occurring, and P(A′) is the probability of event A not occurring | Tip: Use this when finding the probability of “not A” is easier than finding P(A) directly. |
| Combined events | P(A ∪ B) = P(A) + P(B) - P(A ∩ B) where P(A) and P(B) are the probabilities of events A and B occurring, and P(A∩B) is the probability of both events occurring together | Remember: If A and B cannot happen together (mutually exclusive), then P(A∩B) = 0, and the formula simplifies to P(A) + P(B). |
| Mutually exclusive events | P(A ∪ B) = P(A) + P(B) where events A and B are mutually exclusive, meaning they cannot occur at the same time | Tip: Use this simplified formula only when A and B cannot happen together. |
| Conditional probability | P(A|B) = P(A ∩ B) / P(B) where P(A|B) is the probability of event A occurring given that event B has occurred, and P(B) ≠ 0 | Remember: For independent events, P(A|B) = P(A). |
| Independent events | P(A∩B) = P(A)P(B) where events A and B are independent, meaning the occurrence of one does not affect the probability of the other | Tip: Use this formula only when A and B are independent; dependence requires a different approach. |
| Expected value of a discrete random variable X | E(X) = ∑xP(X = x) where X is a discrete random variable, x represents its possible values, and P(X=x) is the probability of each value | Common mistake: Forgetting to multiply each value x by its probability P(X=x) before summing. |
Binomial distribution X ≈ (n, p) Mean | E(X) = np where n is the number of trials, p is the probability of success for each trial, and E(X) is the expected value (mean) of the distribution | Tip: The mean np gives the average number of successes over many repetitions of the experiment. |
Binomial distribution Variance | Var(X) = np(1 - p) where X∼B(n,p), n is the number of trials, p is the probability of success, and Var(X) is the variance of the binomial distribution | Tip: The variance measures the spread of the distribution around the mean. |
Topic 4: Statistics and probability – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Linear transformation of a single random variable | E(aX + b) = aE(X) + b Var(aX + b) = a2Var(X) where X is a random variable a and b are constants | Quick check: Variance only changes with scaling (a), not with translation (b). |
| Linear combinations of n independent random variables, X1, X2, ..., Xn | E(a1X1±a2X2±...±anXn) = a1E(X1)±a2E(X2)±...±anE(Xn) Var(a1X1±a2X2±...±anXn) =a12 Var(X1)+a22 Var(X2)+...+an2 where X are independent random variables and a are constants | Common mistake: Forgetting that the variance formula only works for independent random variables. |
Sample statistics Unbiased estimate of population variance s2n-1 | s2n-1 = n/n-1 s2n where n is the sample size | Common mistake: Forgetting the n−1 in the denominator; using n gives a biased estimate. |
Poisson distribution X ~ Po(m) Mean | E(X) = m where X is a Poisson random variable and m is the mean number of occurrences | Quick check: For large m, the Poisson can be approximated by a normal distribution N(m, m). |
Poisson distribution X ~ Po(m) Variance | Var(X) = m where X is a Poisson random variable and m is the mean number of occurrences | Remember: The model assumes independent events occurring at a constant average rate. |
| Transition matrices | Tns0 = sn , where s0 is the initial state where T is the transition matrix s₀ is the initial state vector, sn is the state vector after n steps, and n is the number of transitions | Remember: Each column of T must sum to 1 if it represents a Markov chain with column vectors. |
Topic 5: Calculus – SL and HL
| Name | Formula | Notes/Annotations |
|---|---|---|
| Derivative of xn | f(x) = xn = f(x) = xn ⇒ f'(x) = nx-1 where n is a real number and f′(x) is the derivative of f(x) with respect to x | Remember: Works for negative and fractional powers too. |
| Integral of xn | ∫xn dx = xn+1 / n+1 + C, n ≠ -1 where n is a real number and C is the constant of integration | Remember: Always add the constant of integration C for indefinite integrals. |
| Area between a curve y = f(x) and the x-axis where f(x) > 0 | A = ∫(b,a) ydx where y = f(x) > 0 is the curve, and a and b are the limits on the x-axis | Remember: The limits a and b must correspond to the x-values of the interval you are considering. |
| The trapezoidal rule | ∫(b,a) ydx ≈ 1/2h((y0 + yn) + 2(y1 + y2 + ...+ yn-1)), where h = b where a, b are the limits of integration, n is the number of sub-intervals, and y are the values of the function at the equally spaced points | Quick check: Increasing n (smaller h) gives a more accurate approximation. |
Topic 5: Calculus – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Derivative of sinx | f(x) = sinx⇒f'(x) = cosx where x is the independent variable | Remember: Works for negative and fractional powers too. |
Derivative of cosx | f(x) = cosx ⇒ x= f'(x)= -sin x where f′(x) is the derivative of f(x) with respect to x | Remember: The limits a and b must correspond to the x-values of the interval you are considering. |
| Derivative of tanx | f(x) = tanx⇒f'(x) = 1/cos2x where x is the independent variable | Quick check: Differentiate tanx as sinx/cosx using the quotient rule to verify the result. |
| Derivative of ex | f(x) = ex⇒f'(x) = ex where x is the independent variable | Tip: The exponential function is unique - its derivative is the same as the function itself. |
| Derivative of lnx | f(x) = lnx⇒f'(x)=1/x where x is the independent variable | Quick check: At x = 1, the derivative equals 1. |
| Chain rule | y = g(u), where u = f(x)⇒dy/dx = dy/du × du/dx where y is a function of u, u is a function of x, and dx/dy is the derivative of y with respect to x | Common mistake: Forgetting to multiply by du/dx after differentiating g(u). |
| Product rule | y = uv ⇒ dy/dx = u dv/dx + v du/dx where u and v are functions of x, dy/dx is the derivative of y with respect to x | Quick check: If one term is constant, it reduces to ordinary differentiation. |
| Quotient rule | y = u/v ⇒ dy/dx = v du/dx - u dv/dx / v2 where u and v are functions of x, dy/dx is the derivative of y with respect to x | Common mistake: Forgetting to square the denominator v2 or reversing the subtraction order. |
| Standard integrals | ∫1/x dx = ln |x| + C ∫sinxdx = -cosx + C ∫cosxdx = sinx + C ∫ex dx = ex dx = ex + C where C is the constant of integration | Quick check: Differentiate your integrated result; it should give you back the original function. |
| Area of a region enclosed by a curve and x-axis | A = ∫(b,a)|y|dx where y = f(x) is the curve, and a and b are the limits on the x-axis. | Tip: Use |y| so areas below the x-axis count positively; otherwise the integral gives signed area. |
| Volume of a revolution about x or y-axes | V = ∫(b,a) πy2dx or V = ∫(b,a)πx2dy where a, b are the limits of integration, and x, y are the variables of the function being rotated | Remember: Always include the factor of π for the circular cross-sections. |
| Acceleration | a = dv/dt = d2 s/dt2 a is the acceleration, v is the velocity, s is the displacement, t is the time | Tip: Acceleration is the rate of change of velocity, not just speed – direction matters. |
| Distance travelled from t1 to t2 | distance = ∫(t2 , t1) |v(t)|dt where a is the acceleration, v is the velocity, s is the displacement, and t is time | Common mistake: Forgetting to split the integral when velocity changes sign (e.g., from positive to negative). |
| Displacement from t1 to t2 | displacement = ∫(t2 , t1) v(t)dt where v(t) is the velocity as a fun ction of time, and t1 and t2 are the initial and final times | Common mistake: Confusing this with total distance; displacement can be zero even if distance travelled is not. |
| Euler's method | yn+1 = yn + h × f(xn , yn) ; xn+1 = xn + h where h is a constant (step length) | Remember: Each step uses the slope at the current point (xn, yn) to estimate the next value. |
| Exact solution for coupled linear differential equtions | x = Aeλ1t p1 + Beλ2t p2 where x is the solution vector A, B are constants λ are numbers (eigenvalues), p are vectors (eigenvectors), and t is the independent variable | Common mistake: Assuming the formula works when eigenvectors are not independent - you need two linearly independent eigenvectors (matrix must be diagonalizable). |
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