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If you're looking for the IB Math AA formula booklet, you've come to the right place! Below you'll find all relevant formulas, with explanations, and notes.
IB Math AA 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 |
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 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 | 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 logaxy = logax + logay loga x/y = logax - logay logaxm = mlogax logax = logbx/logba 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 |
| 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 |
| Binomial theorem | (a+b)n = an + nC1 an−1b+...+nCran−rbr +...+bn nCr = n!/r!(n - r)! where n is a positive integer, r is the term number, a and b are terms in the expansion, and (n r) is the binomial coefficient | Remember: the powers of a and b always add up to n in each term; check carefully to avoid mismatched exponents |
Topic 1: Number and algebra – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Combinations | nCr = n!/r!(n - r)! where n is the total number of items, r is the number of items chosen, and n! denotes the factorial of n. | Common mistake: avoid canceling factorials incorrectly – always write them out step by step to prevent errors |
| Permutations | nPr = n!/(n - r)! where n is the total number of items, r is the number of items arranged, and n! denotes the factorial of n. | Tip: nPr counts arrangements where order matters; don’t confuse it with combinations |
| 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 |
| Modulus-argument (polar) and exponential (Euler) form | z = r (cosθ + isinθ) = reiθ = r cisθ where r is the modulus (magnitude) of z and θ is the argument (angle) 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 |
| De Moivre's theorem | [r(cosθ + isinθ)]n = rn(cosnθ + isinnθ) = rneinθ = rncisnθ where r is the modulus, θ is the argument of the complex number, and n is a positive integer | Tip: De Moivre’s theorem is useful for powers and roots of complex numbers – don’t forget to apply it separately to each term in expansions |
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 ⇒ axis of symmetry is x = -b/2a where a,b, and c are constants of the quadratic function | Tip: the axis of symmetry always passes through the vertex of the parabola – use it to quickly check your graph |
| Solutions of a quadratic equation | ax2 + bx + c = 0 ⇒ x = -b ± √b2 - 4ac/2a, a ≠ 0 where a,b, and c are constants of the quadratic equation | Common mistake: be careful with the ± sign – both plus and minus give distinct solutions |
| 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 |
| Exponential and logarithmic functions | ax = exlna ; logaax = x = alogax where a, x > 0, a ≠ 1 where a is the base, x is the exponent or input, and lna is the natural logarithm of a. | Tip: remember the base restrictions – base a must be positive and not equal to 1, and x must be positive; check parentheses carefully when using ln |
Topic 2: Functions – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Sum and product of the roots of polynomial equations of the form ∑(n, r=0) = arxr = 0 | Sum is -an-1/an ; product is (-1)na0/an where an is the coefficient of the highest degree term, an-1 is the coefficient of the next term, a0 is the constant term, n is the degree of the polynomial | Common mistake: pay attention to the signs and powers of -1; using the wrong exponent or forgetting the negative can give incorrect sum or product |
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 = 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 |
| Identity for tan θ | tanθ = sinθ/cosθ where θ is an angle in a right-angled triangle or in standard position on the unit circle | Tip: tanθ can be negative depending on the quadrant; don’t assume it’s always positive |
| Pythagorean identity | cos2θ + sin2θ = 1 where θ is an angle in a right-angled triangle or in standard position on the unit circle | Common mistake: students sometimes apply it only in right-angled triangles, but it’s valid for any angle in standard position |
| Double angle identities | sin2θ + 2sinθcosθ = 1 cos2θ = cos2θ − sin2θ = 2cos2θ - 1= 1 - 2sin2θ where θ is an angle | Common mistake: when solving equations, watch the range of θ; double angle formulas can introduce extra solutions if not careful |
Topic 3: Geometry and Trigonometry – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Reciprocal trigonometric identities | secθ = 1/cosθ cosecθ = 1/sinθ where θ is an angle and cosθ ≠ 0 | Tip: always check that the denominator isn’t zero; secθ is undefined when cosθ = 0 |
| Pythagorean identities | 1 + tan2 θ = sec2θ 1 + cot2 θ = cosec2θ where θ is an angle, tanθ = sinθ/cosθ, and cotθ = cosθ/sinθ | Tip: when solving equations, avoid multiplying through by cos 2θ or sin2θ too early – rewrite tan and sec (or cot and csc) in terms of sin and cos first to reduce extraneous solutions |
| Compound angle identities | sin(A±B) = sinA cosB ± cosA sinB cos(A±B) = cosA cosB ∓ sinA sinB tan(A±B) = tanA ± tanB / 1 ∓ tanA tanB where A, B, and θ are angles
| Common mistake: don’t confuse the double angle for tangent with the compound angle formulas for sine and cosine; tan2θ has its own distinct form |
| Double angle identity for tan | tan2θ = 2tanθ / 1 - tan2 θ where θ is an angle and cos2θ ≠ 0 | Note: be careful with undefined values – tan2θ is undefined when cos2θ = 0, so check the angle before calculating |
| 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 |
| 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 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. |
| Cartesian equations of a line | x - x0 / l , y - y0 / m ' z - z0 / n 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. | Tip: If two lines have proportional direction vectors, they may be parallel; check their points to confirm. |
| 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. |
| Vector equation of a plane | r =a +λb+μc where r is the position vector of a point on the plane, a is a fixed point on the plane, b and c are two non-parallel vectors lying on the plane, and λ, μ are scalar parameters. | Tip: Make sure b and c are not parallel; otherwise, they won’t define a plane. |
| Equation of a plane (using the normal vector) | r⋅n = a⋅n where r is the position vector of a point on the plane, a is a fixed point on the plane, and n is the normal vector to the plane. | Quick check: If you know three points on the plane, you can find n by taking the cross product of two vectors lying on the plane. |
| Cartesian equation of a plane | ax + by + cz = d where a, b, c are the components of the normal vector to the plane,(x, y, z) is any point on the plane, and d is a constant. | Remember: (x, y, z) can be any point on the plane, not just specific ones. |
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. |
| Standardised normal variable | z = x - μ / σ where x is the observed value, μ is the mean, σ is the standard deviation, and z is the standardised normal variable | Remember: Positive z means above the mean, negative z means below the mean. |
Topic 4: Statistics and probability – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Bayes' theorem | P(B|A) = P(B) P(A|B) / P(B) P(A|B) + P(B′) P(A|B') P(Bi|A) = P(Bi) P(A|Bi) / P(B1) P(A|B1) + P(B2) P(A|B2) + P(B3) P(A|B3) where P(B|A) is the probability of event B given A, P(A|B) is the probability of A given B, and B′ is the complement of B. In the second formula, B1, B2, B3 are mutually exclusive and exhaustive events. | Common mistake: Forgetting to include all mutually exclusive events in the denominator for the general case. |
| Standard deviation σ | σ = ∑(K, i = 1) f1 (xi - μ)2 / n where fi is the frequency of each data value xi, μ is the mean, k is the number of distinct data values, and n = ∑(k, i = 1) fi is the total frequency. | Remember: Always divide by the total frequency n for a population, or (n−1) for a sample. |
| Linear transformation of a single random variable | E(aX + b) = aE(X) + b Var(aX + b) = a2 Var(x) where X is a random variable, a and b are constants, E(X) is the expected value, and Var(X) is the variance of X. | Common mistake: Forgetting to square a when calculating the variance. |
| Expected value of a continuous random variable X | E(X) = μ = ∫(∞, -∞) x f(x)dx where X is a continuous random variable, f(x) is its probability density function (PDF), and μ is the mean (expected value). | Common mistake: Forgetting that f(x) must satisfy ∫f(x)dx = 1 over the entire range. |
| Variance | Var(x) = E(X - μ)2 = E(X2) - [E(X)]2 where X is a random variable, μ=E(X) is the mean, and Var(X) is the variance. | Remember: Variance is always non-negative and measures the spread of X around its mean. |
| Variance of a discrete random variable X | Var(X) = ∑(x - μ)2 P(X = x) = ∑x2 P = (X = x) - μ2 where X is a discrete random variable, x represents its possible values, P(X=x) is the probability of each value, and μ=E(X) is the mean | Common mistake: Forgetting to multiply each squared difference by the probability P(X=x). |
| Variance of a continuous radom variable X | Var(X) = ∫(∞, -∞) (x - μ)2 f(x)dx = ∫(∞, -∞) x2 fx(dx)- μ2 where X is a continuous random variable, f(x) is its probability density function (PDF), and μ=E(X) is the mean | Quick check: Variance is always non-negative, and a larger variance indicates a wider spread of values. |
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. |
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 ex | Remember: Valid for all real x. | |
| Derivative of lnx | ||
| 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. |
| 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. |
| 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. |
Topic 5: Calculus – HL only
| Name | Formula | Notes/Annotations |
|---|---|---|
| Derivative of f(x) from first principles | y = f(x) ⇒ dy/dx = f'(x) = lim(h→0)(f(x+h) - f(x) / h) f(x) is the function, f’(x) is the derivative of f(x) with respect to x, h is a small change in x approaching 0 | Tip: First principles is the definition of the derivative, useful for proving basic rules. |
| Standard derivatives tanx | f(x)=tanx ⇒ f′(x)= sec2x f(x) is the tangent of x, f′(x) is the derivative of f(x) with respect to x. | Common mistake: Confusing sec²x with sec x – don’t forget it’s squared. |
| Standard derivatives secx | f(x) = secx ⇒ f′(x)= secxtanx f(x) is the secant of x, f'(x) is the derivative of f(x) with respect to x | Quick check: Multiplying sec x by tan x gives the correct growth rate of the secant curve. |
| Standard derivatives cosecx | f(x) = cosecx ⇒ f'(x) = - cosecxcotx where x is the variable, cosecx is the cosecant of x, and cotx is the cotangent of x. | Quick check: Multiplying cosec x by cot x gives the slope of the curve at any point. |
| Standard derivatives cotx | f(x) = ⇒ cotx − = f′(x) cosec2 x where x is the variable, cotx is the cotangent of x, cosecx is the cosecant of x, and it is valid for x≠nπ, where n is an integer, because cotx is undefined there | Remember: Valid for x ≠ nπ, where cot x is undefined. |
| Standard derivatives ax | f(x) = ax ⇒ f′(x) = ax (lna) where x is the variable, a is a constant (with a>0 and a≠1), and lna is the natural logarithm of a. | Tip: The derivative of a^x involves the natural logarithm of the base: multiply by ln a. |
| Standard derivatives logax | f(x) = logax ⇒ f'(x) = 1/xlna where x is the variable, a is the base of the logarithm (with a > 0 a>0 and a≠1), and lna is the natural logarithm of a. | Remember: Valid for x > 0 and a > 0, a ≠ 1. |
| Standard derivatives arcsinx | f(x) =arcsinx ⇒ f′(x) = 1/√(1-x2) where x is the variable, and the formula is valid for −1 ≤ x ≤ 1 | Remember: Valid for −1 < x < 1, the domain of arcsin. |
| Standard derivatives arccosx | f(x) = arccosx ⇒ f'(x) = - 1/√1-x where x is the variable, and the formula is valid for −1 ≤ x ≤ 1 | Tip: The domain of arccosx is − 1 ≤ x ≤ 1; outside this interval, the function is not real. |
| Standard derivatives arctanx | f(x) = arctanx ⇒ f'(x) = 1/1+x2 where x is the variable, and the derivative is valid for all real values of x
| Remember: Valid for all real x, unlike arcsin or arccos. |
| Standard integrals | ×∫ax dx = 1/lna ax + C ∫1/a2 + x2 dx = 1/a arctan(x/a) + C ∫1/√a2-x2 dx = arcsin(x/a) + C, |x| < a where x is the variable, a is a positive constant (a>0), lna is the natural logarithm of a, and C is the constant of integration; the third formula is valid for ∣x∣<a so that the square root √a2-x2 remains real. | Quick check: Differentiating your result should give back the original integrand – a good way to verify. |
| Integration by parts | ∫u dv/dx = uv - ∫v du/dx or ∫u dv = uv - ∫v du where u and v are differentiable functions of x, du is the derivative of u with respect to x, and dv is the differential of v with respect to x | Common mistake: Forgetting the negative sign or swapping u and dv incorrectly. |
| Area of region enclosed by a curve and y-axis | A = ∫(b,a) |x| dy | Quick check: If the curve lies entirely on one side of the y-axis, the absolute value may not change the result, but it’s safe to include. |
| Volume of a revolution about the x on the y-axes | V = ∫(b,a)πy2 dx or V = ∫(b,a) πx2 dy | Common mistake: Mixing up dx and dy with the axis of rotation. |
| 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. |
| Integrating factor for y' + P(x)y = Q(x) | e∫P(x)dx y is the unknown function of x, P(x) and Q(x) are given functions of x, μ(x) is the integrating factor, and C is an arbitrary constant of integration. | Common mistake: Forgetting to multiply the entire equation by μ(x) before integrating. |
| Maclaurin series | f(x) = f(0) + xf'(0) + x2/2! f''(0) + ... | Remember: Only use this if the function is differentiable at x = 0. |
| Maclaurin series for special functisno | ex = 1 + x + x2/2! + ... ln(1+x) = x - x2/2 + x3/3 - ... sinx = x - x3/3! + x5 /5! - ... cosx = 1 - x2/2! + x4/4! - ... arctanx = x - x3/3 + x5/5 - ... x is the variable (real number), n! denotes factorial of n, the series for e3, sinx, cosx converge for all real x (radius of convergence ∞), the series for ln(1+x) has radius of convergence ∣x∣<1 (and it converges at x=1 conditionally). | Remember: The factorial appears only in sin x and cos x series, not in ln(1+x) or arctan x. |
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