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Calculus and functions questions, answered
Derivatives and integrals, limits, functions and their inverses, and the notation that goes with them.
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Bessel Functions Calculator
What is the Bessel Functions Calculator?
Calculate Bessel functions of the first and second kind for any order and argument. Useful for solving differential equations in cylindrical coordinates and wave physics.
Is the Bessel Functions Calculator free to use?
Yes. The Bessel Functions Calculator is free to use on Calculate.co.nz, with no sign-up, paywall or account required.
How does the calculator compute Bessel functions?
It uses the standard series and recurrence definitions for Bessel functions of the first and second kind, the same mathematical formulas used everywhere. The computation runs entirely in your browser, so you can evaluate any order and argument without installing anything, and the results are numerical approximations to the precision shown.
Composite Function Calculator
What does f(g(x)) mean?
Apply g to x first, then apply f to the result. It is read f of g of x.
Is composition commutative?
No. In general f(g(x)) is different from g(f(x)); the order of composition matters.
Is the composite of two linear functions linear?
Yes. The composite is linear too, with slope equal to the product of the two slopes.
Daily Light Integral (DLI) Calculator
What is Daily Light Integral (DLI)?
Daily Light Integral (DLI) is the total amount of photosynthetically active radiation (PAR) a plant receives over one day, measured in moles of light per square metre per day (mol/m2/day). It combines light intensity (PPFD) with the length of time the light is shining (photoperiod), which matters more to plant growth than intensity alone, since a dim light for a long time and a bright light for a short time can deliver the same total DLI.
How do you calculate DLI from PPFD?
DLI (mol/m2/day) = PPFD (micromol/m2/s) x photoperiod (hours) x 3600 (seconds per hour) / 1,000,000 (to convert micromol to mol). For example, a PPFD of 400 micromol/m2/s over a 12 hour photoperiod gives DLI = 400 x 12 x 3600 / 1,000,000 = 17.28 mol/m2/day.
What DLI do most plants need?
Target DLI varies widely by crop. Low-light houseplants and leafy greens generally do well on 10 to 15 mol/m2/day. Most fruiting vegetables such as tomatoes and cucumbers want 20 to 30 mol/m2/day. High-light crops like cannabis in flowering and some fruiting crops can use 35 to 45 mol/m2/day or more. Outdoor full sun in New Zealand summer can reach 40 to 55 mol/m2/day, while a cloudy winter day may only provide 5 to 10 mol/m2/day, which is why supplemental lighting is common in greenhouses over autumn and winter.
Definite Integral Calculator
What does adjustable precision mean?
The precision field sets the number of subintervals used in Simpson's rule. More subintervals means more function evaluations and higher accuracy, at the cost of a tiny extra computation time. For most smooth functions 100 subintervals gives excellent results; for rapidly oscillating functions you may want 1000 or more. The number is rounded up to the nearest even integer automatically.
What is the difference between this calculator and the Integral Calculator?
Both use Simpson's rule for numerical integration. This calculator lets you change the number of subintervals, shows the step size h, and is designed for users who want to study how precision affects the result. The Integral Calculator uses a fixed 1000 subintervals for convenience.
Can I integrate sin(x) from 0 to pi?
Yes. Enter sin(x) as the function, 0 as the lower bound, and use 3.14159265358979 (or PI, which is available in the expression as Math.PI) as the upper bound. The exact result is 2, and this calculator will return a value very close to 2.0000 with 100 or more subintervals.
Degrees to Radians Calculator
How do you convert degrees to radians?
Multiply the degree value by pi divided by 180. So radians = degrees times (pi / 180). For example, 90 degrees = 90 times pi/180 = pi/2 radians, which is approximately 1.5708.
Why are radians used instead of degrees?
Radians are the natural unit for angles in mathematics and physics because they relate directly to arc length. An angle of 1 radian subtends an arc equal in length to the radius of the circle. Many formulas in calculus and trigonometry are simpler in radians. Degrees are a historical convention based on dividing a circle into 360 parts.
What is a gradian (grad)?
A gradian (also called a grad or gon) divides a full circle into 400 equal parts. A right angle is exactly 100 gradians. Gradians are sometimes used in surveying and civil engineering. To convert from degrees, multiply by 10/9: gradians = degrees times (400/360).
Derivative Calculator
What is a numerical derivative?
A numerical derivative estimates the rate of change of a function at a specific point without finding a symbolic formula. The central difference method evaluates the function slightly to the left and right of the point and divides the change in output by the change in input. The result approximates f prime at that point with high accuracy for smooth functions.
What functions can I enter?
You can enter any expression using standard JavaScript Math syntax: x*x or pow(x,2) for powers, sin(x), cos(x), tan(x) for trig, log(x) for natural log, exp(x) for e raised to x, sqrt(x) for square root, and combinations like sin(x)*exp(-x). Use * for multiplication and avoid spaces around operators.
How accurate is the central difference method?
The central difference formula has an error proportional to h squared, where h is the step size (this calculator uses h = 0.0000001). For most smooth functions the result is accurate to at least five or six significant figures. Accuracy may be lower near discontinuities, very steep regions or points where the function is undefined.
Differential Pressure Calculator NZ
What is differential pressure?
Differential pressure (often written dP or ΔP) is simply the difference between two pressure readings, calculated as P1 minus P2. It is used throughout engineering and building services to measure things such as the pressure drop across a filter, the flow through an orifice plate, the level in a sealed tank, or the pressure loss along a length of pipe or duct. A single absolute pressure reading tells you very little on its own; the difference between two points tells you about flow, restriction, or level.
How do you convert differential pressure between units?
Differential pressure uses the same units as absolute or gauge pressure: pascals (Pa), kilopascals (kPa), bar, pounds per square inch (psi), or millimetres of water column (mmH2O). The common conversions are 1 kPa = 1,000 Pa, 1 bar = 100 kPa = 14.5038 psi, 1 psi = 6.89476 kPa, and 1 kPa is approximately equal to 101.97 mmH2O. This calculator converts your ΔP result into all of these units automatically.
How does differential pressure relate to flow rate?
For flow through a fixed restriction such as an orifice plate, venturi, or filter, flow rate is approximately proportional to the square root of the differential pressure across it (Q2/Q1 = sqrt(dP2/dP1)), assuming the fluid, restriction geometry, and discharge coefficient stay the same. This means doubling the differential pressure only increases flow by about 41%, not 100%. This calculator uses that relationship to estimate the new flow rate if you know a reference flow and pressure drop and want to find the flow at a different dP.
Divisor Function Calculator
What is the divisor function sigma?
The sum of all positive divisors of n, including 1 and n. For 12 it is 1 + 2 + 3 + 4 + 6 + 12 = 28.
What is the number of divisors?
How many positive divisors n has. For a factorisation into primes, multiply each exponent plus one together.
How does it classify a number?
If the proper divisor sum (sigma minus n) equals n the number is perfect; more is abundant, less is deficient.
Function Inverse Calculator
How do I find the inverse of a linear function?
Swap x and y, then solve for y. For mx + b this gives (x minus b) / m.
When does a linear function have an inverse?
Whenever the slope m is non-zero, so the function is one-to-one. A horizontal line (m = 0) has no inverse.
What does the inverse do?
It reverses the function: if f sends a to b, the inverse sends b back to a.
Implicit Differentiation Calculator
What is implicit differentiation?
Implicit differentiation finds dy/dx when y is not isolated as an explicit function of x, but both x and y appear together in an equation such as x squared plus y squared equals 25. By differentiating both sides with respect to x and treating y as a function of x, you can solve for dy/dx using the implicit function theorem: dy/dx = minus Fx divided by Fy, where Fx and Fy are the partial derivatives of F with respect to x and y.
How do you find dy/dx implicitly for x squared plus y squared equals 25?
Write F(x,y) = x squared plus y squared minus 25 = 0. Then Fx = 2x and Fy = 2y. So dy/dx = minus Fx divided by Fy = minus 2x divided by 2y = minus x divided by y. At the point (3, 4), dy/dx = minus 3 divided by 4 = minus 0.75.
What is the tangent line at a point found by implicit differentiation?
Once you have the slope dy/dx at a point (x0, y0), the tangent line is y minus y0 = m times (x minus x0), where m = dy/dx at that point. For x squared plus y squared = 25 at (3, 4), the slope is minus 0.75 and the tangent line is y minus 4 = minus 0.75 times (x minus 3), which simplifies to y = minus 0.75x + 6.25.
Integral Calculator
What is numerical integration?
Numerical integration estimates the area under a curve (the definite integral) by dividing the interval into many small pieces and summing approximate areas for each piece. Unlike symbolic integration, it works for functions that have no closed-form antiderivative, making it widely useful in engineering, physics and data analysis.
What is Simpson's rule?
Simpson's rule approximates each small strip of area using a parabola (a quadratic curve) fitted to three points instead of a rectangle or trapezoid. This gives much more accurate results for the same number of subintervals. With 1000 subintervals the error for most smooth functions is extremely small, typically less than 0.0001 percent.
What expressions can I use for f(x)?
You can use any JavaScript Math expression: x*x for x squared, pow(x,3) for x cubed, sin(x), cos(x), exp(x), log(x) for the natural logarithm, sqrt(x), abs(x), and combinations of these. Use * for multiplication and avoid division where the denominator could be zero within your interval.
Limit Calculator
What is a function limit?
The limit of f(x) as x approaches a value c is the number that f(x) gets arbitrarily close to as x gets closer and closer to c from either side, without necessarily equalling c. Limits form the basis of calculus: they define the derivative (rate of change) and the integral (accumulated area), and they describe the behaviour of functions at points where they may be undefined.
What is the difference between the left and right limit?
The left limit evaluates the function as x approaches c from values smaller than c (from the left). The right limit evaluates it as x approaches from values larger than c (from the right). For the two-sided limit to exist, both one-sided limits must exist and be equal. If they differ, the two-sided limit does not exist.
How does numerical limit estimation work?
This calculator evaluates the function at four points very close to c: c plus 0.001, c plus 0.0001 from the right, and c minus 0.001, c minus 0.0001 from the left. If the values at the two right points are converging to the same number, that number is the right limit. The same test is applied for the left. If both one-sided limits agree to four decimal places, the two-sided limit is reported.
Mobius Function Calculator
What does the Mobius function do?
It returns 0 for numbers with a squared prime factor, and plus or minus 1 for squarefree numbers depending on whether they have an even or odd number of prime factors.
What is a squarefree number?
One not divisible by any perfect square above 1, equivalently no prime appears twice in its factorisation.
Where is it used?
In Mobius inversion, in counting problems, and in deep links between primes and the Riemann zeta function.
Partial Derivative Calculator
What is a partial derivative?
A partial derivative measures how a multivariable function changes when you vary one input while holding all others constant. For f(x,y), the partial with respect to x treats y as fixed and differentiates with respect to x only, giving the instantaneous rate of change in the x direction at a specific point.
How is a partial derivative calculated numerically?
This calculator uses the central difference formula: df/dx is approximately (f(x+h, y) minus f(x-h, y)) divided by 2h, where h is 1e-7. Central difference is more accurate than forward or backward difference because errors cancel symmetrically, giving around 12 significant figures for smooth functions.
What is the gradient vector?
The gradient vector of f(x,y) at a point is (df/dx, df/dy). It points in the direction of steepest ascent on the surface z = f(x,y). Its magnitude is the rate of change in that steepest direction. Gradients are used in optimisation and machine learning to find minima and maxima.
Radians to Degrees Calculator
How do you convert radians to degrees?
Multiply the radian value by 180 divided by pi. So degrees = radians times (180 / pi). For example, pi radians = pi times (180/pi) = 180 degrees. And 1 radian = 180/pi = approximately 57.296 degrees.
What is 1 radian in degrees?
1 radian equals 180 divided by pi degrees, which is approximately 57.2958 degrees. This is the angle subtended at the centre of a circle by an arc equal in length to the radius.
How many radians are in a full circle?
A full circle (360 degrees) equals 2 times pi radians, which is approximately 6.2832 radians. A half circle (180 degrees) equals pi radians (approximately 3.1416). A quarter circle (90 degrees) equals pi over 2 radians (approximately 1.5708).
Radians to Revolutions Calculator
How many radians in one revolution?
Two pi radians, about 6.283185. So revolutions = radians divided by two pi.
What is a revolution?
One full turn around a circle, equal to 360 degrees or two pi radians. It is also called a turn or cycle.
How do I also get degrees?
Multiply radians by 180 divided by pi. The calculator shows the degree value beside the result.
Taylor Series Calculator
What is a Taylor series?
A Taylor series is an infinite polynomial expansion of a function about a centre point a. The series is f(a) + f'(a)(x-a) + f''(a)(x-a) squared divided by 2! + f'''(a)(x-a) cubed divided by 3! and so on. A Maclaurin series is a Taylor series centred at a = 0. By taking more terms you can approximate the function to any desired accuracy over a given interval.
What are the first five terms of the Taylor series for sin(x)?
The Maclaurin series for sin(x) (centred at zero) is x minus x cubed divided by 6 plus x to the fifth divided by 120 minus x to the seventh divided by 5040 plus x to the ninth divided by 362880, and so on. The general term is (minus 1) to the power n times x to the power (2n+1) divided by (2n+1) factorial.
How accurate is a Taylor series approximation?
Accuracy depends on how many terms you include and how far x is from the centre a. For sin(x) at x = 0.5 radians, five terms give an approximation of 0.47943, matching the true value to five decimal places. Adding more terms extends the range of good accuracy further from the centre.
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Answers are gathered from the calculators and guides listed above and are general information, not advice. Last reviewed 2026-09-06. See also the finance glossary, the guides and the reference data.