DerivativeLimitIntegralFunctionsEquations and inequalitiesPolar coordinates

The derivative is the instantaneous rate of change of a function. It tells how steeply the graph rises or falls at a given point; graphically, it is the slope of the tangent drawn to the curve.

The limit is the number that the values of a function approach as x approaches a. It tells what height the graph is heading towards, even if the function is not defined at a; graphically, it is the height the graph approaches from both sides.

The definite integral is the area of the region between the curve and the x-axis, with the part below the x-axis counted as negative. It is estimated with rectangles, and as their number grows, the estimate approaches the integral. With an antiderivative, the integral is calculated exactly.

An equation is a statement that two expressions are equal. Its solution is a number for which the statement is true. The solution can be seen in the graph: when the left side f(x) and the right side g(x) are drawn in the same picture, the solutions are the x-coordinates of the intersection points of the graphs. An inequality asks where one side is larger, and the solution is a set of numbers. In Calculations the equation is solved step by step and the solutions are checked.

Polar coordinates describe the position of a point with two numbers: the angle θ and the distance, or radius, r from the origin. The curve r(θ) appears when the radius depends on the angle: a constant radius gives a circle, a growing radius a spiral and an oscillating radius a rose. The graph shows how the angle and the radius together draw the curve, and in the Calculations section the point is converted to Cartesian coordinates.

A function assigns to each value x of the variable exactly one value f(x). The graph shows where the function is zero, positive or negative, where it increases and where it has its largest or smallest value. The coefficients a, b, c and d stretch and shift the graph.

Electricity

DC circuits

From Ohm's law and power to connections and Kirchhoff's laws. Each section has a simulator where you can enter your own numbers, a worked example step by step and practice questions that are generated anew each time.

Electricity

Electric field and capacitors

From charges and the electric field to voltage and capacitors. Each section has a simulator where you can enter your own numbers, a worked example step by step and practice questions that are generated anew each time.

Electricity

Magnetism and induction

From the magnetic force to the magnetic field of a current and electromagnetic induction. Each section has a simulator where you can enter your own numbers, a worked example step by step and practice questions that are generated anew each time.

Electricity

Oscillations and alternating current

From oscillation and resonance to waves, electromagnetic radiation and AC circuits. Each section has a simulator where you can enter your own numbers, a worked example step by step and practice questions that are generated anew each time.

Exercise simulator

Solve derivative and limit exercises one step at a time. A calculation exercise gives a ready-made expression; a word problem gives a practical situation from electrical engineering. At every step you can get a hint when needed, for example which rule to use, and a button shows the graph of the exercise. The exercises start from the basics and get harder level by level.

Reference

Mathematics glossary

The most important mathematical concepts and notations. Each row shows the symbol first, then the name, a value or example and a short explanation. Search for a word or symbol, or jump to a group using the links.

Reference

Key formulas

Basic formulas of geometry, trigonometry, equations, percentages, derivatives and physics. Each row shows the formula, its quantities and often an example. The rules for powers, fractions and logarithms are on the page Key calculation rules.

Reference

Key calculation rules

Order of operations, laws of arithmetic, signs, fractions, powers, brackets and logarithms in brief.

Reference

Branches of mathematics

The main fields of mathematics and their branches: what each field studies. The link on a row takes you to the part of the app where you can practise, explore or read more about the branch.

Reference

Mathematical models

A mathematical model describes a real situation in the language of mathematics. The building blocks of a model (expression, equation, inequality, function, system of equations and algorithm) are explained with the same taxi example so that their differences are clear. At the end are the most common types of model.

Electricity

Quantities of electricity

The most common quantities of electricity: symbol, unit and main formula in a separate table for each topic. Hover over (or tap) the name of a quantity to see an explanation and an example.

Electricity

Laws of electricity

The most important laws of electricity and their formulas: circuits, the electric field, the magnetic field and induction. Hover over (or tap) the name of a law to see a more detailed explanation and an example.

Limit at a

Left-hand limit
Right-hand limit
Function value

As x approaches a

Function values on both sides of a. Tap a value to move the points there.
h	f(a − h)	f(a + h)

Derivative at a

Point
Slope of the tangent
Angle of the tangent to the x-axis
Instantaneous rate of change

Equation of the tangent

Difference quotient

The slope of the secant, i.e. the average rate of change of the function on the interval [a, a + h].

As h approaches zero

The difference quotient with small steps. Tap a value to set the step h to it.
\|h\|	h > 0	h < 0

Increasing, decreasing and extrema

On the visible interval

Definite integral

Bounds
Area above the x-axis
Area below the x-axis
Total area
Meaning

Calculation with an antiderivative

Fundamental theorem of calculus: the integral is the difference between the values of the antiderivative at the upper and lower bounds.

Riemann sum

The interval is divided into n subintervals of equal width, and the area is estimated with rectangles.

As n grows

Lower and upper sums with more and more subintervals. Tap a value to set the number of subintervals and the type of sum.
n	lower sum	upper sum

Function value at point P

Point
Sign of the value
Increasing or decreasing here

Properties

On the visible interval

Domain
Zeros
Intersection with the y-axis
Symmetry
Vertical asymptotes
Horizontal asymptotes

Sign, increasing/decreasing and extrema

Graph transformation

Table of values

Step

0.1 0.5 1 2

Intersection points

Point P on the curve

Angle θ
Radius r
Cartesian coordinates
Area of the sector
Arc length

From polar to Cartesian coordinates

The same point is described by an angle and a radius (r, θ) or by a horizontal and a vertical distance (x, y).

Properties of the curve

Largest distance from the origin
Passes through the origin
Symmetry
Closing
Area enclosed by the curve
Length of the curve

Table of values

Angle step

15° 30° 45° 90°

Try point P

Left side
Right side
Difference (left − right)

Solution

Solution step by step

Table of values

Step

0.1 0.5 1 2

Cheat sheet

Limit

limx→a f(x) = L

The values f(x) can be made as close to L as we like by choosing x close enough to a. The value of the function at a does not matter, and the function need not even be defined there.

One-sided limits

limx→a⁻ f(x) from the left, x < a
limx→a⁺ f(x) from the right, x > a

The limit exists only if the one-sided limits exist and are equal. For example, |x|/x approaches −1 from the left and 1 from the right, so there is no limit at 0.

Continuity

limx→a f(x) = f(a)

A function is continuous at a if its limit equals the value of the function. Polynomials and rational functions, as well as root, exponential, logarithmic and trigonometric functions, are continuous on their domains, so their limits can be found by substitution.

Calculating a limit

Substitute x = a. If you get a number, that is the limit.
Form 0/0: factorise and cancel, or multiply by the conjugate.
Form c/0 with c ≠ 0: the values increase or decrease without bound. The sign of the denominator on each side of a decides whether the result is ∞, −∞ or neither.
Absolute value: calculate the left-hand and right-hand limits separately.

Limit at infinity

limx→∞ cx^k = 0 when k > 0

When x grows without bound and the values approach L, the line y = L is a horizontal asymptote of the graph. In a rational expression, divide the numerator and the denominator by the highest power of the denominator. Then the degrees decide:

→ 0 the degree of the numerator is smaller
→ a/b the degrees are equal (a and b are the leading coefficients)
→ ±∞ the degree of the numerator is larger, and there is no horizontal asymptote

Definite integral

∫baf(x) dx

The area between the curve y = f(x) and the x-axis from a to b, with the part below the x-axis counted as negative. The numbers a and b are the lower and upper bounds of integration.

Lower and upper sums

A_n ≤ ∫baf(x) dx ≤ Y_n

The interval is divided into n subintervals of width Δx = (b − a)/n. In the lower sum, the height of each rectangle is the smallest value of the function on the subinterval; in the upper sum, the largest. As n grows without bound, both sums approach the integral.

Antiderivative

∫ f(x) dx = F(x) + C, where F′(x) = f(x)

Integration is the inverse of differentiation: the derivative of an antiderivative is the original function. The constant C can be any number, because the derivative of a constant is zero. You can always check the result by differentiating.

Fundamental theorem of calculus

∫baf(x) dx = F(b) − F(a)

When f is continuous on the interval, the integral is calculated by substituting the bounds into any antiderivative. The notation [F(x)]_a^b means the same difference.

Integration rules

∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ −1

∫ 1/x dx = ln |x| + C

∫ e^x dx = e^x + C

∫ sin x dx = −cos x + C

∫ cos x dx = sin x + C

∫ (f + g) dx = ∫ f dx + ∫ g dx

∫ cf dx = c ∫ f dx

∫ f(kx + b) dx = F(kx + b)/k + C

Area using the integral

f(x) ≥ 0 the area is the integral
f(x) ≤ 0 the area is the negative of the integral

If the function changes sign on the interval, the interval is split at the zeros, and the areas of the parts are calculated separately. For example, sin x on 0 … 2π: the integral is 0, but the area is 4.

Polar coordinates

(r, θ)

A point is given by its distance r from the origin and the angle θ, measured counter-clockwise from the positive x-axis. The polar curve r = r(θ) tells how far the point is in each direction.

Conversion to Cartesian

x = r cos θ, y = r sin θ

Back: r = √(x² + y²) and tan θ = y/x. The tangent does not tell the quadrant, so the angle is also deduced from the signs of x and y.

Degrees and radians

180° = π rad

30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2
angle in radians = angle in degrees · π/180
a full turn is 360° = 2π

Negative radius

(−r, θ) = (r, θ + π)

If the radius is negative, the point is drawn on the opposite side of the direction of the angle. That is why a curve can pass through the origin and continue on the other side.

Well-known curves

r = a circle
r = a sin θ circle through the origin
r = a(1 + cos θ) cardioid
r = a cos kθ rose: k petals when k is odd and 2k when it is even
r = aθ Archimedean spiral
r = a e^bθ logarithmic spiral
r = a sec θ the line x = a

Area and length

A = ½ ∫ r² dθ

The area of a sector is found by integration. The length of the curve is L = ∫ √(r² + r′²) dθ. For a circular sector A = ½r²θ.

Symmetry

r(−θ) = r(θ): symmetric about the x-axis
r(π − θ) = r(θ): symmetric about the y-axis
r(θ + 2π) = r(θ): the curve repeats after one turn

Solution of an equation

f(x) = g(x)

A solution is a number for which both sides of the equation are equal. In the graphs, the solutions are the x-coordinates of the intersection points. A solution can always be checked by substitution: if the sides are equal, the number is a solution.

Equivalent transformations

add or subtract the same number on both sides
multiply or divide both sides by the same non-zero number
simplify each side separately

These transformations do not change the solutions. When other transformations are used, such as squaring or multiplying by the denominator, the solutions must be checked.

First-degree equation

ax + b = 0 ⇒ x = −b/a

Move the unknowns to one side and the numbers to the other, and divide by the coefficient. The graphs are lines, so there is one intersection point, none (parallel lines) or infinitely many (the same line).

Quadratic equation

ax² + bx + c = 0 ⇒ x = −b ± √(b² − 4ac)2a

D > 0 two solutions
D = 0 one solution (a double root)
D < 0 no real solutions

The discriminant is D = b² − 4ac. If b = 0 or c = 0, the quadratic formula is not needed.

Zero-product rule

ab = 0 ⇒ a = 0 or b = 0

A product is zero only if one of its factors is zero. An equation in factored form (x − 1)(x + 2) = 0 is solved immediately: x = 1 or x = −2. The right side must be zero.

Fractional and radical equations

Condition: denominator ≠ 0, under a square root ≥ 0
Fractions: multiply by the common denominator
Square root: isolate the root and square both sides
Always check the solutions: a false root does not satisfy the original equation

Exponential and logarithmic equations

a^x = c ⇒ x = ln c / ln a

An unknown in an exponent is brought out with a logarithm. Conversely, ln x = c gives x = e^c. An absolute value equation |u| = c splits into the cases u = c and u = −c.

Inequality

First solve the corresponding equation: its solutions are the boundaries
Test one point from each interval
When multiplying or dividing by a negative number, the direction of the inequality reverses: −2x < 6 ⇒ x > −3
The signs < and > leave out the boundary, ≤ and ≥ include it

Function and graph

x ↦ f(x)

A function assigns to each number x in its domain exactly one value f(x). The graph consists of all the points (x, f(x)). The domain tells which values of the variable are allowed, the range which values are obtained.

Zeros and sign

f(x) = 0

At a zero, the graph crosses or touches the x-axis. The zeros are found by solving the equation f(x) = 0. The sign of a function can change only at a zero or at a point where the function is not defined.

Increasing, decreasing and extrema

increasing: the value grows as x increases
decreasing: the value falls as x increases

At a local maximum the graph turns from rising to falling, and at a minimum from falling to rising. The extrema are found exactly at the zeros of the derivative.

Graph transformations

y = a · f(b(x − c)) + d

a stretches vertically (a negative value reflects in the x-axis)
b scales horizontally by the factor 1/b: when |b| > 1, the graph gets narrower (a negative value reflects in the y-axis)
c shifts right (a negative value shifts left)
d shifts up (a negative value shifts down)

Even and odd

f(−x) = f(x) even, symmetric about the y-axis
f(−x) = −f(x) odd, symmetric about the origin

For example, x², |x| and cos x are even, while x³, 1/x and sin x are odd. Most functions are neither.

Asymptotes

x = c vertical: the values increase or decrease without bound as x → c
y = L horizontal: the values approach L as x → ±∞

A rational function usually has a vertical asymptote at a zero of the denominator. For the exponential function 2^x, the horizontal asymptote is the x-axis.

Definition of the derivative

f′(a) = limh→0 f(a + h) − f(a)h

The difference quotient is the slope of the secant. As h approaches zero, point Q slides towards point P, the secant turns into the tangent and the difference quotient approaches the derivative.

Derivative and tangent

The derivative f′(a) is the slope of the tangent drawn to the curve at the point (a, f(a)). It is the instantaneous rate of change of the function.

y − f(a) = f′(a)(x − a)

The slope is k = tan α, where α is the angle between the tangent and the x-axis.

Sign of the derivative and increasing/decreasing

f′(x) > 0 the function is increasing
f′(x) < 0 the function is decreasing
f′(x) = 0 the tangent is horizontal

If the sign of the derivative changes at a zero, there is an extremum: + → − maximum, − → + minimum. If the sign does not change, the point is a stationary point of inflection.

Differentiation rules

D c = 0	D xⁿ = nxⁿ⁻¹
D e^x = e^x	D ln x = 1/x
D sin x = cos x	D cos x = −sin x
D [f + g] = f′ + g′ D [cf] = cf′
D [fg] = f′g + fg′
D f(g(x)) = f′(g(x))·g′(x)

Calcuralab – interactive mathematics and electricity simulators

DerivativeLimitIntegralFunctionsEquations and inequalitiesPolar coordinates

DC circuits

Electric field and capacitors

Magnetism and induction

Oscillations and alternating current

Exercise simulator

Mathematics glossary

Key formulas

Key calculation rules

Branches of mathematics

Mathematical models

Quantities of electricity

Laws of electricity

Function to exploreEquationCurve to explore

Example functionsExample equationsExample curves

Derivative of the functionLimit of the functionAntiderivative of the functionExpression of the functionEquationEquation of the curve

Show in the graph

Sign chart

Derivative functionAntiderivativeDifference functionRadius as a function of the angle

Cheat sheet

Limit

One-sided limits

Continuity

Calculating a limit

Limit at infinity

Definite integral

Lower and upper sums

Antiderivative

Fundamental theorem of calculus

Integration rules

Area using the integral

Polar coordinates

Conversion to Cartesian

Degrees and radians

Negative radius

Well-known curves

Area and length

Symmetry

Solution of an equation

Equivalent transformations

First-degree equation

Quadratic equation

Zero-product rule

Fractional and radical equations

Exponential and logarithmic equations

Inequality

Function and graph

Zeros and sign

Increasing, decreasing and extrema

Graph transformations

Even and odd

Asymptotes

Definition of the derivative

Derivative and tangent

Sign of the derivative and increasing/decreasing

Differentiation rules

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