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.
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.
Mathematics
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.
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.
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
lim
Bounds
Transformation y = a · f(b(x − c)) + d
Parameters a, b, c, k (write them as letters in the expression)
Show in the graph
function f(x)
curve r(θ)
approaching points x = a ± happroaching point
limitlimit, i.e. horizontal asymptote
right side g(x)
Zoom by holding down Ctrl and scrolling
Drag point Pthe bounds a and bthe point athe approaching point or tap the graph.
Pinch with two fingers to zoom.
Drag point P or Q directly in the graph. Dragging from an empty spot moves the view, and Ctrl + scroll or the buttons zoom (in full-screen mode, just scroll). To enter an exact value, click the value of a or h. Drag point P or Q, or tap the graph to move P to where you tapped. Dragging with one finger moves the view, and pinching with two fingers zooms. To enter an exact value, tap the value of a below the graph. The vertical dashed line connects the point a in all the graphs. A reading underlined with a dotted line explains itself when you hover over it or click it. Tap a reading underlined with a dotted line to see what it means.
Drag the upper bound b or the lower bound a directly in the graph (the orange points on the x-axis). The slider changes the number of subintervals n, and n → ∞ increases it until the rectangles fill the region. To enter an exact bound, click its value. A reading underlined with a dotted line explains itself when you hover over it or click it. Drag the bounds a and b (the orange points on the x-axis), or tap the graph to move the upper bound there. The slider below the graph changes the number of subintervals n. To enter an exact bound, tap its value. Tap a reading underlined with a dotted line to see what it means.
Drag point P or click the graph to see the value of the function at that point. The coefficients a, b, c and d change the graph, and the ▶ button moves a coefficient back and forth. You can type a sum or a composite function directly into the function field, e.g. x^2 + sin(x).
Drag point P or tap the graph to see the value of the function at that point. The coefficients a, b, c and d change the graph, and the ▶ button moves a coefficient back and forth.
The solutions of the equation f(x) = g(x) are the x-coordinates of the intersection points of the graphs of the left and right sides. Drag point P or click the graph to see whether the sides are equal at that point. Change the sign to <, ≤, > or ≥ and the solution set is coloured on the x-axis. You can also type the whole equation in the top field. The solutions of the equation are the x-coordinates of the intersection points of the graphs. Drag point P or tap the graph to see whether the sides are equal at that point. The sign <, ≤, > or ≥ turns the equation into an inequality whose solution set is coloured.
The point P is on the curve at the angle θ: the radius r is measured from the origin in the direction of the angle. Drag the point or click the graph and the angle follows the direction you point to. Change the parameters a, b, c and k with the sliders or write your own expression, e.g. a(1 + cos(θ)). The graph below shows the same curve as a function of the angle.
The point P is on the curve at the angle θ. Drag the point or tap the graph and the angle follows the direction you point to. Change the parameters with the sliders or write your own expression, e.g. a(1 + cos(θ)).
Press x → ∞ and the approaching point moves further and further away, and the view zooms out to keep it visible. As the curve settles towards the dashed line, the limit is the height of the dashed line. You can also drag the point, and the numbers in the table of values move it to 10, 100, … The ⟲ button resets the view. Drag the point a or the approaching points directly in the graph. Dragging from an empty spot moves the view, and Ctrl + scroll or the buttons zoom (in full-screen mode, just scroll). You can type an exact point under the lim sign. A reading underlined with a dotted line explains itself when you hover over it or click it. Drag the point a or the approaching points, or tap the graph to move the point a to where you tapped. Dragging with one finger moves the view, and pinching with two fingers zooms. You can type an exact point under the lim sign on the Calculations tab. Tap a reading underlined with a dotted line to see what it means.
Sign chart
The sign of the derivative tells whether the function is increasing or decreasing.
Derivative functionAntiderivativeDifference functionRadius as a function of the angle
The same curve in a Cartesian coordinate system: the angle θ is on the horizontal axis and the radius r on the vertical axis. The points of the polar curve correspond to the points of this graph, and when r < 0, the point in the polar picture is on the opposite side of the angle.
When everything is moved to the left, the equation becomes f(x) − g(x) = 0. The zeros of the difference function are the solutions of the equation, and for an inequality we look at where the difference is positive or negative.
Accumulated area from the lower bound a to x. It increases where f(x) > 0 and decreases where f(x) < 0, i.e. F′(x) = f(x).
Cheat sheet
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→∞ cxk = 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
An ≤ ∫baf(x) dx ≤ Yn
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)]ab means the same difference.
Integration rules
| ∫ xn dx = xn+1/(n + 1) + C, n ≠ −1 |
| ∫ 1/x dx = ln |x| + C |
| ∫ ex dx = ex + 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 ebθ 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
ax = c ⇒ x = ln c / ln a
An unknown in an exponent is brought out with a logarithm. Conversely, ln x = c gives x = ec. 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 2x, 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 ex = ex | 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) | |
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