Variables, Functions and Limits

READING FORMULAS

(Чтение формул )

a b = c a divided by b is equal to c

2 x 2 = 4 twice two is four

c x d = b c multiplied by d equals b

dx differential of x

= a plus b over a minus b is equal to c plus d over c minus b

y_a-b ∙ x_b-c = 0 y sub a minus b multiplied by x sub b

minus c is equal to zero

+ [1 + b(s)]y = 0 the second derivative of y with respect to s

plus y times open bracket one plus b of s

in parentheses, close bracket is equal to

zero

∫f(x)dx the integral of f(x) with respect to x

the definite integral of f(x) with respect to

x from a to b (between limits a and b)

c(s) = K_abc of s is equal to K sub ab

x_a-b = c x sub a minus b is equal to c

a b a varies directly as b

a : b :: c : d; a is to b as (equals) c is to d

a : b = c : d

x 6 + 42 1) x times six is forty two;

2) x multiplied by six is forty two

10 ÷ 2 = 5 1) ten divided by two is equal to five;

2) ten over two is five

a squared over c equals b

1) a raised to the fifth power is c;

2) a to the fifth degree is equal to c

a³ = log_cb a cubed is equal to the logarithm of b

to the base c

the logarithm of b to the base a is equal

to c

x sub a minus b is equal to c

= 0 the second partial derivative of u w

with respect to t equals zero

c : d = e : 1 c is to d as e is to 1

15 : 3 = 45 : 9 1) fifteen is to three as forty five is to nine;

2) the ratio of fifteen to three is equal to

the ratio of forty five to nine

p T p is approximately equal to the sum of x

sub i delta x sub i and it changes from zero

to n minus one

the square root of a

squared plus b squared minus the square

root of a squared plus b sub one squared

by absolute value is less or equal to b

minus b sub one by absolute value (by

modulus)

a to the power z sub n is less or equal to

the limit a to the power z sub n where n

tends (approaches) the infinity

the sum of n terms a sub j, where j runs

from l to n

= 3 the fourth root of 81 is equal to three

c d c varies directly as d

sin = a sine angle is equal to a

integral of dx divided by (over) the square

root out of a square minus x square

d over dx of the integral from x sub 0 to x

of capital x_dx

2 : 5 1) two is to five;

2) the ratio of two to five

5:4::20:16 1) five is to four as twenty is to sixteen;

2) the ratio of five to four equals the ratio of twenty to sixteen;

3) five has the same ratio to four as twenty

has to sixteen

a b 1) a is not equal to b;

2) a differs from b;

3) a is different from b

1) a approximately equals b;

2) a is approximately equal to b

p plus (or) minus q

m > n m is greater than n

m < n m is less than n

1) a is greater than or equal to b;

2) a is greater than or equals to b

1) a is less than or equals to b;

2) a is less than or is equal to b

y → r y approaches r

y r y approaches r

Note: Some authors use the notation .

The symbol signifies an approach to a

limit. Thus

may be read “x approaches 2” (as a

limit). If the words “as a limit” are not

expressed, they must be always

understood.

two times thee to the n-th (power) minus

2 is not less than five hundred

Note: Just the symbol means

“not equal to”, so the symbol means

“not less than”.

B = capital B is equal to infinity

1) the modulus of a;

2) the absolute value of a;

3) the numerical value of a

the modulus of the quantity x minus b is

greater than zero and less than or equal to

capital C

the interval a to b

28^o 28 degrees (angular measure and

temperature measure)

56' 1) 56 minutes (angular measure);

2) 56 feet (linear measure)

45'' 1) 45 seconds (angular measure);

2) 45 inches (linear measure)

the square root (out) of 7

5% 5 per cent

2/9 % 1) two ninths per cent;

2) two ninths of one per cent

½ % 1) a half per cent;

2) a half of one per cent

0.47% 1) point four seven per cent;

2) zero point forty-seven per cent;

3) nought point forty-seven per cent;

4) o point four seven of one per cent

7 %₀ 7%₀ seven per mille

c is equal to (dash, line of division) a

over (divided by, by) b

Note: The words dash and line of division

are often omitted.

c(a+b) 1) c parenthesis a plus b close parenthesis;

2) c round brackets opened a plus b round

brackets closed;

3) c times (multiplied by) the quantity

a plus b

3[(4+5)6-20] 1) three, square brackets, parenthesis, four

plus five, close parenthesis, times

(multiplied by) six minus twenty, close

square brackets

2) three, square and round brackets

opened, four plus five, round brackets

closed, six, minus twenty, square

brackets closed;

3) three times (multiplied by) the whole

quantity: the quantity four plus five,

times six, minus twenty

2{70-3[(4+5)6-20]} 1) two, braces, seventy minus… close

braces

2) two, braces opened, seventy minus …

braces closed

ABC EDF the triangle ABC is congruent to the

triangle EDF

BD AC BD is perpendicular to AC

AB is parallel to CD

the angle A is equal to the angle B

3! = 1 x 2 x 3 = 6 factorial three is equal to one times two

times three is equal to six

* the fourth binomial coefficient

the number of combinations of seven

things taken two at a time is equal to seven

times six over factorial two is equal to

twenty-one

the number of permutations of seven

things taken five at a time

1) the derivative of y with respect to x;

2) d over (by) dx of y

the derivative of the quantity two x square plus five with respect to x

1) second derivative of y with respect to x;

2) d two over (by) dx of y

d over (by) dx of the integral from a to b

of f of x dx

x is equal to the logarithm of capital N to

the base q

arc sin a 1) the angle whose sine is a;

2) the inverse sine of a;

3) the anti-sine of a;

4) the arc sine of a

Note: The symbol arc sin a is

sometimes written sin^-1a.

sin (arc sin a) the sine of the angle whose sine is a

sin 23º the sine of 23º

cos 47º the cosine of 47º

sec 80º the secant of 80º

the tangent of a (one) half (of) A

sin α the sine of (the angle) α

the cosine of the angle of one half A

minus B (the difference of A and B)

sin(α-β) the sine of (the angle) α minus β

cot(α+β) the cotangent of (the angle) alpha plus β

sin ABC the sine of the angle ABC

cos ABC the cosine of the angle ABC

tan β the tangent of (the angle) β

Subscripts and Superscripts

' a prime

b'' 1) b double prime;

2) b second prime;

3) b twice dashed

1) c first;

2) c sub one

1) d sub zero;

2) d zero-th

1) f prime, sub one;

2) f sub one, prime

1) f prime, m-th;

2) f m-th, prime

1) capital A, r-th;

2) capital A, sub r

capital L

1) capital P sub m minus two, prime;

2) capital P prime, sub m minus two

first derivative of y

second derivative of y

h triple prime

1) c sub q

2) c q-th

Exponents

1) a square;

2) a squared;

3) a to the second;

4) a to the second power;

5) a raised to the second power;

6) the square of x;

7) the second power of x

1) b cube;

2) b cubed;

3) b to the third;

4) b to the third power;

5) b raised to the third power;

6) the cube of b;

7) the third power of b

c to the sixth

Note: The variants given below are possible with all the exponents.

1) d to the y-th;

2) d to the y-th power;

3) d raised to the y-th power;

4) the y-th power of d

m to the minus first

n to the minus seventh

c to the m-th

two to the pi-th

x to the minus q-th

c to the power m minus n

1) z to the power m over (divided by, by) n;

2) z to the m by n-th power;

3) z to by (power)

k to the power m over n minus q

minus capital B to the n minus one

1) the quantity five minus a over b to the

fourth…;

2) parenthesis five minus a over b close

parenthesis to the fourth;

3) round brackets opened, five minus a over

b round brackets closed to the fourth

Roots

the square root (out) of two

the square root (out) of a

the cube root (out) of b

the seventh root (out) of d

the q-th root (out) of a to the m-th power

five times the fourth root (out) of three

two-thirds times the square root (out) of b

capital C is equal to the square root (out) of

x square plus y square

the twelfth root (out) of p cube plus q cube

the fourth root (out) of (dash, line of

division) capital A first (sub one) plus

capital B over (by, divided by) three a

second (sub two) b double prime (twice

dashed)

the minus m-th root (out) of p to the fourth

(power)

Variables, Functions and Limits

1) f of x;

2) a function of x

Note: Not f times x

capital F first (sub one) of x

1) a function of (two variables) x and y;

2) f of y and y

1) a function of x, y and z;

2) f of x, y and z

1) a function of the quantity x plus delta x;

2) f of the quantity x plus delta x

1) the limit of f of x is equal to l;

2) the limit of a function of x is equal to l

the limit of x as x tends to (approaches) is

equal to l

the limit of z, as v approaches l, is equal to a

the limit of f of x, as x tends to infinity (x

increases without bound, x becomes infinity)

is equal to l

The point P with coordinates x and y

1) The point A with coordinates minus two

and minus three;

2) The point A whose abscissa is minus two

and ordinate minus three

EXAMPLES OF READING FORMULAS

Algebraic Formulas

The square of the sum of two numbers (a binomial) is equal to the square of the first term, plus twice the product of the first and last terms, plus the square of the last term.

The square of the difference of two numbers (of a binomial) is equal to the square of the first term, minus twice the product of the first and last terms, plus the square of the last term.

The product of the sum and difference of two numbers is equal to the difference of the squares of the numbers.

The product of two binomials having a common term is equal to the product of the first two terms, plus the sum of the last terms multiplied by the common term, plus the product of the last terms.

The square of the sum of three numbers (a trinomial) is equal to the sum of the squares of each term of the trinomial and twice the product of each term by each of the other terms.

The sum of two cubes is factored into the sum of the cube roots and the incomplete square of the difference.

The difference of two cubes is factored into the difference of the cube roots and the incomplete square of the sum.

The derivative of a function is the limit of the ratio of the increment of the function to the increment of the independent variable, when the latter varies and approaches zero as a limit.

SOME GEOMETRICAL FORMULAS

Notation Used in Formulas

A area

a apothem

a, b, c sides of a triangle

b and b’ bases or areas of bases

C circumference

d diameter

h height, altitude

l length

w width

p perimeter

r radius

s slant height

s half the perimeter of a triangle =

3.1416

V volume

Formulas for lines

Right triangle

The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides

The square on the hypotenuse of a right triangle is equal to the sum of the squares on the other two sides (the Pythagorean theorem).

Circle

The circumference of a circle is equal to times the diameter.

Triangle

The altitude drawn to one side of a triangle is equal to the product of either one of the two other sides and the sine of the angle adjacent to it.

Areas

Square A = b²

The area of a square equals the length of the side of the square multiplied by itself, that is, the side squared.

Rectangle A = bh

The area of a rectangle is equal to the product of its base and altitude.

Parallelogram A = bH

The area of a parallelogram is equal to the product of the base and altitude.

A = ab sin A

The area of a parallelogram is equal to the product of the two sides and the sine of the included angle.

Triangle

The area of a triangle is equal to one-half the product of the base and altitude.

The area of a triangle is equal to one-half the product of two sides and the sine of the included acute angle.

Circle

The area of a circle is equal to one-half the product of the circumference and the radius.

The area of a circle is equal to times the square of the radius.

Trapezoid

The area of a trapezoid is equal to one-half the sum of the parallel bases times the altitude.

Zone

The area of a zone is equal to its altitude multiplied by the circumference of a great circle.

Lateral area of a cone

The lateral area of a right cone is equal to one-half the product of the circumference of the base and the slant height.

Lateral area of a pyramid

The lateral surface of a right pyramid is equal to one-half the product of the perimeter of the base and the slant height.

Lateral area of a cylinder

The lateral area of a cylinder is equal to the circumference of the base times the height.

Lateral area of a prism

The lateral area of a right prism is equal to the product of the perimeter of the base and the height.

Sector

The area of a sector is equal to one-half the product of the radius and the length of the arc.

Regular polygon

The area of a polygon is equal to one-half the product of the perimeter and apothem.

Rhombus

The area of a rhombus is equal to one-half the product of the two diagonals.

Circumscribed polygon

The area of a circumscribed polygon is equal to one-half the perimeter times the radius of the inscribed circle.

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Taken from: Английский язык. Reading Numerals and Formulas. Методический материал для чтения и перевода текстов, знаков, символов на английском языке / авт.-сост. Н.В.Вадбольская, А.В.Лучникова, И.В.Сальникова; Перм. ун-т. – Пермь, 2010. – 80 с. – На англ. яз.

Ellipse

The area of an ellipse equals times the product of the long and short diameters, or times the product of the long and short radii.

Surface of a sphere

The surface of a sphere is four times the area of a circle of the same diameter.

Volumes

Rectangular solid

The volume of a rectangular solid is equal to the product of its length, width and height.

The volume of a rectangular solid is equal to the area of the base times the height.

Prism or cylinder

The volume of any right prism is equal to the product of the base and altitude.

The volume of any right cylinder is equal to the product of the base and altitude.

The volume of any parallelepiped is equal to the product of the base times the altitude.

Oblique prism or cylinder

The volume of any oblique prism is equal to the product of a right section and the slant height.

Pyramid or cone

The volume of a pyramid (cone) is equal to one-third the product of the base and altitude.

Frustum of a pyramid or a cone

The volume of a frustum of a pyramid is equal to one-third the altitude multiplied by the sum of the area of the lower base, the area of the upper base and the square root of the product of the two bases.

Spherical sector

The volume of a spherical sector is equal to one-third the product of the area of the zone and the radius of the sphere.

TRIGONOMETRIC FORMULAS

In any oblique triangle the square of any side is equal to the sum of the squares of the other two sides diminished by (minus) twice their product times the cosine of the included angle (The law of cosines).

In any triangle the difference of any two sides is to their sum as the tangent of one half the difference of their respective opposite angles is to the tangent of one half of the sum of these angles (The law of tangents).

a plus b over a minus b is equal to c plus d over c minus d.

a cubed is equal to the logarithm of d to the base c.

1) of z is equal to b, square brackets, parenthesis, z divided by c sub m plus 2, close parenthesis, to the power m over m minus 1, close square brackets;

2) of z is equal to b multiplied by the whole quantity: the quantity two plus z over c sub m, to the power m over m minus 1, minus 1.

The absolute value of the quantity sub j of t one, minus sub j of t two, is less than or equal to the absolute value of the quantity M of t₁ minus over j, minus M of t₂ minus over j.

j = 1, 2, … n)

k is equal to the maximum over j of the sum from i equals one to i equals n of the modulus of of t, where t lies in the closed interval a b and where j runs from one to n.

The limit as n becomes infinite of the integral of f of s and of s plus delta n of s, with respect to s, from to t, is equal to the integral of f of s and of s, with respect to s, from to t.

sub n minus r sub s plus 1 of t is equal to p sub n minus r sub s plus 1, times e to the power t times sub plus s.

L sub n adjoint of g is equal to minus 1 to the n, times the n-th derivative of a sub zero conjugate times g, plus, minus one to the n minus 1, times the n minus first derivative of a sub one conjugate times g, plus … a sub n conjugate times g.

The partial derivative of F of lambda sub i of t, with respect to lambda, multiplied by lambda sub i prime of t, plus the partial derivative of F with arguments lambda sub i of t and t, with respect to t, is equal to 0.

The second derivative of y with respect to s, plus y, times the quantity 1 plus b of s, is equal to zero.

( arg z = )

f of z is equal to sub mk hat, plus big O of one over the absolute value of z, as absolute z becomes infinite, with the argument of z equal to gamma.

D sub n minus 1 prime of x is equal to the product from s equal to zero to n of, parenthesis, 1 minus x sub s squared, close parenthesis, to the power epsilon minus 1.

K of t and x is equal to one over two , times the integral of K of t and z, over w minus w of x, with respect to w along curve of the modulus of w minus one half, is equal to rho.

(a > 0)

The second partial (derivative) of u with respect to t, plus a to the fourth power, times the Laplacian of the Laplacian of u, is equal to zero, where a is positive.

(c > 1)

D sub k of x is equal to one over two , times integral from c minus i infinity to c plus i infinity of dzeta to the k of w, x to the w divided by w, with respect to w, where c is greater than 1.

4 c plus W third plus 2 m first a prime plus R a-th equals thirty-three and one-third.

A is equal to one half mu by r p-th omega L second omega L first over (by) the square root out of R second round brackets opened R first plus omega square L first square by r p-th round brackets closed.

Capital M is equal to R sub one multiplied by x minus capital P sub one, round brackets opened, x minus a sub one, round brackets closed, minus capital P sub two, round brackets opened, x minus a sub two, round brackets closed.

is equal to B divided by six point four five multiplied by two point five four.

Capital F is equal to capital C sub s, A, I L sine theta

Tangent r is equal to, dash (line of division), tangent i over (by) e.

A v-th is equal to mu omega m omega square L square (divided) by r p-th square brackets opened omega square m square plus R second round brackets opened R first plus omega square L square (divided) by r p-th round and square brackets closed.

Therefore cotangent r is equal to e cotangent i.

Therefore M sub t is equal to G theta, dash d to the fourth power divided by thirty-two.

P critical is equal to square E I divided by four 1 square.

* There are various notations for them, one common form being and so forth.

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