Calculus

x+3-x= x-x+3 = 3

"Any light beam moving vertically downwards in the concavity of the parabola (parallel to the axis of symmetry) will bounce off the parabola moving directly towards the focus." http://en.wikipedia.org/wiki/Parabola

y=1+log2x

implies

-1=log2x

which can be solved by raising both sides as an exponent of 2 (since the logarithm is base 2).

2-1=2log2x

This finally implies:

1/2=x

For explanations of why this is:

Anything to a negative exponent simply equals that thing to the positive of that exponent under 1

5-1=1/51=1/5

x-2=1/x2

2-3=1/23=1/8

and so 2-1=1/2

Any number raised to a logarithm of the same base as itself will equal the number inside the logarithm.

5log520=20

xlogx2y=2y

eln 7=7

and so 2log2x=x

x9/x2 = x9-2 = x7

40

it equals 1.637156443824977731300036754897665332 to the power of 49

x = +/- sqrt(y/3)

24

The solution of the system of the two equations, 4x - y = 33 and 5y = 24 may be found by first solving the second equation for y: y = 24/5. then substituting this value into the first equation to produce 4x - (24/5) = 33, or x = [33 + (24/5)]/4 = [(189/5)/4] = (189/20).

It is very similar to solving an equation. You can add or subtract the same term from both sides. In addition, you can multiply or divide both sides by the same thing. A couple of things to note, though, if you multiply or divide both sides by a negative number, the inequality must be flipped [greater than (>) becomes less than (<) and < would become >]. The same is true if you take the reciprocal of both sides.

Spectator

Inspect

Respect

Speculate

Spectacular

x = 0

No, 4,8,12,16,20,24,28,32,36 and 40 are.

x3 - 1 = (x - 1)(x2 + x + 1)

so (x3 - 1) will be zero if either (x - 1) is zero or (x2 + x + 1) is zero.

To get the complex solutions, solve the quadratic equation x2 + x + 1 = 0

x2 + x + 1 = (x + 1/2)2 + 3/4 = 0

(x + 1/2)2 = -3/4

So

x + 1/2 = i (square-root-of-3) / 2

or

x + 1/2 = - i (square-root-of-3) / 2

The complex solutions are therefore

(1 / 2) (-1 + i (3)(1/2))

and

(1 / 2) (-1 - i (3)(1/2))

The inverse is the opposite. If you're using 'x' it will change to division.

x = 59

x2 + 5x - 5 = 0

∴ x2 + 5x + 25/4 = 45/4

∴ (x + 5/2)2 = 45/4

∴ x + 5/2 = ±√(45/4)

∴ x = -5/2 ± 3√(5)/2

∴ x = (-5 ± 3√5)/2

∴ x ∈ {(-5 - 3√5)/2, (-5 + 3√5)/2}

A line integral is a simple integral.

they look like:

integral x=a to b of (f(x)).

A surface integral is an integral of two variables.

they look like:

integral x=a to b, y=c to d of (f(x,y)).

or

integral x=a to b of (integral y=c to d of (f(x,y))).

The second form is the nested form. A pair of line integrals, one inside the other. This is the easiest way to understand surface integrals, and, normally, solve surface integrals.

A volume integral is an integral of three variables.

they look like:

integral x=a to b, y=c to d, z=e to f of (f(x,y,z)).

or

integral x=a to b of (integral y=c to d of (integral z=e to f of f(x,y,z))).

the above statement is wrong, the person who wrote this stated the first 2 types of integrals as regular, simple, scalar integrals, when line and surface integrals are actually a form of vector calculus. in the previous answer, it is stated that the integrand is just some funtion of x when it is actually usually a vector field and instead of evaluating the integral from some x a to b, you will actually be evaluating the integral along a curve that you will parametrize to get the upper and lower bounds of the integral. as you can see, these are a lot more complicated.

looking at your question tho, i dont think you want the whole expanation on how to solve these problems, but more so what they are and what they are used for, because these can be a pain to solve and there are also several ways to solve them indirectly.

line integrals have an important part in physics because they alow us to calculate things such as work that have vector values rather than just scalar values as you can use these integrals to describe a particles path along a curve in a force field. surface integrals help us calculate things like flux, or how fluid flows over a surface.

if you want to learn more, look into things like greens theorem, or the divergence theorem.

p.s. his definition of a surface integral is acutally how you find the volume of a region

5x + 2 = x + 7

Subtract x from both sides: 4x + 2 = 7

Subtract 2 from both sides: 4x = 5

Divide by 4 on both sides: x = 5/4 = 1.25

Double-check by putting the value of x back into the equation:

(5 x 1.25)+2 = 8.25

1.25 + 7 + 8.25

Therefore x = 1.25

There is no solution. Your system looks like this:
x + 7y = 2
x = 9 - 7y

The first can be rewritten as x = 2 - 7y,
and then join the two as 9 - 7y = 2 - 7y

Adding 7y to both yields 9 = 2 which is false, so there's no solution.

You can convert these to slope/intercept form as:
y = -x/7 + 2/7
y = -x/7 + 9/7

These are parallel lines, and will never intersect, so there is no solution that satisfies both equations.

-2x = 14

(divide both sides by -2)


x = -7