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Calculus
The branch of mathematics that deals with the study of continuously changing quantities, with the use of limits and the differentiation and integration of functions of one or more variables, is called Calculus. Calculus analyzes aspects of change in processes or systems that can be modeled by functions. The English physicist, Isaac Newton, and the German mathematician, G. W. Leibniz, working independently, developed calculus during the 17th century.
25,068 Questions
Why is X to the minus 2nd power equal to 1 over x to the 2nd power?
when a variable is to a negative power, the answer is always the variable with the additive inverse of the power, (ex. -2 changes to 2) and is divided by one.
such as X to the -5 power equals 1 over X to the 5th power. I will use the symbol "^" for power. Let's se a numeric example, with 10 instead of "x". Consider this sequence:
10^3 = 1000
10^2 = 100
10^1 = 10
10^0 = ???
10^(-1) = ???
10^(-2) = ???
How should this sequence be completed? In the left-hand column, the exponent decreases one at a time; in the right-hand column, the number decreases by a factor 10 each time. It seems logical to continue this pattern - so the missing question marks can be replaced by 1, 0.1, and 0.01.
More specifically, several laws for powers continue being valid if these definitions are used (a^0 = 1; a^(-b) = 1/a^b). For example, the addition of exponents: x^a * x^b = x^(a+b) continues being valid, even if a or b, or both, are negative.
It is only defined that way; there is no way to "prove" it. The common definition, however, turns out to be very useful.
What is the slope x 4y equals 12?
I presume that since you labeled this as a Calculus problem that you mean x * 4y = 12?
x * 4y = 12 --->
4y = 12 / x --->
y = 3 / x --->
y = 3 * x^(-1)
You will notice that this function is not a line, but a curve. The slope will be different at different points on the line. Thus, we can't find the slope of the entire function, but we CAN find a function which gives us the slope of a tangent line at any point on the function. We do this by taking the derivative.
For f(x) = a * x^(n)
f'(x) = a * n * x^(n-1)
Using a = 3 and n = -1, we have:
y = f(x) = 3 * x^(-1)
dy/dx = f'(x) = 3 * -1 * x^(-1 - 1) = -3 * x^(-2)
So your answer will be:
dy/dx = -3 * x^(-2)
How to solve for the Quadractic formula 5x2-3x-7 equals 0?
5x2 - 3x - 7 = 0 where a = 5, b = -3, and c = -7
x = [-b ± √b2 - 4(a)(c)]/2a
x = [-(-3) ± √(-3)2 - 4(5)(-7)]/2(5)
x = [3 ± √(9 + 140)]/10
x = (3 ± √149)/10
x = (3 ± 12.21)/10
x = (3 + 12.21)/10 = 15.21/10 = 1.521 or
x = (3 - 12.21)/10 = -9.21/10 = -0.921
The EXACT answer would be x = 3/10 ± √149 /10
Note: 149 is not divisable by a perfect square
What are some real life application of exponents and logarithms?
Logarithms are used in the royal navy in sonars
They are but it's far wider than that.
The application you mean is the "decibel", the 10X or 20X logarithm of the ratio of two signal intensities or powers - not just in military and commercial sonar, but in acoustics generally, and in electrical engineering such as amplifier design.
The Richter Scale and the magnitude scale of star brightness are logarithmic. Common and hyperbolic logarithms crop up in many places - the latter control the expansion or compression of gas in an engine or compressor cylinder, for example.
Exponents also facilitate handling very large & very small numbers by turning them into multiples of plus or minus powers of 10.
What is the answer to 3x-4 equals 2x plus 1?
The 'answer' is a number that produces a true statement when it's inserted in place of 'x' . It
can be identified by careful implementation of the following arcane, complex, highly technical
procedure:
3x - 4 = 2x + 1
Subtract 2x from each side of the equation:
x - 4 = 1
Add 4 to each side:
x = 5
What is the y intercept of the line y equals 2x 3?
It is 3 with the sign which should have accompanied the number in the question.
What does x plus 1 mean in relation to functions?
If the question is what I think it is, then it means a translation of the graph by 1 unit to the left.
What are limits and why you use them?
Limits give upper and lower bounds for integration. One simple example is in finding an enclosed area. The upper and lower limits form vertical lines which enclose an area between the function and the x-axis and then integration from the lower limit (smaller x boundary) to the upper limit (larger x boundary).
How do you solve 3x plus seventeen?
set the equation equal to zero: 3x + 17=0
subtract 17 from both sides so you end up with : 3x=-17
divide both sides by 3 to isolate x: x=-17/3
your answer is -17/3
Who remembered the most digits for pi?
By memory, it is Hiroyuki Goto, who memorized and recited 42,195 in seventeen hours and twenty one minutes in 1995.
-2x squared plus x plus 1 divided by x-1?
(-2x2 + x + 1) / (x - 1)
= (-2x - 1)(x - 1)/(x - 1)
= -2x - 1
If 3x, 5y and 7z are equal terms, them each of them will equal one third of 357, or 119.
3x = 119
x = 39 and 2/3
5y = 119
y = 23 and 4/5
7z = 119
z = 17
