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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
Is a gram the same a a cubic centimeter?
No, it is not. A gram measures mass (like ounces or pounds), and a cubic centimeter measures volume (like cubic inches or quarts). They do not measure the same kind of value.
What is 7x plus 6 equals 41 what is x?
Here is the work for that problem.
7x+6= 41
7x=41-6 (subtracting 6 from both sides)
7x=35
x=5
What is the solution of y equals x plus 2 and negative 5x plus 6y equals 15?
y=x+2 -5x+6y=15
You use the substitution property to plug y into the 2nd equation
-5x+6(x+2)=15
Then, you solve the equation
That should leave you with x=3
Then you plug that into the first equation, and that should leave you with a final answer of
x=3 and y=5
Take the exponent and multiply it by the coefficient (or 1 if there is no coefficient) then subract 1 from the exponent.
For example, the derivative of 2x^3 is 6x^2
If there is no exponent, for example, 2x the derivative is 2 because the exponent is actually 1 which produces the same coefficient and the exponent 0 meaning there is no x.
What the solution of 2x plus 5y equals 16 and 2x plus 3y equals 14?
2x + 5y = 16
-2x - 3y = -14
2y = 2
y = 1 / x = 5.5 (5.5,1) or (5 1/2,1)
What is the simplest form of the expression w plus 2x plus 3w - 2xw?
w + 2x + 3w - 2xw = 2x + 4w - 2xw, in simplest form
4x+7=25-5x
(subtract 4x from each side to isolate the variables)
7=25-x
(add 25 to each side to get the numbers on the opposite side or the variable)
32=-x
(divide or multiply both sides by -1 to make the x positive)
x=-32
Where does y equals x2 plus 9x plus 20 cross the x and y axis?
The y-intercept is easy to find. Substitute 0 for x. It crosses the y-axis at 20, (0,20).
To find the x- intercepts substitute 0 for y and solve. Since this creates a quadratic equation, it is a little harder.
0 = x2 + 9x + 20, can be solved by several methods, factoring is easiest.
(x+5)(x+4) = 0
x = -5 and x = -4
So it crosses the x-axis at two points, -5 and -4, (-5,0) and (-4,0)
What is 1 fourth plus y equals x what equation would be solved in this?
That statement itself is an equation, and there are an infinite number
of (x, y) pairs of numbers that satisfy it.
To draw a graph of the equation, draw a straight line that rises toward the
right at an angle of 45° and passes through [ y = -1/4 ] on the y-axis.
The x-value and y-value of every point on that line satisfy your equation.
(And we all know how many points there are on a line. Particularly on a line
that has no ends.)
The sum of x and 8 is less than 72 find all possible values of x?
-1 < x < 64, but if x is less than 0 then x can be any value.
12x - 5y equals 30 and y equals 2x - 6 what is the y value?
12x - 5(2x - 6) = 30
12 x - 10x + 30 = 30
2x = 0, x = 0
y = -6
Find gf of 5 if f of x equals x plus 1 and g of x equals 3 x - 2?
gf(5) = g(f(5)) = g(5+1) since f(x) = x+1
and then g(6) = 3*6 - 2 = 18 - 2 = 16
What are the Objectives of teaching calculus to high school?
Objectives: Students will:
- evaluate derivatives for complexly constructed elementary functions;
- evaluate definite and indefinite integrals; and
- evaluate limits using algebraic, geometric, analytic techniques.
2. Learning Goal: Mathematics majors will learn and retain basic knowledge in the core branches of mathematics.
Objectives: Students will, during their senior year:
- demonstrate proficiency in calculus;
- demonstrate proficiency in linear algebra; and
- demonstrate proficiency in algebra.
3. Learning Goal: Mathematics majors will be able to learn and explain mathematics on their own.
Objectives: Students will:
- read a mathematics journal article and explain it, orally or in writing, to an audience of math majors and
- after graduation, be able to master new mathematics necessary for their employment.
4. Learning Goal: Mathematics majors will be able to read and construct rigorous proofs.
Objectives: Students will:
- construct clearly written proofs which use correct terminology and cite previous theorems;
- construct proofs using mathematical induction;
- construct proofs by contradiction; and
- judge whether a proof is sound, and identify errors in a faulty proof.
5. Learning Goal: Mathematics majors will be able to obtain employment in their area of mathematical interest or gain admittance to a graduate program in mathematics.
Objectives: Students who:
- seek admission to graduate schools in mathematics will succeed in gaining admission, and perform adequately in these programs;
- seek entry-level employment in math-related fields will obtain it;
- specialize in actuarial science will obtain entry-level work as actuaries, if they seek it;
- specialize in secondary education will demonstrate proficiency in mathematics needed to obtain Initial Certification in New York State; or
- seek jobs in secondary or elementary education will obtain jobs at the appropriate grade level.
6. Learning Goal: Master's students will recognize connections between different branches of mathematics.
Learning Objectives: Students will:
- correctly incorporate specific examples from one branch of mathematics into their study of another branch of mathematics (e.g., Lp-spaces as an example in linear algebra) and
- identify and explain cases in which major results of one branch of mathematics rely nontrivially on results from another branch (e.g., the application of linear algebra to solving systems of differential equations).
7. Learning Goal: Graduating master's degree students will be able to obtain employment in their area of mathematical interest or gain admittance to a doctoral program in mathematics.
Learning Objectives: Students who:
- seek admission to doctoral programs in mathematics, applied mathematics, mathematical finance, mathematics education or other math-related fields will succeed in gaining admission to such programs, and perform adequately in these programs;
- seek employment as full-time instructors at community colleges or as part-time instructors at four-year colleges or universities will obtain it; and
- seek employment in other math-related fields will obtain it.
How do you work out the value of 32.7 x 27 without a calculator?
Those of us past a certain age were taught ancient sacred techniques and rituals
at the feet of our wise and learned gurus when we were in school. You see, we had
much more time back then, because mp3, ipods, ipads, CDs, DVDs, cellphones, earbuds,
and even TV had not been invented yet, and nobody had any of these. Facebook and
Twitter didn't exist, and nobody had a computer, if you can just imagine that! So life
was empty and boring, with a lot of time to spare, and for people who were interested,
there were lots of wonderful and mysterious things to learn. Something like Hogwart's,
but not so scary.
But where were we before we started telling stories of a land long ago and far away ?
Oh yes . . .
To "work out" the value of 32.7 x 27 without a calculator . . . (we also didn't have
calculators when we were in school, and had to do all our math without them) . . .
With a pencil and a piece of paper:
-- Multiply 32.7 by 20. Write the answer off to the side.
-- Multiply 32.7 by 7. Write the answer off to the side.
-- Add up the two numbers that you find off to the side.
The sum is the answer to (32.7 x 27).
If you can't handle that multiplication with only a pencil, then try it this way:
-- In one long, straight column, write 32.7 on each line, 27 times.
-- Carefully add up the long column of 27 numbers.
-- The sum is the answer to (32.7 x 27).
