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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
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)
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.
What is 228 divided by 10 x 12?
That depends where the brackets fall in the equation:
(228/10) x 12 = 273.6
228 / (10 x 12) = 1.9
What is the equation for 12 plus y equals 22?
The equation is exactly what you just said: " 12 + y = 22 " .
As long as we're both here, let's go ahead and find the solution to it.
It could get complicated. Hold on to your hat, and try to stay with me:
12 + y = 22
Subtract 12 from each side:
y = 10
(598 + x) can have an infinite number of different values,
depending on the value of 'x'.
Find the equation of x plus 3y equals 4?
This equation is unsolvable since there are two unknowns and only one equation. You would require a second equation in order to solve 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).
What is the integral of sine to the 7th power 3x dx?
If you look in most calculus books there is a table of intergals. Your problem is one of the trigonometric forms. Below is given the reduction formula,
∫sinn(u) du = (-1/n)sinn-1(u)cos(u) + ((n-1)/n)∫sinn-2(u) du
where n ≥ 2 is an integer.
Basically yours would be n=7 & u=3x. You keep integrating until you lose the last integral sign on the right side. Pretty tedious work, but if I am using the right integral form & didn't make any mistakes the answer should be:
(-1/7)sin6(3x)cos(3x) - (6/35)sin4(3x)cos(3x) - (8/35)sin2(3x)cos(3x) - (16/35)cos(3x) + C
Here is the work:
Let u = 3x
u' = 3 dx, so that dx = du/3
∫sin7 3x dx = (1/3)∫sin7(u) du
∫sin7(u) du = (-1/7)sin6(u)cos(u) + (6/7)∫sin5(u) du
= (-1/7)sin6(u)cos(u) + (6/7)[(-1/5)sin4(u)cos(u) + (4/5)∫sin3(u) du]
= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) + (24/35)∫sin3(u) du
= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) + (24/35)[(-1/3)sin2(u)cos(u) + (2/3)∫sin(u) du]
= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) - (8/35)sin2(u)cos(u) + (16/35)∫sin(u) du
= (-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) - (8/35)sin2(u)cos(u) - (16/35)cos(u) + C
so that
∫sin7 3x dx = (1/3)∫sin7(u) du
= (1/3 [(-1/7)sin6(u)cos(u) - (6/35)sin4(u)cos(u) - (8/35)sin2(u)cos(u) - (16/35)cos(u)] + C
= -(1/21)sin6(u)cos(u) - (2/35)sin4(u)cos(u) - (8/105)sin2(u)cos(u) - (16/105)cos(u) + C
= -(1/21)sin6(3x)cos(3x) - (2/35)sin4(3x)cos(3x) - (8/105)sin2(3x)cos(3x) - (16/105)cos(3x) + C
Or
∫sin7 3x dx = ∫sin63x sin 3x dx = ∫(sin2 3x)3 sin 3x dx =∫(1 - cos2 3x)3 sin 3x dx
Let u = cos 3x
u' = (cos 3x)'
du = -sin 3x*3 dx, and dx = du/-3sin 3x
= (-1/3)∫(1 - u2)3 du = ∫(1 - 3u2 + 3u4 + u6) du = (-1/3) [u - (3/3)u3 + (3/5)u5 - u7/7] + C
= (-1/3)u + (1/3)u3 - (1/5)u5 + (1/21)u7 + C
= (-1/3)cos 3x + (1/3)cos3 3x - (1/5)cos5 3x + (1/21)cos7 3x + C
Is it possible to find the inverse equation of y equals x squared plus 5x minus 6?
Perhaps, using the Quadratic Equation...
y=x2+5x-6
Solve for x:
x2+5x+(-6-y)=0
Using the Quadratic Equation:
x = .5 *(-5 (+-) (25 + (24+4y)).5) = -2.5 (+-) .5*(49+4y).5
Substitute y for x, x for y:
y = -2.5 (+-) .5*(49+4x).5
The answer to 3 plus X plus -7 times 6 equals-38. This is considered to be a math problem.
What is the answer to 3x plus 4 equals 4x-5?
3x + 4 = 4x - 5
Add 5 to both sides: 3x + 9 = 4x
Subtract 3x from both sides: 9 = x
What is the step by step solution to -6p-5 equals -9 plus 8p?
-6p -5 = -9 +8p
Add 6p to both sides: -5 = -9 + 14p
Add 9 to both sides: +4 = 14p
Divide both sides by 14: p =4/14 = 2/7 or 0.2857..
Are you sure that the second term in the equation was -5 and not +5 (as seems more likely)?
2x + 21 = 9 x - 7
Add 7 to each side
2x + 21 = 9x
Subtract 2x from each side
21 = 7x
Divide each side by 7
3 = x
Sorted.
What is the diameter of a cylinder?
The diameter is the length of a line segment perpendicular to the side of the cylinder, passing through the center of the cylinder, from a point on one side to a point on the opposite side of the cylinder. It is also twice the radius and is related to the circumference by a factor of pi.
Explain how to solve x2 plus 8x -2 equals 0 by completing the square?
x² + 8x -2 = 0
x² + 8x = 2
x² + 8x + (8/2)² = 2 + (8/2)²
x² + 8x + 16 = 2 + 16
(x + 4)² = 18
x + 4 = ±√18
x = -4 ±√18
x = -4 + √18 or x = -4 - √18
if you are still confused, i want you to follow the related link that explains the concept of completing the square clearly.
