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### Selecting Centrifugal Pumps - KSB

by marechausse 0 Comments 1 Viewed 0 Times

### Soal dan Pembahasan Matematika IPA SNMPTN 2011

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### NYS Mathematics Glossary* – Algebra 2/Trig - Regents Exam Prep ...

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NYS Mathematics Glossary* – Algebra 2/Trig *This glossary has been amended from the full SED Commencement Level Glossary of Mathematical Terms (available at http://www.emsc.nysed.gov/ciai/mst/math/glossary/home.html) to list only terms indicated to be at the Algebra 2/Trig level.) This Glossary, intended for teacher use only, provides an understanding of the mathematical terms used in the Regents-approved course entitled Algebra 2/Trig (as reflected in the NYS Mathematics Core Curriculum). A a + bi form The form of a complex number where a and b are real numbers, and i = −1 . abscissa The horizontal or x-coordinate of a two-dimensional coordinate system. absolute value The distance from 0 to a number n on a number line. The absolute value of a number n is indicated by n . Example: −3 = 3 , +3 = 3 , and 0 = 0 . absolute value equation An equation containing the absolute value of a variable. Example: x+3 = 9 absolute value function A function containing the absolute function of a variable. ⎧ x, x ≥ 0 ⎫ Example: f ( x) = x = ⎨ ⎬ ⎩ − x, x < 0 ⎭ absolute value inequality An inequality containing the absolute value of a variable. Example: x + 3 < 9 adjacent angles Two coplanar angles that share a common vertex and a common side but have no common interior points. Example: In the figure, ∠AOB and ∠BOC are a pair of adjacent angles, but ∠AOC and ∠BOD are not adjacent. A B C O D 2 adjacent sides Two sides of any polygon that share a common vertex. algebraic equation A mathematical statement that is written using one or more variables and constants which contains an equal sign. Examples: 3y + 5 = 1 2 x − 5 = 11 log 5 ( x − 3) = 2 2x = 1 8 algebraic expression A mathematical phrase that is written using one or more variables and constants, but which does not contain a relation symbol ( <, >, ≤, ≥, =, ≠ )...

### ALGEBRA 2 and TRIGONOMETRY

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### Review of Algebra, Geometry, and Trigonometry

by Jimakon 0 Comments 4 Viewed 0 Times

Review of Algebra, Geometry, and Trigonometry Properties and Measurement D.1 Review of Algebra, Geometry, and Trigonometry ■ Algebra ■ Properties of Logarithms ■ Geometry ■ Plane Analytic Geometry ■ Solid Analytic Geometry ■ Trigonometry ■ Library of Functions Algebra Operations with Exponents 1. x nx m ϭ x nϩm ΂x΃ y xn yn 7. cx n ϭ c͑x n͒ 4. n 2. xn ϭ x nϪm xm 3. ͑xy͒ n ϭ x ny n 5. ͑x n͒ m ϭ x nm ϭ 8. x n ϭ x͑n m 6. Ϫx n ϭ Ϫ ͑x n͒ ͒ m Exponents and Radicals (n and m are positive integers) 1. x n ϭ x и x и x . . . x 2. x 0 ϭ 1, x 0 n factors 1 3. xϪn ϭ n , x x n 5. x1͞n ϭ Ίx 0 n *4. Ίx ϭ a x ϭ an n 6. x m͞n ϭ ͑x1͞n͒m ϭ ͑Ίx͒ m͞n m 1͞n n m x ϭ ͑x ͒ ϭ Ίx m 2 7. Ίx ϭ Ίx Operations with Fractions bc ϩ c ΂ ΃ ϩ d ΂b΃ ϭ ad ϩ bd ϭ ad bd bc b bd c bc Ϫ ΂ ΃ Ϫ d ΂b΃ ϭ ad Ϫ bd ϭ ad bd bc b bd c a d a ϩ ϭ b d b d a c a d 2. Ϫ ϭ b d b d ac a c 3. ϭ b d bd a͞b a d 4. ϭ c͞d b c a͞b a͞b ϭ ϭ c c͞1 1. ΂ ΃΂ ΃ ΂ ΃΂ ΃ ϭ ad bc a ΂a΃΂1΃ ϭ bc b c 5. ab b ϭ ac c ab ϩ ac a͑b ϩ c͒ b ϩ c ϭ ϭ ad ad d * If n is even, the principal nth root is defined to be positive. D1 9781133109280_App_D1.qxp D2 12/3/11 Appendix D ■ 1:34 PM Page D2 Properties and Measurement Quadratic Formula ax 2 ϩ bx ϩ c ϭ 0 xϭ Ϫb ± Ίb2 Ϫ 4ac 2a Factors and Special Products 1. x 2 Ϫ a2 ϭ ͑x Ϫ a͒͑x ϩ a͒ 2. x3 Ϫ a3 ϭ ͑x Ϫ a͒͑x 2 ϩ ax ϩ a2͒ 3. x3 ϩ a3 ϭ ͑x ϩ a͒͑x 2 Ϫ ax ϩ a2͒ 4. x 4 Ϫ a 4 ϭ ͑x Ϫ a͒͑x ϩ a͒͑x 2 ϩ a2͒ Factoring by Grouping acx3 ϩ adx 2 ϩ bcx ϩ bd ϭ ax 2͑cx ϩ d͒ ϩ b͑cx ϩ d͒ ϭ ͑ax2 ϩ b͒͑cx ϩ d͒ Binomial Theorem 1. ͑x ϩ a͒2 ϭ x 2 ϩ 2ax ϩ a2 2. ͑x Ϫ a͒2 ϭ x 2 Ϫ 2ax ϩ a2 3. ͑x ϩ a͒3 ϭ x3 ϩ 3ax 2 ϩ 3a2x ϩ a3 4. ͑x Ϫ a͒3 ϭ x3 Ϫ 3ax 2 ϩ 3a2x Ϫ a3 5. ͑x ϩ a͒4 ϭ x 4 ϩ 4ax3 ϩ 6a2x 2 ϩ 4a3x ϩ a4 6. ͑x Ϫ a͒4 ϭ x 4 Ϫ 4ax3 ϩ 6a2x 2 Ϫ 4a3x ϩ a4 n͑n Ϫ 1͒ 2 nϪ2 n͑n Ϫ 1͒͑n Ϫ 2͒ 3 nϪ3 . . . ax ϩ ax ϩ ϩ nanϪ1x ϩ an 2! 3! n͑n Ϫ 1͒ 2 nϪ2 n͑n Ϫ 1͒͑n Ϫ 2͒ 3 nϪ3 . . . ϩ ax Ϫ ax ϩ ± nanϪ1x ϯ an 2! 3! 7. ͑x ϩ a͒n ϭ x n ϩ nax nϪ1 ϩ 8. ͑x Ϫ a͒n ϭ x n Ϫ nax nϪ1 Miscellaneous 1. If ab ϭ 0, then a ϭ 0 or b ϭ 0. 2. If ac ϭ bc and c 0, then a ϭ b. 3. Factorial: 0! ϭ 1, 1! ϭ 1, 2! ϭ 2 и 1, 3! ϭ 3 и 2 и 1, 4! ϭ 4 и 3 и 2 и 1, etc. Sequences 1. Arithmetic: a, a ϩ b, a ϩ 2b, a ϩ 3b, a ϩ 4b, a ϩ 5b, . . . 2. Geometric: ar 0, ar1, ar 2, ar 3, ar 4, ar 5, . . . a͑1 Ϫ r nϩ1͒ ar 0 ϩ ar1 ϩ ar 2 ϩ ar 3 ϩ . . . ϩ ar n ϭ 1Ϫr 1 1 1 ...

### Laporan lengkap

by Hermawan 0 Comments 55 Viewed 0 Times

### CHALLENGING PROBLEMS FOR CALCULUS STUDENTS 1 ...

by luan 0 Comments 17 Viewed 0 Times

In what follows I will post some challenging problems for students who have had some calculus, preferably at least one calculus course. All problems require a proof. They are not easy but not impossible. I hope you will ﬁnd them stimulating and challenging. 2. Problems (1) Prove that eπ > π e . (2.1) Hint: Take the natural log of both sides and try to deﬁne a suitable function that has the essential properties that yield inequality 2.1. 1 1 1 4 1 2 (2) Note that = but = . Prove that there exists inﬁnitely many 4 2 pairs of positive real numbers α and β such that α = β; but αα = β β . Also, ﬁnd all such pairs. Hint: Consider the function f (x) = xx for x > 0. In particular, focus your attention on the interval (0, 1]. Proving the existence of such pairs is fairly easy. But ﬁnding all such pairs is not so easy. Although such solution pairs are well known in the literature, here is a neat way of ﬁnding them: look at an article written by Jeﬀ Bomberger1, who was a freshman at UNL enrolled in my calculus courses 106 and 107, during the academic year 1991-92. 1 4 1 ; 2 (3) Let a0 , a1 , ..., an be real numbers with the property that a1 a2 an a0 + + + ... + = 0. 2 3 n+1 Prove that the equation a0 + a1 x + a2 x2 + ...an xn = 0 1Jeﬀrey Bomberger, On the solutions of aa = bb , Pi Mu Epsilon Journal, Volume 9(9)(1993), ...

### Calculus I Practice Problems 1: Answers 1. Find the equation of the ...

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### Appendix: Using Calculus to Solve Problems in Mechanics - SCIPP

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In most realistic situations forces and accelerations are not fixed quantities but vary with time or displacement. In these situations algebraic formulas cannot do better than approximate the situation, but the tools of calculus can give exact solutions. The derivative gives the instantaneous rate of change of displacement (velocity) and of the instantaneous rate of change of velocity (acceleration). The integral gives an infinite sum of the product of a force that varies with displacement times displacement (work), or similarly if the force varies with time (impulse). The Key Concepts: Acceleration is the derivative of velocity with respect to time. The slope of the tangent to the line of a graph of velocity vs. time is the acceleration. Velocity is the derivative of displacement with respect to time. The slope of the tangent to the line of a graph of displacement vs. time is the velocity Work is the integral of force as a function of displacement times displacement. The area under the curve of a graph of force vs. displacement is the work. Impulse is the time integral of force as a function of time. The area under the curve of a graph of force vs. time is the impulse. Other Derivatives include rotational velocity—angle with respect to time; angular acceleration—rotational velocity with respect to time Other Integrals include moment of inertia, where mass varies with radius and rotational work, where torque varies with angle Harmonic Motion can be written as a differential equation.

### 1101 Calculus I 4.7 Optimization Problems - facultypages.morris ...

by luan 0 Comments 78 Viewed 0 Times

The Method of Solution: 1. Understand the problem. 2. Draw a diagram. 3. Introduce notation (Q is to be maximized or minimized) 4. Find relation between quantities (Q and all others) 5. Make the relation look like Q = f (x) (one variable) 6. Solve f (x) = 0 for x. 7. Explain whether you have found a max or min, and if possible if it is an absolute extrema (Closed Interval Method, First Derivative Test, Second Derivative Test, argue based on the geometry of the problem) 8. Write a concluding statement. Example A farmer has 2400 ft of fencing. What are the dimensions of the rectangular pen that produce the largest area? • Understand the problem: We need a rectangle. The rectangle should have maximum area for a given perimeter. • Draw a diagram : A x y • Introduce notation and ﬁnd relations: The perimeter is P = 2x + 2y. The area is A = xy. This is what we want to maximize. We need to eliminate y from the equation for A. Use P = 2x + 2y = 2400, −→ y = 1200 − x. Therefore, A = xy = x(1200 − x) = 1200x − x2 . If x < 0, the area would be negative. This is unphysical. If x > 1200, the area would be negative. This is unphysical. The domain for the area is 0 ≤ x ≤ 1200. • Find the maximum of A(x) = 1200x − x2 , 0 ≤ x ≤ 1200. A = 1200 − 2x. A = 0 = 1200 − 2x → x = 600 ft. This is a maximum since A (600) = −2 < 0 and A will be concave down by the second derivative test. Check endpoints: A(0) = 0 = A(1200) < A(600) = 360 000. The absolute maximum is 360,000 ft2 when the rectangle is a square of side 600 ft...

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