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... Of course, it may happen that after doing one of these steps the problem still complicated, and we may have to try one or more of the steps again, repeating until we get down to a problem that we can solve. Or, it may happen that after several such simplifications, we realize that we’ve gone down t ...
... Of course, it may happen that after doing one of these steps the problem still complicated, and we may have to try one or more of the steps again, repeating until we get down to a problem that we can solve. Or, it may happen that after several such simplifications, we realize that we’ve gone down t ...
Two Inequalities for Differentiable Mappings and
... It is clear that if the mapping f is not twice differentiable or the second derivative is not bounded on (a, b), then (4.2) cannot be applied. In recent papers [9-11], Dragomir and Wang have shown that the remainder term E ( f , d) can be estimated in terms of the first derivative only. These estima ...
... It is clear that if the mapping f is not twice differentiable or the second derivative is not bounded on (a, b), then (4.2) cannot be applied. In recent papers [9-11], Dragomir and Wang have shown that the remainder term E ( f , d) can be estimated in terms of the first derivative only. These estima ...
Lies My Calculator and Computer Told Me
... standard functions in double precision. For example, the number 4 tan11, whose representation with sixteen decimal digits should be 3.14159 26535 89793, appears as 3.14159 29794 31152; this is off by more than 3 107. What is worse, the cosine function is programmed so badly that its “cos” ...
... standard functions in double precision. For example, the number 4 tan11, whose representation with sixteen decimal digits should be 3.14159 26535 89793, appears as 3.14159 29794 31152; this is off by more than 3 107. What is worse, the cosine function is programmed so badly that its “cos” ...
Products of Sums of Squares Lecture 1
... (b) Write the 4 × 4 matrix A corresponding to Euler’s [4, 4, 4] identity. (c) Construct an 8-square identity. This can be viewed as an 8 × 8 matrix A with orthogonal rows, where each row is a signed permutation of (x1 , . . . , x8 ). (Another method is described in Lecture 2.) EXERCISE 2. Proof of t ...
... (b) Write the 4 × 4 matrix A corresponding to Euler’s [4, 4, 4] identity. (c) Construct an 8-square identity. This can be viewed as an 8 × 8 matrix A with orthogonal rows, where each row is a signed permutation of (x1 , . . . , x8 ). (Another method is described in Lecture 2.) EXERCISE 2. Proof of t ...
2 = 2cos 2 θ −1= 1−2 sin 2 θ = 1 + cosu 2 1− cosu 2
... Substitute the identities in part 1b) into the double angle formula that only has “sin” in it. so that u is the only variable. ...
... Substitute the identities in part 1b) into the double angle formula that only has “sin” in it. so that u is the only variable. ...
9 - SFU Computing Science
... Let be a vocabulary. A model appropriate to is a pair M=(U,) consisting of • the universe of M, a non-empty set U • the interpretation, a function that assigns - to each predicate symbol P a concrete predicate P M on U M - to each function symbol f a concrete function f on U • the equality pr ...
... Let be a vocabulary. A model appropriate to is a pair M=(U,) consisting of • the universe of M, a non-empty set U • the interpretation, a function that assigns - to each predicate symbol P a concrete predicate P M on U M - to each function symbol f a concrete function f on U • the equality pr ...
Week 9: Differentiation Rules. - MA161/MA1161: Semester 1 Calculus.
... The derivative of a function f at a point p is the instantaneous rate of change of f at p. Derivatives can be computed from first principles, that is directly from the above definition. Differentiation rules are formulas which simplify differentiation tasks, and apply to wide ranges of functions. ...
... The derivative of a function f at a point p is the instantaneous rate of change of f at p. Derivatives can be computed from first principles, that is directly from the above definition. Differentiation rules are formulas which simplify differentiation tasks, and apply to wide ranges of functions. ...
Final Exam topics - University of Arizona Math
... If f’’(x) is positive on an interval then f’(x) is increasing, so the graph of f(x) is concave up If f’’(x) is negative on an interval then f’(x) is decreasing, so the graph of f(x) is concave down Functions that are not differentiable have/are: Cusps (sharp corners), piecewise, or oscillate to infi ...
... If f’’(x) is positive on an interval then f’(x) is increasing, so the graph of f(x) is concave up If f’’(x) is negative on an interval then f’(x) is decreasing, so the graph of f(x) is concave down Functions that are not differentiable have/are: Cusps (sharp corners), piecewise, or oscillate to infi ...
SCHOOL OF MATHEMATICS MATHEMATICS FOR PART I
... This module covers some of Programme 15 in S. Turn to p.824 and read frame 1. Move on to the next frame and results 1-4, 6-8 and 13 in the table. Note that inverse functions will be studied in module 7, and for the present it is only necessary to know that if y = tan x then reversing the operation g ...
... This module covers some of Programme 15 in S. Turn to p.824 and read frame 1. Move on to the next frame and results 1-4, 6-8 and 13 in the table. Note that inverse functions will be studied in module 7, and for the present it is only necessary to know that if y = tan x then reversing the operation g ...
Week 24 Geometry Assignment
... Notes on Assignment: Pages 462-463: General notes for this section: These problems are all based on the formulas for the volume of a cube or rectangular prism. Work to show: #All problems: Write down the formulas, fill in, and work out. #9-14: Leave all answers in simplified radical form. #9: This i ...
... Notes on Assignment: Pages 462-463: General notes for this section: These problems are all based on the formulas for the volume of a cube or rectangular prism. Work to show: #All problems: Write down the formulas, fill in, and work out. #9-14: Leave all answers in simplified radical form. #9: This i ...
INTRODUCTION TO LIE ALGEBRAS. LECTURE 2. 2. More
... By definition of ad, one has Z(L) = Ker(ad). For example, Z(n3 ) = hzi. 2.5. Simplicity of sl2 . Definition 2.5.1. A Lie algebra L is simple if it is not one-dimensional and if it has no non-trivial ideals. Our aim is to prove the following Theorem 2.5.2. sl2 is simple. 2.5.3. Some linear algebra Le ...
... By definition of ad, one has Z(L) = Ker(ad). For example, Z(n3 ) = hzi. 2.5. Simplicity of sl2 . Definition 2.5.1. A Lie algebra L is simple if it is not one-dimensional and if it has no non-trivial ideals. Our aim is to prove the following Theorem 2.5.2. sl2 is simple. 2.5.3. Some linear algebra Le ...
Automatic differentiation
In mathematics and computer algebra, automatic differentiation (AD), also called algorithmic differentiation or computational differentiation, is a set of techniques to numerically evaluate the derivative of a function specified by a computer program. AD exploits the fact that every computer program, no matter how complicated, executes a sequence of elementary arithmetic operations (addition, subtraction, multiplication, division, etc.) and elementary functions (exp, log, sin, cos, etc.). By applying the chain rule repeatedly to these operations, derivatives of arbitrary order can be computed automatically, accurately to working precision, and using at most a small constant factor more arithmetic operations than the original program.Automatic differentiation is not: Symbolic differentiation, nor Numerical differentiation (the method of finite differences).These classical methods run into problems: symbolic differentiation leads to inefficient code (unless carefully done) and faces the difficulty of converting a computer program into a single expression, while numerical differentiation can introduce round-off errors in the discretization process and cancellation. Both classical methods have problems with calculating higher derivatives, where the complexity and errors increase. Finally, both classical methods are slow at computing the partial derivatives of a function with respect to many inputs, as is needed for gradient-based optimization algorithms. Automatic differentiation solves all of these problems.