Deriving the Formula for the Sum of a Geometric Series

... expression x can also be written x 1 . So the geometric series can also be written x 0 + x 1 + x 2 + ... + x n . The word geometric comes from the fact that each term is obtained from the preceding one by multiplication by the quantity x. The trick for adding up the series was observed by mathematic ...

... expression x can also be written x 1 . So the geometric series can also be written x 0 + x 1 + x 2 + ... + x n . The word geometric comes from the fact that each term is obtained from the preceding one by multiplication by the quantity x. The trick for adding up the series was observed by mathematic ...

Core 2 Self-Assessment Tick List

... To understand and use the laws of indices Knowledge of the effect of simple transformations on the graph of y=f(x) as represented by y=af(x), y=f(x)+a, y =f(x +a), y=f (ax) . Sequences, including those given by a formula for the nth term. Including using the Σ notation. Sequences generated b ...

... To understand and use the laws of indices Knowledge of the effect of simple transformations on the graph of y=f(x) as represented by y=af(x), y=f(x)+a, y =f(x +a), y=f (ax) . Sequences, including those given by a formula for the nth term. Including using the Σ notation. Sequences generated b ...

Math 237. Calculus II Solutions to the HW on Newton`s Method (3.7

... 14. We seek the smallest positive root of 2 cot(x) = x. Let f (x) = 2 cot(x) – x. In order to begin the process, I need a first guess. A look at the graph of f . . . y ...

... 14. We seek the smallest positive root of 2 cot(x) = x. Let f (x) = 2 cot(x) – x. In order to begin the process, I need a first guess. A look at the graph of f . . . y ...

Regan drew a number of different shapes with 5 sides

... n= the number of terms in the sequence divided by 2 ...

... n= the number of terms in the sequence divided by 2 ...

9.6 The Quadratic Formula and the Discriminant.notebook

... 9.6 The Quadratic Formula and the Discriminant.notebook ...

... 9.6 The Quadratic Formula and the Discriminant.notebook ...

2. 2 2 = 1 1 n i=1 n−1 i=1

... only if x1 = x/b, and the remaining xi are defined recursively by xi+1 = ri xi . ...

... only if x1 = x/b, and the remaining xi are defined recursively by xi+1 = ri xi . ...

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.