1. Write down, and simplify wherever possible
... 1. Use the auxiliary equation method to find a formula for un when the sequence (un ) is defined by (i) u0 = 1, u1 = 3, un+2 − 3un+1 − 4un = 0 (n ≥ 0) (ii) u0 = 2, u1 = 0, u2 = −2, un+3 − 6un+2 + 11un+1 − 6un = 0 (n ≥ 0) (iii) u0 = 1, u1 = 0, u2 = 0, un+3 − 3un+1 + 2un = 0 (n ≥ 0). 2. Professor McBr ...
... 1. Use the auxiliary equation method to find a formula for un when the sequence (un ) is defined by (i) u0 = 1, u1 = 3, un+2 − 3un+1 − 4un = 0 (n ≥ 0) (ii) u0 = 2, u1 = 0, u2 = −2, un+3 − 6un+2 + 11un+1 − 6un = 0 (n ≥ 0) (iii) u0 = 1, u1 = 0, u2 = 0, un+3 − 3un+1 + 2un = 0 (n ≥ 0). 2. Professor McBr ...
Secondary Math 3 I Can Statements.docx
... I can extend polynomial identities to the complex numbers. I know the Fundamental Theorem of Algebra. I can show that the Fundamental Theorem of Algebra is true for polynomials with real ...
... I can extend polynomial identities to the complex numbers. I know the Fundamental Theorem of Algebra. I can show that the Fundamental Theorem of Algebra is true for polynomials with real ...
Proving Sine Addition and Subtraction Identities Task RTF
... is not as straight forward as adding or subtracting the sine of the individual angles. Convince yourself of this by performing the following operation: sin(30) sin(60) sin(90)? ...
... is not as straight forward as adding or subtracting the sine of the individual angles. Convince yourself of this by performing the following operation: sin(30) sin(60) sin(90)? ...
Trigonometry #1
... Ex4: Find the value of y so that the distance between the points is 5. (5,y) & (8,-1) ...
... Ex4: Find the value of y so that the distance between the points is 5. (5,y) & (8,-1) ...
Calculus I Midterm II Review Materials Solutions to the practice
... Solutions to the practice problems are provided in another sheet, it’s recommended that you first do the problems by yourself and then check with the answer. 1, Definition of Derivative, Differentials? Relation with change rate, tangent, velocity? Express a limit as √ a derivative and evaluate. ...
... Solutions to the practice problems are provided in another sheet, it’s recommended that you first do the problems by yourself and then check with the answer. 1, Definition of Derivative, Differentials? Relation with change rate, tangent, velocity? Express a limit as √ a derivative and evaluate. ...
17 Complex Numbers Addendum– Lay Appendix B 2
... Basic facts about the set C of complex numbers are: 1. The set C contains an element, usually called i, which is not a real number. 2. Every member of C can be written uniquely in the form x + iy. where x and y are real numbers. (Here ‘uniqueness’ means that, if x1 + iy1 and x2 + iy2 represent the s ...
... Basic facts about the set C of complex numbers are: 1. The set C contains an element, usually called i, which is not a real number. 2. Every member of C can be written uniquely in the form x + iy. where x and y are real numbers. (Here ‘uniqueness’ means that, if x1 + iy1 and x2 + iy2 represent the s ...
Molecular Stoichiometry
... 3. Divide each by the smallest number of moles to obtain the simplest whole number ratio. 4. If whole numbers are not obtained* in step 3), multiply through by the smallest number that will give all whole numbers * Be ...
... 3. Divide each by the smallest number of moles to obtain the simplest whole number ratio. 4. If whole numbers are not obtained* in step 3), multiply through by the smallest number that will give all whole numbers * Be ...
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.