Calculus BC Review Book
... 9.2- Convergent and Divergent Series Sn = a1 + a2 +…+ an If the sequence of partial sums {Sn} converges to S, then the series converges. (the nth partial sum). If {Sn} diverges, then the series diverges. (sum of the series) Theorem 9.6: Convergence of a Geometric Series ...
... 9.2- Convergent and Divergent Series Sn = a1 + a2 +…+ an If the sequence of partial sums {Sn} converges to S, then the series converges. (the nth partial sum). If {Sn} diverges, then the series diverges. (sum of the series) Theorem 9.6: Convergence of a Geometric Series ...
4.10 Applications of Hessians
... Because the Hessian of an equation is a square matrix, its eigenvalues can be found (by hand or with computers – we’ll be using computers from here on out). Because Hessians are also symmetric (the original and the transpose are the same), they have a special property that their eigenvalues will alw ...
... Because the Hessian of an equation is a square matrix, its eigenvalues can be found (by hand or with computers – we’ll be using computers from here on out). Because Hessians are also symmetric (the original and the transpose are the same), they have a special property that their eigenvalues will alw ...
Section 2.4 - Analytic Methods for Special Systems
... = 3x is an exponential model with general solution x(t) = k1 e3t , The differential equation dx dt dy and dt = −2y is an exponential model with general solution y(t) = k2 e−2t . Hence, the general solution to the system is (x(t), y(t)) = (k1 e3t , k2 e−2t ). Since Y(0) = (x(0), y(0)) = (k1 , k2 ) = ( ...
... = 3x is an exponential model with general solution x(t) = k1 e3t , The differential equation dx dt dy and dt = −2y is an exponential model with general solution y(t) = k2 e−2t . Hence, the general solution to the system is (x(t), y(t)) = (k1 e3t , k2 e−2t ). Since Y(0) = (x(0), y(0)) = (k1 , k2 ) = ( ...
x - Piazza
... 2. The wave function must be single valued. 3. The wave function must be twice differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.) 4. In order to normalize a wave function, it must approach zero as x ...
... 2. The wave function must be single valued. 3. The wave function must be twice differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.) 4. In order to normalize a wave function, it must approach zero as x ...
Mathematical Methods (10/24
... homogeneous problems with homogeneous boundary conditions. The idea is to assume that the original function of two variables can be written as a product of two functions, each of which is only dependent upon a single independent. The separated form of the solution is inserted into the original linea ...
... homogeneous problems with homogeneous boundary conditions. The idea is to assume that the original function of two variables can be written as a product of two functions, each of which is only dependent upon a single independent. The separated form of the solution is inserted into the original linea ...
Space-time fractional derivative operators
... model predicts. Anomalous diffusion is observed in many physical situations, motivating the development of new mathematical and physical models [5, 6, 7, 13, 16, 20]. Some of the most successful models employ fractional derivatives [21, 27] in place of the usual integer order derivatives in the diff ...
... model predicts. Anomalous diffusion is observed in many physical situations, motivating the development of new mathematical and physical models [5, 6, 7, 13, 16, 20]. Some of the most successful models employ fractional derivatives [21, 27] in place of the usual integer order derivatives in the diff ...
AP Calculus
... The “average” slope is the slope of a line drawn from endpoint to endpoint. No matter how much the function may change between endpoints, we say that its “average” change is simply the ratio of changes in y to changes in x. For average slope, use the slope formula. The MVT says that there has to be ...
... The “average” slope is the slope of a line drawn from endpoint to endpoint. No matter how much the function may change between endpoints, we say that its “average” change is simply the ratio of changes in y to changes in x. For average slope, use the slope formula. The MVT says that there has to be ...
2.4 Continuity
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
Dougherty Lecture 3
... Most texts call (8) the standard form of (7). Solving (8) is the subject of Farlow’s Section 2.1. Note that we divided by a1 (x), which may occasionally be zero. We have not yet discussed the topic of just where we can find a solution, i.e., for which x’s can we solve such an equation. Thus anytime ...
... Most texts call (8) the standard form of (7). Solving (8) is the subject of Farlow’s Section 2.1. Note that we divided by a1 (x), which may occasionally be zero. We have not yet discussed the topic of just where we can find a solution, i.e., for which x’s can we solve such an equation. Thus anytime ...
f(x)
... 13) Is f defined at x = 2 ? Is f continuous at x = 2? 14) At what values of x is f continuous? 15) What value should be assigned to f(2) to make the extended function continuous at x = 2 . 16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 . 17) Is it pos ...
... 13) Is f defined at x = 2 ? Is f continuous at x = 2? 14) At what values of x is f continuous? 15) What value should be assigned to f(2) to make the extended function continuous at x = 2 . 16) What new value should be assigned to f(1) to make the extended function continuous at x = 1 . 17) Is it pos ...
On Malliavin`s proof of Hörmander`s theorem
... manifold M, recall that E ⊂ T M is a smooth subbundle of dimension d if Ex ⊂ Tx M is a vector space of dimension d at every x ∈ M and if the dependency x 7→ Ex is smooth. (Locally, Ex is the linear span of finitely many smooth vector fields on M.) A subbundle is called integrable if, whenever U, V a ...
... manifold M, recall that E ⊂ T M is a smooth subbundle of dimension d if Ex ⊂ Tx M is a vector space of dimension d at every x ∈ M and if the dependency x 7→ Ex is smooth. (Locally, Ex is the linear span of finitely many smooth vector fields on M.) A subbundle is called integrable if, whenever U, V a ...
Fixed Point Theorems, supplementary notes APPM
... constructive. It is due to Banach, 1922, and implies the implicit function theorem. The Brouwer fixed-point theorem is one of the most important results in modern mathematics, and is easy to state but hard to prove. It its most basic form, it is completely topological. Theorem 2.1 (Brouwer, topologi ...
... constructive. It is due to Banach, 1922, and implies the implicit function theorem. The Brouwer fixed-point theorem is one of the most important results in modern mathematics, and is easy to state but hard to prove. It its most basic form, it is completely topological. Theorem 2.1 (Brouwer, topologi ...
Math 500 – Intermediate Analysis Homework 8 – Solutions
... of convergence for this power series is − 65 , 56 . 24.2 For x ∈ [0, ∞), let fn (x) = nx . (a) Find f (x) = limn→∞ fn (x). Solution: For a given x0 ∈ [0, ∞), the numerical sequence fn (x0 ) = xn0 clearly converges to zero as n → ∞. Thus, the sequence of functions (fn ) converges pointwise to the fun ...
... of convergence for this power series is − 65 , 56 . 24.2 For x ∈ [0, ∞), let fn (x) = nx . (a) Find f (x) = limn→∞ fn (x). Solution: For a given x0 ∈ [0, ∞), the numerical sequence fn (x0 ) = xn0 clearly converges to zero as n → ∞. Thus, the sequence of functions (fn ) converges pointwise to the fun ...
Poisson`s remarkable calculation
... evaluated directly by the fundamental theorem of calculus. The familiar computation of the Gaussian integral is via the following remarkable trick, attributed to Poisson. One forms the square of I, interprets it as a double integral in the plane, transforms to polar coordinates and the answer magica ...
... evaluated directly by the fundamental theorem of calculus. The familiar computation of the Gaussian integral is via the following remarkable trick, attributed to Poisson. One forms the square of I, interprets it as a double integral in the plane, transforms to polar coordinates and the answer magica ...
continuity
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
SPACE-TIME FRACTIONAL DERIVATIVE OPERATORS 1
... In classical diffusion, particles spread in a normal bell-shaped pattern whose width grows like the square root of time. Anomalous diffusion occurs when the growth rate or the shape of the particle distribution is different than the classical model predicts. Anomalous diffusion is observed in many physi ...
... In classical diffusion, particles spread in a normal bell-shaped pattern whose width grows like the square root of time. Anomalous diffusion occurs when the growth rate or the shape of the particle distribution is different than the classical model predicts. Anomalous diffusion is observed in many physi ...
Slide
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
... In fact, the change in f(x) can be kept as small as we please by keeping the change in x sufficiently small. If f is defined near a (in other words, f is defined on an open interval containing a, except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if f is not co ...
AP Calculus AB
... Steps for Solving Differential Equations: “Find a solution (or solve) the separable differentiable equation…” 1. Separate the variables 2. Integrate each side 3. Make sure to put C on side with independent variable (normally x) 4. Plug in initial condition and solve for C (if given) 5. Solve for the ...
... Steps for Solving Differential Equations: “Find a solution (or solve) the separable differentiable equation…” 1. Separate the variables 2. Integrate each side 3. Make sure to put C on side with independent variable (normally x) 4. Plug in initial condition and solve for C (if given) 5. Solve for the ...
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MATH 409, Fall 2013 [3mm] Advanced Calculus I
... continuous on [a, b]. By assumption, there exists ε > 0 such that for any δ > 0 we can find two points x, y ∈ [a, b] satisfying |x − y | < δ and |f (x) − f (y )| ≥ ε. In particular, for any n ∈ N there exist points xn , yn ∈ [a, b] such that |xn − yn | < 1/n while |f (xn ) − f (yn )| ≥ ε. By constru ...
... continuous on [a, b]. By assumption, there exists ε > 0 such that for any δ > 0 we can find two points x, y ∈ [a, b] satisfying |x − y | < δ and |f (x) − f (y )| ≥ ε. In particular, for any n ∈ N there exist points xn , yn ∈ [a, b] such that |xn − yn | < 1/n while |f (xn ) − f (yn )| ≥ ε. By constru ...
Page 1 MCV4U0 PROBLEM SET V: Derivatives II In order to receive
... For critical points, we need A ′( x ) = 0 . Hence, A ′( x ) = 8 x − 3x 2 which equals 0 when x = 0 and x = 83 . Since the function is continuous and differentiable over the given interval, then any absolute maximum or minimum must come from the endpoints and/or critical points as guaranteed by the e ...
... For critical points, we need A ′( x ) = 0 . Hence, A ′( x ) = 8 x − 3x 2 which equals 0 when x = 0 and x = 83 . Since the function is continuous and differentiable over the given interval, then any absolute maximum or minimum must come from the endpoints and/or critical points as guaranteed by the e ...
- Deer Creek High School
... Semester Exam: 10% Chapter Tests: Chapter tests will be given at the conclusion of each chapter. Daily Work: Students will receive a reasonable amount of independent practice problems. Time may be given at the end of the class period to work on these problems. You are responsible for utilizing the t ...
... Semester Exam: 10% Chapter Tests: Chapter tests will be given at the conclusion of each chapter. Daily Work: Students will receive a reasonable amount of independent practice problems. Time may be given at the end of the class period to work on these problems. You are responsible for utilizing the t ...
ANTIDERIVATIVES AND AREAS AND THINGS 1. Integration is
... What we just used to calculate the area under a curve between two points is a big result called The Fundamental Theorem of Calculus and understanding this theorem will be one of our main goals for the course. That said, let’s just go a head and state: Theorem 1 (The Area Theorem). If f is a function ...
... What we just used to calculate the area under a curve between two points is a big result called The Fundamental Theorem of Calculus and understanding this theorem will be one of our main goals for the course. That said, let’s just go a head and state: Theorem 1 (The Area Theorem). If f is a function ...
Section 3 Integral Equations
... Equation (3.2) may be solved by reduction to a set of simultaneous linear algebraic equations as we shall now show. Substituting (3.3) into (3.2) gives ...
... Equation (3.2) may be solved by reduction to a set of simultaneous linear algebraic equations as we shall now show. Substituting (3.3) into (3.2) gives ...
