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שיטות דיפרנציאליות ואינטגרליות
Differential and Integral Methods |
0509-1842-22 | ||||||||||||||||||||||||
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| הנדסה | תואר ראשון - קורסי שירות | |||||||||||||||||||||||||
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סילבוס מקוצר
לעיון בסילבוס המקוצר נא לעיין בסילבוס המפורט
Course description
Credit points: 6
Functions: domain, range, graph, translations and reflections of graphs, monotonicity, inverse functions, odd/even functions. Composite function. Linear functions and the straight line, quadratic functions and their properties, polynomials, the circle equation, the ellipse equation, the hyperbola equation. The exponential function, logarithms. Trigonometric functions: periodicity, amplitude, frequency, phase, inverse trigonometric functions. Hyperbolic and inverse hyperbolic functions. The limit concept and examples. The number “e” as a limit, calculating lim sinx/x. The derivative as a slope and as velocity. Equation of a tangent and of a normal. Derivatives of polynomials, negative and rational powers. The chain rule. Derivatives of logarithmic, trigonometric and hyperbolic functions and their inverses. Parammetric curves and their derivatives, the tangent and normal expressions. Curvature, velocity and acceleration in the plan and in space. Linearization and differentials, arithmetics of differentials. Applications of L’Hopital’s rule. Applications of Taylor’s formula with Taylor series and remainder. Investigation of functions, sufficient condition for extrema, Newton’s binomial formula, expansions of the elementary functions. The definition of “I” trigonometric representation of complex numbers, Euler’s formula. Complex representation of complex numbers, Euler’s formula. Complex representation of the trigonometric functions. Indefinite integrals, arithmetics of integrals. Definite integrals and areas, the fundamental theorem of calculus. Variable transformations in definite and indefinite integrals. Integration techniques: substitution, integration by parts, Mathematica, Maple, indefinite integrals. Numerical integration: the trapezoidal rule and Simpson’s rule. Leibniz’s rule for taking the derivative of integral with respect to a parameter. Calculating integrals by expansion in series. First order differential equations: separable, linear, particular and general solutions, integrating factor, transformation of variables. Second order linear equations with constant coefficients, particular and general solutions. Integration of mx”=f(x), mathematical and physical pendulum. Calculating the length of an arc, area, volumes of solids of revolution, moments, center of mass, Pappus rules. Integral of the first type along a work curve. Partial derivatives, gradients, tangent planes and normals to a surface, the chain rule, differentials, implicit differentiation. Taylor’s formula for functions of two variables extreme points, Lagrange multipliers. Double and triple integrals in Cartesian coordinates, the connection to Iterated integrals and order changing. Integrals of the second type along a curve and flux, Green-Gauss-Stokes formula in the plane and the independence of a line integral on the trajectory. Area element on a surface, surface area, surface integral of the first and second type. Green, Gauss, stokes formula. Vector analysis.
Functions: domain, range, graph, translations and reflections of graphs, monotonicity, inverse functions, odd/even functions. Composite function. Linear functions and the straight line, quadratic functions and their properties, polynomials, the circle equation, the ellipse equation, the hyperbola equation. The exponential function, logarithms. Trigonometric functions: periodicity, amplitude, frequency, phase, inverse trigonometric functions. Hyperbolic and inverse hyperbolic functions. The limit concept and examples. The number “e” as a limit, calculating lim sinx/x. The derivative as a slope and as velocity. Equation of a tangent and of a normal. Derivatives of polynomials, negative and rational powers. The chain rule. Derivatives of logarithmic, trigonometric and hyperbolic functions and their inverses. Parammetric curves and their derivatives, the tangent and normal expressions. Curvature, velocity and acceleration in the plan and in space. Linearization and differentials, arithmetics of differentials. Applications of L’Hopital’s rule. Applications of Taylor’s formula with Taylor series and remainder. Investigation of functions, sufficient condition for extrema, Newton’s binomial formula, expansions of the elementary functions. The definition of “I” trigonometric representation of complex numbers, Euler’s formula. Complex representation of complex numbers, Euler’s formula. Complex representation of the trigonometric functions. Indefinite integrals, arithmetics of integrals. Definite integrals and areas, the fundamental theorem of calculus. Variable transformations in definite and indefinite integrals. Integration techniques: substitution, integration by parts, Mathematica, Maple, indefinite integrals. Numerical integration: the trapezoidal rule and Simpson’s rule. Leibniz’s rule for taking the derivative of integral with respect to a parameter. Calculating integrals by expansion in series. First order differential equations: separable, linear, particular and general solutions, integrating factor, transformation of variables. Second order linear equations with constant coefficients, particular and general solutions. Integration of mx”=f(x), mathematical and physical pendulum. Calculating the length of an arc, area, volumes of solids of revolution, moments, center of mass, Pappus rules. Integral of the first type along a work curve. Partial derivatives, gradients, tangent planes and normals to a surface, the chain rule, differentials, implicit differentiation. Taylor’s formula for functions of two variables extreme points, Lagrange multipliers. Double and triple integrals in Cartesian coordinates, the connection to Iterated integrals and order changing. Integrals of the second type along a curve and flux, Green-Gauss-Stokes formula in the plane and the independence of a line integral on the trajectory. Area element on a surface, surface area, surface integral of the first and second type. Green, Gauss, stokes formula. Vector analysis.