book

Calculus Problem Solutions with MATLAB®

Name: Calculus Problem Solutions with MATLAB®
ISBN: 9783110663754

by Dingyü Xue, Tsinghua University Tsinghua University Press

March 2020

Intermediate to advanced

312 pages

9h 35m

English

De Gruyter

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Preface
1 Introduction to calculus problems
1.1 A brief history of calculus1.2 Main topics in the book
2 Functions and sequences
2.1 Functions and mappings2.1.1 Definitions of functions2.1.2 MATLAB computation of commonly used transcendental functions2.1.3 MATLAB representation of functions2.1.4 Curves and surfaces of functions2.2 Descriptions of various functions2.2.1 Inverse functions2.2.2 Composite functions2.2.3 Describing piecewise functions2.2.4 Implicit functions2.2.5 Parametric equations2.2.6 Polar functions2.3 Odd and even functions2.4 Complex-valued functions and mapping2.4.1 Complex matrices and manipulations2.4.2 Mapping of complex-valued functions2.4.3 Riemann surfaces2.5 Numeric and functional sequences2.6 Exercises
3 Limits
3.1 Limits of univariate functions3.1.1 The ε–δ definition3.1.2 Limit computing with MATLAB3.1.3 Limits of composite functions3.1.4 Limits of sequences3.1.5 Limits of piecewise functions3.1.6 Infinitesimals and infinity3.2 Single-sided limits and continuity of functions3.2.1 Left and right limits3.2.2 Continuity of functions3.2.3 Interval limits3.2.4 Applications of continuity – assessment of equation solutions3.3 Singularities, poles and residues of complex functions3.3.1 Computation of singularities and poles3.3.2 Residues of complex functions3.4 Limits of multivariate functions3.4.1 Sequential limits3.4.2 Multiple limits and computations3.5 Exercises
4 Derivatives and differentials
4.1 Derivatives and high-order derivatives4.1.1 Derivatives and differentials4.1.2 Higher-order derivatives4.1.3 Derivatives of composite functions4.1.4 Derivatives of piecewise functions4.1.5 Derivatives of matrices4.2 Derivatives of parametric equations4.3 Partial derivatives of multivariate functions4.3.1 Partial derivatives4.3.2 Total differential4.3.3 Derivatives of composite functions4.4 Introduction to fields4.4.1 Scalar and vector fields4.4.2 Gradient, divergence, and curl4.4.3 Potentials of a vector field4.5 Derivative matrices4.5.1 Jacobian matrix4.5.2 Hessian matrix4.5.3 Laplacian operators for scalar fields4.6 Partial derivatives of implicit functions4.6.1 First-order derivative of an implicit function4.6.2 Higher-order derivatives of implicit functions4.6.3 Partial derivatives of simultaneous implicit functions4.7 Applications of derivatives and differentials4.7.1 Extreme value problem4.7.2 Newton–Raphson iterative algorithm4.7.3 Tangent planes and normal lines4.8 Exercises
5 Integrals
5.1 Indefinite integrals of univariate functions5.2 Definite and improper integrals5.2.1 Definite integrals5.2.2 Infinite and improper integrals5.3 Multiple integrals5.3.1 Multiple indefinite integrals5.3.2 Constructions of undetermined polynomials5.3.3 Computation of multiple integrals5.3.4 Conversions of integration regions5.4 Applications of definite integrals5.4.1 Computation of arc length5.4.2 Computation of volume5.4.3 Volume and mass computation5.4.4 Computation of probability distribution5.4.5 An introduction to integral transforms5.5 Path integrals5.5.1 Type I path integral5.5.2 Type II path integral5.6 Surface integrals5.6.1 Type I surface integrals5.6.2 Type II surface integrals5.7 Exercises
6 Series and function fitting
6.1 Series sums6.1.1 Number series sums6.1.2 Sum of infinite series6.1.3 Sum of functional series6.1.4 Special infinite term problems6.2 Convergence tests for infinite series6.2.1 General description of a positive series6.2.2 Convergence tests for positive series6.2.3 Convergence test for alternating series6.2.4 Convergence interval of a functional series6.3 Products of sequences6.3.1 Products of number sequences6.3.2 Products of functional sequences6.3.3 Convergence test for the products of positive sequences6.4 Taylor series6.4.1 Taylor series expansions for univariate functions6.4.2 Taylor series for multivariate functions6.5 Fourier series expansions6.5.1 Mathematical description of Fourier series6.5.2 MATLAB implementation of Fourier series6.6 Rational function approximation of univariate functions6.6.1 Continued fraction expansions6.6.2 Padé approximations6.7 Laurent series6.7.1 Laurent series expansion6.7.2 Laurent series of rational functions6.8 Exercises
7 Numerical derivatives and differentials
7.1 Numerical derivative algorithms7.1.1 Forward and backward difference algorithms7.1.2 Central difference algorithms with o(h2) precision7.1.3 Central difference algorithm with o(h4) precision7.1.4 Central difference algorithms of higher precision7.1.5 Deriving high-order high precision algorithms7.1.6 High precision forward and backward difference algorithms7.2 MATLAB implementations of the numerical derivative algorithms7.2.1 Implementation of the o(h2) algorithms7.2.2 Implementation of the seven-point central difference algorithms7.2.3 Implementation of forward difference numerical derivative algorithms7.3 Numerical derivatives of any orders7.4 Numerical partial derivatives of 2D functions7.4.1 Gradient computation7.4.2 High precision algorithms for derivatives with respect to a single variable7.4.3 Numerical computation of mixed partial derivatives7.4.4 Numerical computation of high-order mixed partial derivatives7.5 Spline interpolation and numerical derivatives7.5.1 Cubic spline7.5.2 B-splines7.5.3 Numerical derivatives with splines7.5.4 Unequally spaced and scattered samples7.6 Exercises
8 Numerical integrals
8.1 Numerical integrals from samples8.1.1 Direct computation of numerical definite integrals8.1.2 Reconstruction of integral function8.1.3 High precision numerical integration algorithm for equally spaced samples8.2 Numerical integrals of univariate functions8.2.1 Simple numerical integral problems8.2.2 MATLAB solutions of numerical integral problems8.2.3 Numerical computation of improper integrals8.2.4 Numerical integrals for integrands with parameters8.2.5 Numerical solutions of integral functions8.3 Numerical computation of double integrals8.3.1 Computing double integrals8.3.2 Computation of double integral functions8.3.3 Evaluations of double integrals in different order8.4 Numerical computation of multiple integrals8.4.1 Numerical triple integrals8.4.2 Triple integrals of integrands with parameters8.4.3 Multiple integrals8.4.4 Numerical solutions of some multiple integrals with variable bounds8.5 Other numerical methods for multiple integrals8.5.1 Numerical integral approximations with Monte Carlo method8.5.2 Spline-based integral evaluations8.5.3 Numerical evaluations of multiple integrals8.6 Exercises
9 Integral transforms
9.1 Laplace transforms and their inverses9.1.1 Definition and properties of Laplace transform9.1.2 Computer solutions of Laplace transforms9.1.3 Solving differential equations with Laplace transform9.2 Numerical solutions of Laplace transform problems9.2.1 Numerical inverse Laplace transform9.2.2 Ideas of feedback control systems9.2.3 Numerical Laplace transforms9.2.4 Computation of irrational systems9.3 Fourier transforms and their inverses9.3.1 Definition and properties of Fourier transforms9.3.2 Computer evaluation of Fourier transform9.3.3 Fourier sine and cosine transforms9.3.4 Discrete Fourier sine and cosine transforms9.3.5 Fast Fourier transform9.4 Computation of other integral transforms9.4.1 Mellin transform9.4.2 Computation of Hankel transforms9.5 The z transform and its inverse9.5.1 Definition and properties of z transforms9.5.2 Computer evaluation of z transforms9.5.3 Bilateral z transform9.5.4 Numerical inverse z transforms for rational functions9.6 Exercises

10 Introduction to fractional calculus
10.1 Definitions in fractional calculus10.1.1 Why fractional calculus?10.1.2 Definitions10.2 Properties and relationships among different fractional calculus definitions10.3 Numerical implementation of Grünwald–Letnikov derivatives10.3.1 Simple Grünwald–Letnikov definition evaluation10.3.2 High precision algorithms and implementation10.3.3 Quantitative comparisons of different algorithms10.4 Numerical computation of Caputo derivatives10.5 Oustaloup filter algorithms and applications10.5.1 Oustaloup filter approximations10.5.2 Filter for Caputo derivative fitting10.5.3 Simulink-based Caputo derivative computation10.6 Numerical computation of even higher-order derivatives and integrals10.7 Exercises
Bibliography
Subject Index
Sachregister

Content preview from Calculus Problem Solutions with MATLAB®

4 Derivatives and differentials

Derivatives and differentials are important topics in differential calculus. The study of derivatives originated from two practical examples.

Example 4.1.

Assume that a particle is moving in a straight line. The displacement at t is described by a function $s(t)$ . In a small period of time h, the displacement changes to $s(t+h)$ . The total displacement in the period of length h is $s(t+h)−s(t)$ . If h is small enough, the instant velocity is the average velocity in the period

\begin{matrix} v (t) = lim_{h \to 0} \frac{s (t + h) - s (t)}{h} . \end{matrix}

Example 4.2.

For a given function $f(x)$ , how can we determine the tangent line at point $x=x0$ ? The function value at $x0$ is known as $f(x0)$ . Find a nearby point x and the function value $f(x)$ . If x approaches $x0$ and ...

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