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- https://math.libretexts.org/Bookshelves/Calculus/Calculus_by_David_Guichard_(Improved)/01%3A_Analytic_Geometry/1.04%3A_FunctionsA function y=f(x) is a rule for determining y when we're given a value of x. Functions can be defined in various ways: by an algebraic formula or several algebraic formulas, by a graph, or by an ex...A function y=f(x) is a rule for determining y when we're given a value of x. Functions can be defined in various ways: by an algebraic formula or several algebraic formulas, by a graph, or by an experimentally determined table of values. (In the latter case, the table gives a bunch of points in the plane, which we might then interpolate with a smooth curve, if that makes sense.)
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/01%3A_Functions_and_GraphsIn this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functio...In this chapter, we review all the functions necessary to study calculus. We define polynomial, rational, trigonometric, exponential, and logarithmic functions. We review how to evaluate these functions, and we show the properties of their graphs. We provide examples of equations with terms involving these functions and illustrate the algebraic techniques necessary to solve them. In short, this chapter provides the foundation for the material to come.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.01%3A_The_Laplace_Transform/5.1E%3A_Exercises_for_Section_5.1This page outlines exercises on finding Laplace transforms and their inverses for various functions, including polynomials, trigonometric, and piecewise functions. It provides examples and answers for...This page outlines exercises on finding Laplace transforms and their inverses for various functions, including polynomials, trigonometric, and piecewise functions. It provides examples and answers for certain problems, illustrating the application of Laplace transform rules. The exercises emphasize techniques such as integration by parts and the use of properties related to exponential and trigonometric functions within the context of Laplace transforms.
- https://math.libretexts.org/Courses/Irvine_Valley_College/Math_4A%3A_Multivariable_Calculus/02%3A_Functions_of_Several_Variables/2.01%3A_Multivariable_FunctionsSketch a graph of a function of two variables. Our first step is to explain what a function of more than one variable is, starting with functions of two independent variables. We also examine ways to ...Sketch a graph of a function of two variables. Our first step is to explain what a function of more than one variable is, starting with functions of two independent variables. We also examine ways to relate the graphs of functions in three dimensions to graphs of more familiar planar functions. We expand this to examples with more than two variables, and also discuss surfaces as an extension of the graph of functions of two variables.
- https://math.libretexts.org/Bookshelves/Algebra/Elementary_Algebra_(Arnold)/05%3A_Polynomial_Functions/5.01%3A_FunctionsA relation is a function if and only if each object in the domain is paired with exactly one object in the range.
- https://math.libretexts.org/Bookshelves/Combinatorics_and_Discrete_Mathematics/Discrete_Mathematics_for_Computer_Science_(Fitch)/03%3A_Functions/3.05%3A_Proof_by_ContradictionThis page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain ...This page discusses direct proof and proof by contradiction, featuring a theorem that the composition of two functions is a function. The proof is divided into two parts: confirming that every domain element maps to a codomain element, and using contradiction to establish uniqueness in mapping. It also introduces mapping composition and explains its operation with two mappings.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/04%3A_Parametric_Equations/4.02%3A_Calculus_of_Parametric_CurvesNow that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a gi...Now that we have introduced the concept of a parameterized curve, our next step is to learn how to work with this concept in the context of calculus. For example, if we know a parameterization of a given curve, is it possible to calculate the slope of a tangent line to the curve? How about the arc length of the curve? Or the area under the curve?
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/01%3A_Integration/1.04%3A_Integration_Formulas_and_the_Net_Change_TheoremThe net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or...The net change theorem states that when a quantity changes, the final value equals the initial value plus the integral of the rate of change. Net change can be a positive number, a negative number, or zero. The area under an even function over a symmetric interval can be calculated by doubling the area over the positive x-axis. For an odd function, the integral over a symmetric interval equals zero, because half the area is negative.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_Variables/1.03%3A_Partial_Derivatives/1.3E%3A_Exercises_for_Section_1.3This page discusses exercises on calculating partial and higher-order derivatives of functions, including limit definitions, surface plot analysis, and practical applications like volume and area. It ...This page discusses exercises on calculating partial and higher-order derivatives of functions, including limit definitions, surface plot analysis, and practical applications like volume and area. It also covers problems in differential calculus, such as finding points where partial derivatives equal zero, verifying Laplace's and heat equations, and analyzing rates of change in contexts like dimensions and productivity functions. Answers to the various exercises are provided throughout.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_IV%3A_Multivariable_Calculus/01%3A_Differentiation_of_Functions_of_Several_VariablesWhen dealing with a function of more than one independent variable, several questions naturally arise. For example, how do we calculate limits of functions of more than one variable? The definition of...When dealing with a function of more than one independent variable, several questions naturally arise. For example, how do we calculate limits of functions of more than one variable? The definition of derivative we used before involved a limit. Does the new definition of derivative involve limits as well? Do the rules of differentiation apply in this context? Can we find relative extrema of functions using derivatives? All these questions are answered in this chapter.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/05%3A_The_Laplace_Transform/5.03%3A_ConvolutionThis page discusses the use of inverse Laplace transforms and convolution in solving ordinary differential and Volterra integral equations. It highlights the simplification of computations through the...This page discusses the use of inverse Laplace transforms and convolution in solving ordinary differential and Volterra integral equations. It highlights the simplification of computations through these methods. An example is provided where a differential equation involving an integral is transformed into the frequency domain, resulting in the expression X(s)=s−1s2−2. The final solution is obtained as x(t)=cosh(√2t)−1√2sinh(√2t).
