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- https://math.libretexts.org/Courses/De_Anza_College/Linear_Algebra%3A_A_First_Course/06%3A_Spectral_Theory/6.05%3A_Markov_Matrices/6.5E%3A_Exercises_for_Section_6.5This page explores exercises on Markov matrices, emphasizing their application in determining population distributions, transition probabilities, and long-term steady states across various scenarios, ...This page explores exercises on Markov matrices, emphasizing their application in determining population distributions, transition probabilities, and long-term steady states across various scenarios, including trailer rentals, political affiliations, and weather conditions. Key topics include calculating steady-state populations and probabilities of returning to specific locations, bolstered by numerical examples.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/03%3A_DerivativesCalculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, an...Calculating velocity and changes in velocity are important uses of calculus, but it is far more widespread than that. Calculus is important in all branches of mathematics, science, and engineering, and it is critical to analysis in business and health as well. In this chapter, we explore one of the main tools of calculus, the derivative, and show convenient ways to calculate derivatives. We apply these rules to a variety of functions in this chapter.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/00%3A_Front_Matter/05%3A_About_the_BookFrom understanding the rate of change to calculating areas and volumes of irregular shapes, this book encourages learners to explore the richness of Calculus in various fields, including physics, econ...From understanding the rate of change to calculating areas and volumes of irregular shapes, this book encourages learners to explore the richness of Calculus in various fields, including physics, economics, and engineering. The online version of the textbook includes 3D figures which gives a three-dimensional visual experience to the learner to understand the concept.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/01%3A_Integration/1.06%3A_Integrals_Involving_Exponential_and_Logarithmic_Functions/1.6E%3A_Exercises_for_Section_1.6This page covers exercises on computing indefinite and definite integrals using calculus techniques, including substitutions and properties of logarithmic functions. It presents specific integrals to ...This page covers exercises on computing indefinite and definite integrals using calculus techniques, including substitutions and properties of logarithmic functions. It presents specific integrals to solve, discusses the natural logarithm function, its properties, and implications, and provides answers for better understanding.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.06%3A_Linear_Equations_and_the_Integrating_FactorOne of the most important types of equations we will learn how to solve are the so-called linear equations. In fact, the majority of the course is about linear equations. In this lecture we focus on t...One of the most important types of equations we will learn how to solve are the so-called linear equations. In fact, the majority of the course is about linear equations. In this lecture we focus on the first order linear equation.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/05%3A_Introduction_to_Differential_Equations/5.03%3A_Separable_EquationsWe now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of discipli...We now examine a solution technique for finding exact solutions to a class of differential equations known as separable differential equations. These equations are common in a wide variety of disciplines, including physics, chemistry, and engineering. We illustrate a few applications at the end of the section.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.10%3A_Numerical_Methods_-_Eulers_MethodThis page elaborates on Euler's method for approximating solutions to differential equations when closed-form solutions are not feasible. It discusses the method's iterative approach and its first-ord...This page elaborates on Euler's method for approximating solutions to differential equations when closed-form solutions are not feasible. It discusses the method's iterative approach and its first-order accuracy, noting the error reduction with smaller step sizes. The text also addresses challenges like numerical instability and the importance of selecting appropriate methods and step sizes.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/02%3A_Applications_of_Integration/2.05%3A_Physical_Applications_of_IntegrationIn this section, we examine some physical applications of integration. Several physical applications of the definite integral are common in engineering and physics. Definite integrals can be used to d...In this section, we examine some physical applications of integration. Several physical applications of the definite integral are common in engineering and physics. Definite integrals can be used to determine the mass of an object if its density function is known. Work can also be calculated from integrating a force function, or when counteracting the force of gravity, as in a pumping problem. Definite integrals can also be used to calculate the force exerted on an object submerged in a liquid.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/01%3A_First_Order_ODEs/1.01%3A_Introduction_to_Differential_EquationsThis page emphasizes the importance of differential equations in science and engineering, focusing on their role in modeling physical laws and solving first-order equations. It includes the derivation...This page emphasizes the importance of differential equations in science and engineering, focusing on their role in modeling physical laws and solving first-order equations. It includes the derivation of a bacteria population growth model, showcasing calculations for growth rates and populations over time. The text also discusses solution types for differential equations, including general and particular solutions, while connecting hyperbolic functions to real-world shapes.
- https://math.libretexts.org/Courses/De_Anza_College/Math_1D%3A_De_Anza/03%3A_Vector_CalculusIn this chapter, we learn to model new kinds of integrals over fields such as magnetic fields, gravitational fields, or velocity fields. We also learn how to calculate the work done on a charged parti...In this chapter, we learn to model new kinds of integrals over fields such as magnetic fields, gravitational fields, or velocity fields. We also learn how to calculate the work done on a charged particle traveling through a magnetic field, the work done on a particle with mass traveling through a gravitational field, and the volume per unit time of water flowing through a net dropped in a river.
