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- https://math.libretexts.org/Courses/City_College_of_San_Francisco/CCSF_Calculus/02%3A_Learning_Limits/2.01%3A_Tangent_Lines_and_VelocityWe begin our exploration of calculus by reconnecting with a topic from our early days in algebra - slope. The concept of slope is fundamentally important in calculus and this section, along with our o...We begin our exploration of calculus by reconnecting with a topic from our early days in algebra - slope. The concept of slope is fundamentally important in calculus and this section, along with our old friend "slope," allows a gentle introduction to a monumentally important subject in mathematics and physics.
- 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_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_I%3A_Differential_Calculus/03%3A_Derivatives/3.06%3A_The_Chain_Rule/3.6E%3A_Exercises_for_Section_3.6This page covers exercises on applying the chain rule for differentiating composite functions and finding derivatives using Leibniz's notation. It includes systematic methods for calculating \(\frac{d...This page covers exercises on applying the chain rule for differentiating composite functions and finding derivatives using Leibniz's notation. It includes systematic methods for calculating dydx and finding tangent and normal lines. Additionally, it addresses problems in physics and calculus, emphasizing rates of change in motion, with applications in velocity, acceleration, and geometric changes.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_373%3A_Trigonometry_for_Calculus/04%3A_Radian_Measure_and_the_Circular_Functions/4.04%3A_Velocities_and_Areas_of_SectorsThis section covers velocities related to circular motion and sector areas. It explains the concepts of angular and linear velocities, providing equations to calculate each, and describes how to deter...This section covers velocities related to circular motion and sector areas. It explains the concepts of angular and linear velocities, providing equations to calculate each, and describes how to determine the area of a sector using the central angle and radius of the circle. Practical examples, such as computing the linear velocity of a Ferris wheel, are included to illustrate these concepts. This foundation is crucial for understanding applications of Trigonometry.
- https://math.libretexts.org/Bookshelves/Calculus/Supplemental_Modules_(Calculus)/Vector_Calculus/2%3A_Vector-Valued_Functions_and_Motion_in_Space/2.5%3A_Velocity_and_AccelerationIn single variable calculus the velocity is defined as the derivative of the position function. For vector calculus, we make the same definition.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_I%3A_Differential_Calculus/01%3A_Functions_and_Graphs/1.04%3A_Inverse_Functions/1.4E%3A_Exercises_for_Section_1.4This page includes exercises on one-to-one functions, inverse functions, and trigonometric evaluations, featuring graphs and real-life applications like temperature conversions and blood velocity anal...This page includes exercises on one-to-one functions, inverse functions, and trigonometric evaluations, featuring graphs and real-life applications like temperature conversions and blood velocity analysis.
- https://math.libretexts.org/Courses/De_Anza_College/Introductory_Differential_Equations/02%3A_Applications_of_First_Order_Equations/2.03%3A_Elementary_MechanicsThis section discusses applications to elementary mechanics involving Newton's second law of motion. The problems considered include motion under the influence of gravity in a resistive medium, and de...This section discusses applications to elementary mechanics involving Newton's second law of motion. The problems considered include motion under the influence of gravity in a resistive medium, and determining the initial velocity required to launch a satellite.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/09%3A_Radian_Measure_and_the_Circular_Functions/9.04%3A_Velocities_and_Areas_of_SectorsThis section covers velocities related to circular motion and sector areas. It explains the concepts of angular and linear velocities, providing equations to calculate each, and describes how to deter...This section covers velocities related to circular motion and sector areas. It explains the concepts of angular and linear velocities, providing equations to calculate each, and describes how to determine the area of a sector using the central angle and radius of the circle. Practical examples, such as computing the linear velocity of a Ferris wheel, are included to illustrate these concepts. This foundation is crucial for understanding applications of Trigonometry.
- https://math.libretexts.org/Courses/Cosumnes_River_College/Math_384%3A_Foundations_for_Calculus/13%3A_Non-Right_Triangle_Trigonometry/13.04%3A_Vectors_-_A_Geometric_ApproachThis section introduces vectors from a geometric perspective, covering definitions, notation, and operations. It explains position vectors, scalar multiplication, vector addition, and the concepts of ...This section introduces vectors from a geometric perspective, covering definitions, notation, and operations. It explains position vectors, scalar multiplication, vector addition, and the concepts of displacement and resultant vectors. The section also addresses vector components, their magnitudes, and applications in velocity and other contexts. Detailed examples and exercises help illustrate these concepts, providing a comprehensive understanding of vectors in geometric terms.
- https://math.libretexts.org/Courses/De_Anza_College/Calculus_II%3A_Integral_Calculus/01%3A_Integration/1.08%3A_Chapter_1_Review_ExercisesThis page features calculus exercises on definite integrals, Riemann sums, and antiderivatives. It includes exercises on evaluating mathematical truths and real-world applications, such as calculating...This page features calculus exercises on definite integrals, Riemann sums, and antiderivatives. It includes exercises on evaluating mathematical truths and real-world applications, such as calculating average costs and velocities. The content ranges from theoretical proofs to practical scenarios, emphasizing the continuity of functions and derivatives. Specific calculations and their answers are provided, demonstrating the connections between theory and application.
