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2.8: The Precise Definition of a Limit
https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/02%3A_Limits/2.08%3A_The_Precise_Definition_of_a_Limit
In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language. The formal definition of a limit is quite possibly one of the most challenging ...In this section, we convert this intuitive idea of a limit into a formal definition using precise mathematical language. The formal definition of a limit is quite possibly one of the most challenging definitions you will encounter early in your study of calculus; however, it is well worth any effort you make to reconcile it with your intuitive notion of a limit. Understanding this definition is the key that opens the door to a better understanding of calculus.
3: Derivatives
https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/03%3A_Derivatives
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, 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.
4.12: Antiderivatives
https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/04%3A_Applications_of_Derivatives/4.12%3A_Antiderivatives
At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a fu...At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function f , how do we find a function with the derivative f and why would we be interested in such a function?
4.12: Antiderivatives
https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/04%3A_Applications_of_Derivatives/4.12%3A_Antiderivatives
At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a fu...At this point, we have seen how to calculate derivatives of many functions and have been introduced to a variety of their applications. We now ask a question that turns this process around: Given a function f , how do we find a function with the derivative f and why would we be interested in such a function?
5.4: Integration Formulas and the Net Change Theorem
https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/05%3A_Integration/5.04%3A_Integration_Formulas_and_the_Net_Change_Theorem
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...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.
3: Derivatives
https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/03%3A_Derivatives
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, 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.
3.2E: Exercises for Section 3.1
https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/03%3A_Derivatives/3.02%3A_Defining_the_Derivative/3.2E%3A_Exercises_for_Section_3.1
For exercises 1 - 10, use the equation $m_{\text{sec}}=\frac{f(x)−f(a)}{x−a}$ to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f(x)$ . For the func...For exercises 1 - 10, use the equation $m_{\text{sec}}=\frac{f(x)−f(a)}{x−a}$ to find the slope of the secant line between the values $x_1$ and $x_2$ for each function $y=f(x)$ . For the functions $y=f(x)$ in exercises 21 - 30, find $f′(a)$ using the equation $\displaystyle f′(a)=\lim_{x→a}\frac{f(x)−f(a)}{x−a}$ . For the following exercises, use the limit definition of derivative to show that the derivative does not exist at $x=a$ for each of the given functions.
2.2: Subtracting Integers
https://math.libretexts.org/Courses/Highline_College/Math_081_091%3A_CAM_Aligned_Textbook/02%3A_Arithmetic/2.02%3A_Subtracting_Integers
In Section 1.2, we stated that “Subtraction is the opposite of addition.” Thus, to subtract 4 from 7, we walked seven units to the right on the number line, but then walked 4 units in the opposite dir...In Section 1.2, we stated that “Subtraction is the opposite of addition.” Thus, to subtract 4 from 7, we walked seven units to the right on the number line, but then walked 4 units in the opposite direction (to the left), as shown in Figure $\PageIndex{1}$ . The key phrase is “add the opposite.” Thus, the subtraction 7 − 4 becomes the addition 7 + (−4), which we would picture on the number line as shown in Figure $\PageIndex{2}$ .
1: Vectors in Space
https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/01%3A_Vectors_in_Space
A quantity that has magnitude and direction is called a vector. Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a...A quantity that has magnitude and direction is called a vector. Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.
7.3: Change of Variables in Multiple Integrals
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Real_Analysis_(Trench)/07%3A_Integrals_of_Functions_of_Several_Variables/7.03%3A_Change_of_Variables_in_Multiple_Integrals
Introduction to Real Analysis (Trench)
https://math.libretexts.org/Bookshelves/Analysis/Introduction_to_Real_Analysis_(Trench)
This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an und...This book introduces readers to a rigorous understanding of mathematical analysis and presents challenging math concepts as clearly as possible. This book is intended for those who want to gain an understanding of mathematical analysis and challenging mathematical concepts.

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