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- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/02%3A_Limits/2.08%3A_The_Precise_Definition_of_a_LimitIn 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.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/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/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/04%3A_Applications_of_Derivatives/4.12%3A_AntiderivativesAt 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?
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/04%3A_Applications_of_Derivatives/4.12%3A_AntiderivativesAt 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?
- 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_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/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/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/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/03%3A_Derivatives/3.02%3A_Defining_the_Derivative/3.2E%3A_Exercises_for_Section_3.1For exercises 1 - 10, use the equation msec=f(x)−f(a)x−a to find the slope of the secant line between the values x1 and x2 for each function y=f(x). For the func...For exercises 1 - 10, use the equation msec=f(x)−f(a)x−a to find the slope of the secant line between the values x1 and x2 for each function y=f(x). For the functions y=f(x) in exercises 21 - 30, find f′(a) using the equation f′(a)=lim. 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.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_221_Calculus_1/02%3A_Limits/2.01%3A_Prelude_to_LimitsWe begin this chapter by examining why limits are so important. Then, we go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we dis...We begin this chapter by examining why limits are so important. Then, we go on to describe how to find the limit of a function at a given point. Not all functions have limits at all points, and we discuss what this means and how we can tell if a function does or does not have a limit at a particular value. The last section of this chapter presents the more precise definition of a limit and shows how to prove whether a function has a limit.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_221_Calculus_1/01%3A_Functions_and_Graphs/1.07%3A_Chapter_1_Review_ExercisesFor the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain. Find the equation C=f(x) that describes the total cost as a functi...For the following problems, determine the largest domain on which the function is one-to-one and find the inverse on that domain. Find the equation C=f(x) that describes the total cost as a function of number of shirts and Carbon dating is implemented to determine how old the skeleton is by using the equation y=e^{rt}, where y is the percentage of radiocarbon still present in the material, t is the number of years passed, and r=−0.0001210 is the decay rate of radiocarbon.
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/02%3A_Vector-Valued_Functions/2.03%3A_Arc_Length_and_Curvature/2.3E%3A_Exercises_for_Section_2.316) Given \vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩, find the unit tangent vector \vecs T(t) evaluated at t=0, \vecs T(0). 18) Given \(\vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩...16) Given \vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩, find the unit tangent vector \vecs T(t) evaluated at t=0, \vecs T(0). 18) Given \vecs r(t)=⟨2e^t,\, e^t \cos t,\, e^t \sin t⟩, find the unit normal vector \vecs N(t) evaluated at t=0, \vecs N(0). 29) Parameterize the curve using the arc-length parameter s, at the point at which t=0 for \vecs r(t)=e^t \sin t \,\hat{\mathbf{i}} + e^t \cos t \,\hat{\mathbf{j}}
- https://math.libretexts.org/Courses/SUNY_Geneseo/Math_223_Calculus_3/02%3A_Vector-Valued_Functions/2.01%3A_Vector-Valued_Functions_and_Space_CurvesOur study of vector-valued functions combines ideas from our earlier examination of single-variable calculus with our description of vectors in three dimensions from the preceding chapter. In this sec...Our study of vector-valued functions combines ideas from our earlier examination of single-variable calculus with our description of vectors in three dimensions from the preceding chapter. In this section, we extend concepts from earlier chapters and also examine new ideas concerning curves in three-dimensional space. These definitions and theorems support the presentation of material in the rest of this chapter and also in the remaining chapters of the text.
