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- 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/Cosumnes_River_College/Math_400%3A_Calculus_I_-_Differential_Calculus/01%3A_Critical_Concepts_for_Calculus/1.08%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=ert, 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/Mission_College/Mission_College_MAT_003B/01%3A_Integration/1.06%3A_Integrals_Involving_Exponential_and_Logarithmic_FunctionsExponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functio...Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functions or logarithms.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/05%3A_Integration/5.07%3A_Integrals_Resulting_in_Inverse_Trigonometric_FunctionsRecall that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions...Recall that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. Also in Derivatives, we developed formulas for derivatives of inverse trigonometric functions. The formulas developed there give rise directly to integration formulas involving inverse trigonometric functions.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Tran)/05%3A_Integration/5.06%3A_Integrals_Involving_Exponential_and_Logarithmic_FunctionsExponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functio...Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functions or logarithms.
- https://math.libretexts.org/Courses/Coastline_College/Math_C180%3A_Calculus_I_(Everett)/05%3A_Integration/5.07%3A_Integrals_Resulting_in_Inverse_Trigonometric_FunctionsRecall that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions...Recall that trigonometric functions are not one-to-one unless the domains are restricted. When working with inverses of trigonometric functions, we always need to be careful to take these restrictions into account. Also in Derivatives, we developed formulas for derivatives of inverse trigonometric functions. The formulas developed there give rise directly to integration formulas involving inverse trigonometric functions.
- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_2410%3A_Calculus_(Open_Stax)_Novick/05%3A_Integration/5.06%3A_Integrals_Involving_Exponential_and_Logarithmic_FunctionsExponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functio...Exponential and logarithmic functions arise in many real-world applications, especially those involving growth and decay. Substitution is often used to evaluate integrals involving exponential functions or logarithms.
- https://math.libretexts.org/Courses/Prince_Georges_Community_College/MAT_2410%3A_Calculus_1_(Beck)/05%3A_Integration/5.01%3A_Approximating_AreasIn this section, we develop techniques to approximate the area between a curve, defined by a function f(x), and the x-axis on a closed interval [a,b]. Like Archimedes, we first approximate the are...In this section, we develop techniques to approximate the area between a curve, defined by a function f(x), and the x-axis on a closed interval [a,b]. Like Archimedes, we first approximate the area under the curve using shapes of known area (namely, rectangles). By using smaller and smaller rectangles, we get closer and closer approximations to the area. Taking a limit allows us to calculate the exact area under the curve.
- https://math.libretexts.org/Courses/Mission_College/Math_3A%3A_Calculus_1_(Sklar)/05%3A_Integration/5.02%3A_The_Definite_IntegralIf f(x) is a function defined on an interval [a,b], the definite integral of f from a to b is given by ∫^b_af(x)dx=\lim_{n→∞} \sum_{i=1}^nf(x^∗_i)Δx, provided the limit exists. If this limit exi...If f(x) is a function defined on an interval [a,b], the definite integral of f from a to b is given by ∫^b_af(x)dx=\lim_{n→∞} \sum_{i=1}^nf(x^∗_i)Δx, provided the limit exists. If this limit exists, the function f(x) is said to be integrable on [a,b], or is an integrable function. The numbers a and b are called the limits of integration; specifically, a is the lower limit and b is the upper limit. The function f(x) is the integrand, and x is the variable of integration.
