Textbook Question
Derivatives of integrals Simplify the following expressions.
d/d𝓍 ∫₀ˣ (√1 + t²) dt (Hint: ∫ˣ₋ₓ (√1 + t²) dt = ∫⁰₋ₓ (√1 + t²) dt + ∫ˣ₋ₓ (√1 + t²) dt ) .
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8. Definite Integrals
Fundamental Theorem of Calculus
2:25 minutesProblem 5.3.2
Textbook Question
Suppose F is an antiderivative of ƒ and A is an area function of ƒ. What is the relationship between F and A?
Verified step by step guidance1
Understand the definitions: An antiderivative F of a function ƒ is a function such that the derivative of F is equal to ƒ, i.e., . An area function A of ƒ represents the accumulated area under the curve of ƒ from a fixed point to a variable point x.
Recall the Fundamental Theorem of Calculus: It states that if A(x) is the area function of ƒ, then A'(x) = ƒ(x). This means the derivative of the area function is the original function ƒ.
Recognize the connection: Since F is an antiderivative of ƒ, and A'(x) = ƒ(x), it follows that A(x) and F(x) differ by a constant. Specifically, , where C is a constant.
Interpret the constant C: The constant C depends on the choice of the lower limit of integration in the area function A(x). If the lower limit is changed, the value of C will adjust accordingly.
Summarize the relationship: The area function A(x) is essentially an antiderivative of ƒ, but it includes a constant term that depends on the lower limit of integration. Both F and A are closely related through this constant adjustment.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Antiderivative
An antiderivative of a function f is another function F such that the derivative of F is equal to f, i.e., F' = f. This means that F represents a family of functions whose slopes at any point correspond to the values of f. Antiderivatives are essential in calculus for solving problems related to integration and finding areas under curves.
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Area Function
An area function A associated with a function f typically represents the accumulated area under the curve of f from a specific point to a variable endpoint. Mathematically, it is defined as A(x) = ∫[a to x] f(t) dt, where a is a constant. The area function is crucial for understanding how the total area changes as the endpoint varies, linking it to the concept of integration.
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of f on an interval [a, b], then the integral of f from a to b can be computed as F(b) - F(a). This theorem establishes that the area function A is directly related to the antiderivative F, as A(x) = F(x) - F(a), illustrating the deep relationship between these concepts.
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