Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dπ β«βΛ£ (tΒ² + t + 1) dt
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8. Definite Integrals
Fundamental Theorem of Calculus
1:29 minutesProblem 5.R.1b
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume Ζ and Ζ' are continuous functions for all real numbers.
(b) Given an area function A(π) = β«βΛ£ Ζ(t) dt and an antiderivative F of Ζ, it follows that A'(π) = F(π) .
Verified step by step guidance1
Step 1: Begin by understanding the problem. The area function A(π) = β«βΛ£ Ζ(t) dt represents the accumulated area under the curve of Ζ(t) from a fixed point 'a' to a variable point 'π'. The goal is to determine whether A'(π) = F(π), where F is an antiderivative of Ζ.
Step 2: Recall the Fundamental Theorem of Calculus, Part 1. It states that if Ζ is continuous on [a, π], then the derivative of the area function A(π) with respect to π is equal to the value of the integrand at π. Mathematically, A'(π) = Ζ(π).
Step 3: Understand the relationship between an antiderivative and the integrand. An antiderivative F of Ζ satisfies F'(π) = Ζ(π). This means that F is a function whose derivative is Ζ.
Step 4: Compare A'(π) = Ζ(π) (from the Fundamental Theorem of Calculus) with the statement A'(π) = F(π). Since F'(π) = Ζ(π), the statement A'(π) = F(π) is incorrect unless F(π) = Ζ(π), which is not generally true. A'(π) equals Ζ(π), not F(π).
Step 5: Provide a counterexample to clarify. Consider Ζ(π) = πΒ². The area function A(π) = β«βΛ£ πΒ² dt has a derivative A'(π) = πΒ². However, an antiderivative F(π) of Ζ(π) could be F(π) = (1/3)πΒ³ + C, which is not equal to Ζ(π). This demonstrates that the statement A'(π) = F(π) is false.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus connects differentiation and integration, stating that if F is an antiderivative of a continuous function f on an interval [a, b], then the integral of f from a to x is given by A(x) = β«βΛ£ f(t) dt = F(x) - F(a). This theorem implies that the derivative of the area function A(x) is equal to the original function f evaluated at x, i.e., A'(x) = f(x).
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Area Function
An area function A(x) represents the accumulated area under the curve of a function f from a fixed point a to a variable point x. Mathematically, it is defined as A(x) = β«βΛ£ f(t) dt. This function is crucial in understanding how the total area changes as x varies, and its derivative A'(x) gives the instantaneous rate of change of this area, which corresponds to the value of the function f at that point.
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Antiderivative
An antiderivative F of a function f is a function whose derivative is f, meaning F' = f. Antiderivatives are essential in calculus as they allow us to reverse the process of differentiation. In the context of the area function, if F is an antiderivative of f, then the area function A(x) can be expressed in terms of F, leading to the conclusion that A'(x) = F'(x) = f(x).
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