Textbook Question
Derivatives of integrals Simplify the following expressions.
d/dz ∫¹⁰ₛᵢₙ ₂ dt /(t⁴ + 1)
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8. Definite Integrals
Fundamental Theorem of Calculus
1:52 minutesProblem 5.R.1c
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.
(c) ∫ₐᵇ ƒ'(𝓍) d𝓍 = ƒ(b) ―ƒ(a) .
Verified step by step guidance1
Step 1: Recall the Fundamental Theorem of Calculus, Part 1, which states that if a function ƒ is continuous on [a, b] and has a derivative ƒ'(𝓍), then the definite integral of ƒ'(𝓍) from a to b is equal to ƒ(b) - ƒ(a).
Step 2: Analyze the given statement ∫ₐᵇ ƒ'(𝓍) d𝓍 = ƒ(b) ― ƒ(a). This matches the conclusion of the Fundamental Theorem of Calculus, Part 1.
Step 3: Verify the conditions for applying the theorem. The problem states that ƒ and ƒ' are continuous functions for all real numbers, which satisfies the requirement for the theorem to hold.
Step 4: Conclude that the statement is true because it directly follows from the Fundamental Theorem of Calculus, given the continuity of ƒ and ƒ'.
Step 5: If needed, provide a counterexample to test the validity of the statement. For instance, if ƒ or ƒ' were not continuous, the theorem would not apply, and the statement could be false. However, in this case, the conditions are met, so the statement is true.
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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 a function is continuous on an interval [a, b], then the integral of its derivative over that interval equals the difference in the function's values at the endpoints. Specifically, it asserts that ∫ₐᵇ ƒ'(𝓍) d𝓍 = ƒ(b) - ƒ(a), which is crucial for evaluating the given statement.
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Continuous Functions
A function is continuous if there are no breaks, jumps, or holes in its graph over its domain. For the Fundamental Theorem of Calculus to hold, both the function ƒ and its derivative ƒ' must be continuous on the interval [a, b]. This ensures that the integral can be computed reliably and that the function behaves predictably across the interval.
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Definite Integral
A definite integral represents the signed area under the curve of a function between two points, a and b. It is denoted as ∫ₐᵇ ƒ(𝓍) d𝓍 and provides a way to calculate the accumulation of quantities. In the context of the question, understanding the properties of definite integrals is essential for verifying the truth of the statement regarding the relationship between the integral of a derivative and the values of the original function.
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