Textbook Question
Area functions for linear functions Consider the following functions Ζ and real numbers a (see figure).
b) Verify that A'(π) = Ζ(π).
Ζ(t) = 4t + 2 , a = 0
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Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 5, Problem 5.3.17b
Area functions for the same linear function Let Ζ(t) = t and consider the two area functions A(π) = β«βΛ£ Ζ(t) dt and F(π) = β«βΛ£ Ζ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (π) , for π β₯ 2.
Verified step by step guidance1
Step 1: Understand the problem. We are given a linear function Ζ(t) = t and two area functions: A(π) = β«βΛ£ Ζ(t) dt and F(π) = β«βΛ£ Ζ(t) dt. The goal is to evaluate F(4) and F(6), and then derive a general expression for F(π) using geometry for π β₯ 2.
Step 2: Recall the geometric interpretation of definite integrals. The integral β«βΛ£ Ζ(t) dt represents the area under the curve Ζ(t) = t from t = 2 to t = π. Since Ζ(t) = t is a linear function, the graph is a straight line, and the area can be calculated as the area of a trapezoid or triangle.
Step 3: Evaluate F(4). To find F(4), compute the definite integral β«ββ΄ t dt. This represents the area under the line Ζ(t) = t from t = 2 to t = 4. Use the formula for the definite integral of t, β« t dt = (1/2)tΒ², and evaluate it at the bounds 2 and 4.
Step 4: Evaluate F(6). Similarly, compute the definite integral β«ββΆ t dt. This represents the area under the line Ζ(t) = t from t = 2 to t = 6. Again, use the formula for the definite integral of t, β« t dt = (1/2)tΒ², and evaluate it at the bounds 2 and 6.
Step 5: Derive a general expression for F(π) for π β₯ 2. Using geometry, note that the area under the line Ζ(t) = t from t = 2 to t = π forms a trapezoid or triangle. The formula for the area of a trapezoid is (1/2)(base1 + base2) Γ height. Alternatively, use the definite integral formula β«βΛ£ t dt = (1/2)πΒ² - (1/2)(2Β²) to express F(π).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Definite Integral
A definite integral represents the signed area under a curve defined by a function over a specific interval. In this context, the area functions A(x) and F(x) are calculated using the definite integral of the function f(t) = t from a lower limit to an upper limit. Understanding how to evaluate definite integrals is crucial for finding the values of F(4) and F(6).
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Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links the concept of differentiation with integration, stating that if F is an antiderivative of f on an interval, then the integral of f from a to b is equal to F(b) - F(a). This theorem is essential for evaluating the area functions A(x) and F(x) and helps in deriving expressions for these functions based on their limits.
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Geometric Interpretation of Integrals
The geometric interpretation of integrals involves visualizing the area under a curve as a physical space that can be calculated. In this problem, using geometry to find an expression for F(x) for x β₯ 2 means recognizing the shape formed by the linear function f(t) = t and calculating the area of the resulting geometric figure, such as a triangle or trapezoid, to derive a formula for F(x).
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Related Practice
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