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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 1 - FunctionsProblema 51a
Capitolo 1, Problema 51a

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


a. Find A(2) .


ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

Guida verificata passo dopo passo
1
Identify the function \( f(t) \) and the intervals: \( f(t) = -2t + 8 \) for \( t \leq 3 \) and \( f(t) = 2 \) for \( t > 3 \).
Since we need to find \( A(2) \), we focus on the interval from \( t = 0 \) to \( t = 2 \), where \( f(t) = -2t + 8 \).
Set up the integral to find the area under the curve from \( t = 0 \) to \( t = 2 \): \( A(2) = \int_{0}^{2} (-2t + 8) \, dt \).
Calculate the integral: Find the antiderivative of \( -2t + 8 \), which is \( -t^2 + 8t \).
Evaluate the definite integral by substituting the limits: \( A(2) = [-t^2 + 8t]_{0}^{2} \).

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Definite Integral

The area function A(x) represents the definite integral of the function f(t) from t=0 to t=x. This integral calculates the net area between the curve and the t-axis, accounting for regions above and below the axis. Understanding how to evaluate definite integrals is crucial for finding specific area values, such as A(2) in this case.
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Introduction to Indefinite Integrals

Piecewise Functions

The function f(t) is defined piecewise, meaning it has different expressions based on the value of t. For t ≤ 3, f(t) = -2t + 8, and for t > 3, f(t) = 2. Recognizing how to work with piecewise functions is essential for correctly evaluating the integral over the specified interval, as the function's behavior changes at t=3.
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Piecewise Functions

Area Under a Curve

The area under a curve can be interpreted as the integral of the function over a given interval. In this problem, calculating A(2) involves finding the area under f(t) from t=0 to t=2. This requires integrating the appropriate expression of f(t) over the specified limits, which is fundamental in applications of calculus to determine physical quantities like area.
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Summary of Curve Sketching Example 2
Pratica correlata
Domanda del libro di testo

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


c. Find a formula for A(x)


ƒ(t) = {-2t+8 if t ≤ 3 ; 2 if t >3 <IMAGE>

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Domanda del libro di testo

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.


c. Find a formula for A(x)


ƒ(t) =6 <IMAGE>

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Domanda del libro di testo

Area functions Let A(x) be the area of the region bounded by the t -axis and the graph of y=ƒ(t) from t=0 to t=x. Consider the following functions and graphs.



b. Find A(6).



ƒ(t) =6 <IMAGE>

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More composite functions Let ƒ(x) = | x | , g(x)= x² - 4 , F(x) = √x , G(x) = (1)/(x-2) Determine the following composite functions and give their domains.


G o g o ƒ

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Use shifts and scalings to transform the graph of ƒ(x)=x2ƒ(x)=x^2 into the graph of g. Use a graphing utility to check your work.


g(x)=ƒ(2x−4)g(x) = ƒ(2x - 4)

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Shifting a graph Use a shift to explain how the graph of ƒ(x)=−x2+8x+9ƒ(x)=\(\sqrt{-x^2+8x+9}\) is obtained from the graph of g(x)=25−x2g(x)=\(\sqrt{25-x^2}\) . Sketch a graph of ƒ.

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