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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 49a
Capitolo 1, Problema 49a

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) =6 <IMAGE>

Guida verificata passo dopo passo
1
Step 1: Understand the problem. We need to find the area under the curve of the function \( f(t) = 6 \) from \( t = 0 \) to \( t = 2 \).
Step 2: Recognize that \( f(t) = 6 \) is a constant function, which means the graph is a horizontal line at \( y = 6 \).
Step 3: The area under a constant function from \( t = 0 \) to \( t = 2 \) is a rectangle with height \( 6 \) and width \( 2 - 0 = 2 \).
Step 4: Calculate the area of the rectangle using the formula for the area of a rectangle: \( \text{Area} = \text{height} \times \text{width} \).
Step 5: Substitute the values into the formula: \( \text{Area} = 6 \times 2 \).

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Concetti chiave

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Area Under a Curve

The area under a curve represents the integral of a function over a specified interval. In this context, the area function A(x) calculates the total area between the t-axis and the graph of the function y = f(t) from t = 0 to t = x. This concept is fundamental in calculus as it connects geometric interpretations with integral calculus.
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Summary of Curve Sketching Example 2

Definite Integral

A definite integral is a mathematical representation that computes the accumulation of quantities, such as area, over a specific interval. It is denoted as ∫[a, b] f(t) dt, where a and b are the limits of integration. In the problem, A(2) can be found by evaluating the definite integral of f(t) from 0 to 2, which gives the area under the curve from t = 0 to t = 2.
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Percorso guidato
05:04
Introduction to Indefinite Integrals

Function Evaluation

Function evaluation involves substituting a specific input value into a function to determine its output. In this case, to find A(2), one must evaluate the integral of the function f(t) at the upper limit of 2. Understanding how to evaluate functions and integrals is crucial for solving problems related to area functions in calculus.
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Percorso guidato
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Evaluating Composed Functions
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) =6 <IMAGE>

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views
Domanda del libro di testo

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.


ƒ o g

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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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Domanda del libro di testo

Slope functions Determine the slope function S (x) for the following functions

ƒ(x)=∣x∣ƒ(x) = | x |

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Domanda del libro di testo

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.


ƒ o G

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views
Domanda del libro di testo

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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