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Ch. 1 - Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 1 - FunctionsProblem 49c
Chapter 1, Problem 49c

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>

Verified step by step guidance
1
Step 1: Understand the problem. We need to find a formula for the area function \( A(x) \), which represents the area under the curve \( y = f(t) \) from \( t = 0 \) to \( t = x \). Here, \( f(t) = 6 \) is a constant function.
Step 2: Set up the integral. Since \( f(t) = 6 \), the area under the curve from \( t = 0 \) to \( t = x \) is given by the definite integral \( A(x) = \int_{0}^{x} 6 \, dt \).
Step 3: Integrate the function. The integral of a constant \( c \) with respect to \( t \) is \( ct \). Therefore, \( \int 6 \, dt = 6t \).
Step 4: Evaluate the definite integral. Substitute the limits of integration into the antiderivative: \( A(x) = [6t]_{0}^{x} = 6x - 6(0) \).
Step 5: Simplify the expression. The formula for \( A(x) \) simplifies to \( A(x) = 6x \). This represents the area under the curve from \( t = 0 \) to \( t = x \).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Definite Integral

The definite integral of a function over an interval gives the net area under the curve of that function between two points. In this context, to find the area function A(x), we use the definite integral of the function f(t) from 0 to x, which mathematically is expressed as A(x) = ∫[0 to x] f(t) dt. This concept is fundamental in calculus as it connects the geometric interpretation of area with the analytical process of integration.
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Introduction to Indefinite Integrals

Area Under a Curve

The area under a curve represents the total space between the curve and the x-axis over a specified interval. For the function f(t), the area A(x) from t=0 to t=x quantifies how much space is enclosed by the graph of f(t) and the t-axis. Understanding this concept is crucial for solving problems related to area functions, as it directly relates to the application of integration.
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Summary of Curve Sketching Example 2

Function Notation

Function notation, such as A(x) or f(t), is a way to represent mathematical functions and their outputs. In this case, A(x) denotes the area as a function of x, while f(t) represents the function whose area we are calculating. Grasping function notation is essential for interpreting and manipulating mathematical expressions correctly, especially in calculus where functions are frequently analyzed and transformed.
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Related Practice
Textbook Question

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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Textbook Question

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>

Textbook Question

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>

Textbook Question

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>

Textbook Question

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

Textbook Question

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