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) =6 <IMAGE>
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Ch. 1 - Functions
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 1, Problem 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>
Verified step by step guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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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05:04Introduction 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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Related Practice
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
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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.
b. Find A(6).
ƒ(t) =6 <IMAGE>
Textbook Question
Slope functions Determine the slope function S (x) for the following functions
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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