Domanda del libro di testo
Working with area functions Consider the function ƒ and the points a, b, and c.
(b) Graph ƒ and A.
ƒ(𝓍) = eˣ ; a = 0 , b = ln 2 , c = ln 4
50
views
Ch. 5 - Integration
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
Non è quello che usi tu?Cambia libro di testoScegli un libro di testo diverso
TOC.tableOfContents
- 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: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 5, Problema 5.3.21b
Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).
(b) Verify that A'(𝓍) = ƒ(𝓍).
ƒ(t) = 3t + 1 , a = 2
Guida verificata passo dopo passo1
Step 1: Understand the problem. The goal is to verify that the derivative of the area function A(x) with respect to x is equal to the given function f(x). The area function A(x) represents the area under the curve y = f(t) from t = a to t = x.
Step 2: Recall the Fundamental Theorem of Calculus. It states that if A(x) is defined as the integral of f(t) from a to x, then the derivative of A(x) with respect to x is equal to f(x). Mathematically, this is expressed as:
Step 3: Define the area function A(x). Using the given function f(t) = 3t + 1 and the lower limit a = 2, the area function is:
Step 4: Differentiate A(x) with respect to x. By the Fundamental Theorem of Calculus, the derivative of the integral with respect to its upper limit x is simply the integrand evaluated at x. Therefore,
Step 5: Verify the result. The derivative of A(x) matches the given function f(x) = 3x + 1, confirming that A'(x) = f(x). This completes the verification.
Risposta video verificata per un problema simile:
Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
3mConcetti chiave
Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.
Area Function
An area function, denoted as A(x), represents the accumulated area under a curve from a fixed point 'a' to a variable point 'x'. In this context, it quantifies the area between the x-axis and the function f(t) = 3t + 1 over the interval [a, x]. Understanding this concept is crucial for analyzing how the area changes as 'x' varies.
Video consigliato:
05:06
Finding Area When Bounds Are Not Given
Fundamental Theorem of Calculus
The Fundamental Theorem of Calculus links differentiation and integration, stating that if A(x) is the area function defined as the integral of f(t) from a to x, then the derivative A'(x) equals f(x). This theorem is essential for verifying the relationship between the area function and the original function, as required in the question.
Video consigliato:
Percorso guidato
06:11Fundamental Theorem of Calculus Part 1
Linear Functions
A linear function is a polynomial function of degree one, typically expressed in the form f(t) = mt + b, where m is the slope and b is the y-intercept. In this case, f(t) = 3t + 1 is a linear function with a slope of 3, indicating a constant rate of change. Understanding linear functions is vital for interpreting the graph and calculating the area under the curve.
Video consigliato:
Percorso guidato
07:17Linearization
Pratica correlata
Domanda del libro di testo
Area functions for linear functions Consider the following functions ƒ and real numbers a (see figure).
b) Verify that A'(𝓍) = ƒ(𝓍).
ƒ(t) = 4t + 2 , a = 0
47
views
Domanda del libro di testo
Using properties of integrals Use the value of the first integral I to evaluate the two given integrals.
I = ∫₀¹ (𝓍³ ― 2𝓍) d𝓍 = ―3/4
(b) ∫₁⁰ (2𝓍―𝓍³) d𝓍
85
views
Domanda del libro di testo
Use Table 5.6 to evaluate the following indefinite integrals.
(b) ∫ sec 5𝓍 tan 5𝓍 d𝓍
103
views
Domanda del libro di testo
{Use of Tech} Midpoint Riemann sums with a calculator Consider the following definite integrals.
(b) Evaluate each sum using a calculator with n = 20, 50, and 100. Use these values to estimate the value of the integral.
∫₀⁴ (4𝓍― 𝓍²) d𝓍
86
views
Domanda del libro di testo
{Use of Tech} Approximating net area The following functions are positive and negative on the given interval.
ƒ(x) = 4 - 2x on [0,4]
(b) Approximate the net area bounded by the graph of f and the x-axis on the interval using a left, right, and midpoint Riemann sum with n = 4.
48
views