Skip to main content
Ch. 4 - Applications of the Derivative
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. 4 - Applications of the DerivativeProblema 4.9.93
Capitolo 4, Problema 4.9.93

Velocity to position Given the following velocity functions of an object moving along a line, find the position function with the given initial position.


v(t) = 2√t; s(0) = 1

Guida verificata passo dopo passo
1
Step 1: Recall that the position function s(t) can be found by integrating the velocity function v(t) with respect to time t. The relationship is given by: s(t) = ∫v(t) dt + C, where C is the constant of integration.
Step 2: Substitute the given velocity function v(t) = 2√t into the integral. This gives: s(t) = ∫2√t dt + C.
Step 3: Simplify the integral. Recall that √t can be written as t^{1/2}. Using the power rule for integration, ∫t^n dt = (t^{n+1})/(n+1), apply this to integrate 2t^{1/2}.
Step 4: After integrating, you will have an expression for s(t) in terms of t and the constant of integration C. Use the initial condition s(0) = 1 to solve for C. Substitute t = 0 into the position function and set s(0) equal to 1.
Step 5: Combine the result of the integration and the value of C to write the final position function s(t).

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:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Velocity and Position Functions

Velocity is the rate of change of position with respect to time, represented mathematically as the derivative of the position function. To find the position function from a given velocity function, one must integrate the velocity function. This relationship is fundamental in calculus, as it connects the concepts of motion and change.
Video consigliato:
Percorso guidato
10:17
Using The Velocity Function

Integration

Integration is the process of finding the integral of a function, which essentially reverses differentiation. In the context of finding a position function from a velocity function, integration allows us to determine the accumulated position over time. The result of integration includes a constant of integration, which can be determined using initial conditions.
Video consigliato:
Percorso guidato
06:18
Integration by Parts for Definite Integrals

Initial Conditions

Initial conditions are specific values that help define a particular solution to a differential equation or integral. In this case, the initial position s(0) = 1 provides the necessary information to solve for the constant of integration after finding the position function. This ensures that the solution accurately reflects the scenario described in the problem.
Video consigliato:
Percorso guidato
05:03
Initial Value Problems
Pratica correlata
Domanda del libro di testo

Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.

lim_x→π/2⁻ (π - 2x) tan x  

164
views
Domanda del libro di testo

Shipping crates A square-based, box-shaped shipping crate is designed to have a volume of 16 ft³. The material used to make the base costs twice as much (per square foot) as the material in the sides, and the material used to make the top costs half as much (per square foot) as the material in the sides. What are the dimensions of the crate that minimize the cost of materials?

286
views
Domanda del libro di testo

{Use of Tech} Special curves The following classical curves have been studied by generations of mathematicians. Use analytical methods (including implicit differentiation) and a graphing utility to graph the curves. Include as much detail as possible.


x³ + y³ = 3xy (Folium of Descartes)

162
views
Domanda del libro di testo

{Use of Tech} Finding roots with Newton’s method For the given function f and initial approximation x₀, use Newton’s method to approximate a root of f. Stop calculating approximations when two successive approximations agree to five digits to the right of the decimal point after rounding. Show your work by making a table similar to that in Example 1.


f(x) = sin x + x - 1; x₀ = 0.5

205
views
Domanda del libro di testo

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = -2x⁴ + x² + 10

165
views
Domanda del libro di testo

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.


ƒ(x) = 3x⁵ - 25x³ + 60x on [-2,3]

175
views