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 59b
Capitolo 4, Problema 59b

Minimizing related functions Complete each of the following parts.
b. What value of x minimizes ƒ(x) = (x- a₁)² + (x - a₂)² + (x - a₃)² , for constants a₁, a₂, and a₃?

Guida verificata passo dopo passo
1
First, recognize that the function ƒ(x) = (x - a₁)² + (x - a₂)² + (x - a₃)² is a sum of squares, which is a quadratic function in terms of x.
To find the value of x that minimizes this function, we need to take the derivative of ƒ(x) with respect to x. The derivative will help us find the critical points where the function could have a minimum.
Calculate the derivative: ƒ'(x) = 2(x - a₁) + 2(x - a₂) + 2(x - a₃). This simplifies to ƒ'(x) = 6x - 2(a₁ + a₂ + a₃).
Set the derivative ƒ'(x) equal to zero to find the critical points: 6x - 2(a₁ + a₂ + a₃) = 0.
Solve for x: x = (a₁ + a₂ + a₃) / 3. This value of x minimizes the function ƒ(x) because it is the point where the derivative is zero, indicating a potential minimum.

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:
3m

Concetti chiave

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

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. They graph as parabolas, which can open upwards or downwards depending on the sign of the coefficient 'a'. The vertex of the parabola represents the minimum or maximum point of the function, which is crucial for optimization problems.
Video consigliato:
6:04
Introduction to Polynomial Functions

Minimization

Minimization in calculus involves finding the lowest point of a function, which can be achieved using techniques such as taking the derivative and setting it to zero to find critical points. The second derivative test can then confirm whether these points are minima or maxima. In the context of the given function, we seek the value of x that minimizes the sum of squared differences.
Video consigliato:
09:07
Example 1: Minimizing Surface Area

Completing the Square

Completing the square is a method used to transform a quadratic equation into a perfect square trinomial, making it easier to analyze its properties. This technique is particularly useful for finding the vertex of a parabola, which directly relates to identifying the minimum value of a quadratic function. In the context of the problem, it can help simplify the expression to find the optimal x value.
Video consigliato:
3:03
Trig Values in Quadrants II, III, & IV Example 1
Pratica correlata
Domanda del libro di testo

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

lim_x→0⁺ (cot x - 1/x) 

264
views
Domanda del libro di testo

{Use of Tech} Modified Newton’s method The function f has a root of multiplicity 2 at r if f(r) = f'(r) = 0 and f"(r) ≠ 0. In this case, a slight modification of Newton’s method, known as the modified (or accelerated) Newton’s method, is given by the formula xₙ + 1 = xₙ - (2f(xₙ)/(f'(xₙ), for n = 0, 1, 2, . . . . This modified form generally increases the rate of convergence.

b. Apply Newton’s method and the modified Newton’s method using x₀ = 0.1 to find the value of x₃ in each case. Compare the accuracy of these values of x₃.

274
views
Domanda del libro di testo

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

lim_x→1⁻ (x/(x-1) - x/(ln x) 

219
views
Domanda del libro di testo

Evaluate the following limits in two different ways: with and without l’Hôpital’s Rule.

lim_x→∞ (2x⁵ - x + 1) / (5x⁶ + x)

287
views
Domanda del libro di testo

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

c. Find the rate at which water flows from the tank and plot the flow rate function. 

195
views
Domanda del libro di testo

The figure shows six containers, each of which is filled from the top. Assume water is poured into the containers at a constant rate and each container is filled in 10 s. Assume also that the horizontal cross sections of the containers are always circles. Let h (t) be the depth of water in the container at time t, for 0 ≤ t ≤ 10 . <IMAGE>


d. For each container, where does h' (the derivative of h ) have an absolute maximum on [0 , 10]?

260
views