Domanda del libro di testo
Find the critical points of the following functions on the given intervals. Identify the absolute maximum and absolute minimum values (if they exist).
ƒ(x) = 4x¹⸍² - x⁵⸍² on [0, 4]
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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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 4, Problema 4.R.56
{Use of Tech} Newton’s method Use Newton’s method to approximate the roots of ƒ(x) = e⁻²ˣ + 2eˣ - 6 to six digits.
Guida verificata passo dopo passo1
Identify the function for which we want to find the roots: \( f(x) = e^{-2x} + 2e^x - 6 \).
Compute the derivative of the function, \( f'(x) \), which is necessary for Newton's method. The derivative is \( f'(x) = -2e^{-2x} + 2e^x \).
Choose an initial guess \( x_0 \) for the root. A good starting point can be found by graphing the function or using prior knowledge about the behavior of exponential functions.
Apply Newton's method formula: \( x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \). Substitute \( f(x) \) and \( f'(x) \) into the formula to get \( x_{n+1} = x_n - \frac{e^{-2x_n} + 2e^{x_n} - 6}{-2e^{-2x_n} + 2e^{x_n}} \).
Iterate the process: Use the formula from the previous step to calculate \( x_1, x_2, \ldots \) until the difference between successive approximations is less than the desired tolerance (e.g., \( 10^{-6} \) for six-digit accuracy).
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Newton's Method
Newton's Method is an iterative numerical technique used to approximate the roots of a real-valued function. Starting with an initial guess, the method uses the function's derivative to refine the guess iteratively. The formula is x_{n+1} = x_n - f(x_n)/f'(x_n), where x_n is the current approximation. This process is repeated until a sufficiently accurate value is found.
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Exponential Functions
Exponential functions, such as e^x, are mathematical functions where the variable is in the exponent. They are characterized by rapid growth or decay, depending on the sign of the exponent. In the given function, e⁻²ˣ and eˣ represent exponential decay and growth, respectively, which influence the behavior and shape of the function's graph.
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Derivative Calculation
Calculating the derivative is essential in Newton's Method as it provides the slope of the tangent line at a given point. For the function ƒ(x) = e⁻²ˣ + 2eˣ - 6, the derivative is found using the rules of differentiation for exponential functions. This derivative is crucial for updating the approximation of the root in each iteration of Newton's Method.
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05:44Derivatives
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{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.
c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.
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Domanda del libro di testo
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{Use of Tech} ƒ(x) = x (x -1)e⁻ˣ
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Domanda del libro di testo
Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
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Domanda del libro di testo
Optimization A right triangle has legs of length h and r and a hypotenuse of length 4 (see figure). It is revolved about the leg of length h to sweep out a right circular cone. What values of h and r maximize the volume of the cone? (Volume of a cone = (1/3) πr²h.) <IMAGE>
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