Example Question - newton's method
Here are examples of questions we've helped users solve.
Solving Equations with Transcendental Functions using Newton's Method
To solve the equation \( e^{x} + 7x + 11 = 20 \), we need to isolate \( x \). However, because this equation combines an exponential term with a polynomial term, it doesn't lend itself to an algebraic solution. The equation can be re-written as: \[ e^{x} + 7x = 9 \] There is no straightforward algebraic approach for solving this equation given that it is transcendental (it involves an exponential function along with a polynomial one). Thus we typically solve these types of equations either graphically or using numerical methods such as Newton's method. However, I can provide a general approach to solving such equations using Newton's method, but please note this will only give an approximate solution. Newton's method is an iterative numerical technique for finding successively better approximations to the roots (or zeroes) of a real-valued function. To apply Newton's method to a function \( f(x) \), you start with an initial guess \( x_0 \) for the root, and then use the iteration: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Here, \( f(x) = e^{x} + 7x - 9 \). To apply Newton's method, we also need the derivative of \( f \), which is \( f'(x) = e^{x} + 7 \). Would you like to try a numerical method to solve the equation, or is this information sufficient for your needs?
Solving Equations with Newton's Method
The equation provided in the image is: \[ e^x + 7x + 11 = 20 \] To solve for \( x \), we’ll first simplify the equation by moving all terms to one side and setting the equation equal to zero: \[ e^x + 7x + 11 - 20 = 0 \] \[ e^x + 7x - 9 = 0 \] This equation isn't easily solvable with elementary algebraic methods because it mixes exponential and polynomial expressions. To solve for \( x \), we typically use numerical methods such as Newton's method, also known as the Newton-Raphson method. This method involves making an initial guess for the value of \( x \) and then iteratively improving that guess until a sufficiently accurate value is found. Here is how the Newton-Raphson method is applied in general: 1. Choose an initial guess \( x_0 \). 2. Calculate the next approximate value using the formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] where \( f(x) \) is the function for which we are trying to find a root, and \( f'(x) \) is its derivative. 3. Repeat step 2 using the new value of \( x \) until the difference between subsequent values of \( x \) is within a desired level of accuracy. The function \( f(x) \) based on the equation is: \[ f(x) = e^x + 7x - 9 \] The derivative of \( f(x) \) is: \[ f'(x) = e^x + 7 \] Without a computer or calculator, it's not feasible to carry out the Newton-Raphson method to a high level of accuracy, but in a general sense, we would begin with a reasonable guess for \( x \) and apply the method until the successive iterations converge on a solution. To properly solve for \( x \), this process would typically be performed using a calculator with capabilities of performing exponential and natural logarithm calculations, or a computer algorithm programmed to apply the Newton-Raphson method.
CamTutor
In regards to math, we are professionals.
Get In Touch
Email: camtutor.ai@gmail.com