Question
upstudy study bank question image url

3. Let \( f(x)=a x^{3}+x-10=0 \). If the Newton method is used starting with the initial value \( x_{0}=1 \) then the first iteration value is found as \( x_{1}=2 \) then what is the value of \( a \) ? \( \begin{array}{llll}\text { a. } 0 & \text { b. } 1 & \text { c. } 2 & \text { d. } 3\end{array} \)

Ask by Cole Daniel. in the United States
Mar 12,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The value of \( a \) is 2.

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To find the value of \(a\) using the information given, we first set up the Newton's method formula: \[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Given \(f(x) = ax^3 + x - 10\), we compute \(f'(x)\): \[ f'(x) = 3ax^2 + 1 \] Now, substituting our initial value \(x_0 = 1\) into \(f(x)\): \[ f(1) = a(1)^3 + 1 - 10 = a - 9 \] Next, we compute \(f'(1)\): \[ f'(1) = 3a(1)^2 + 1 = 3a + 1 \] Applying the Newton's method: \[ x_1 = 1 - \frac{f(1)}{f'(1)} = 1 - \frac{a - 9}{3a + 1} \] We know \(x_1 = 2\), so we set up the equation: \[ 2 = 1 - \frac{a - 9}{3a + 1} \] Rearranging gives us: \[ 1 = -\frac{a - 9}{3a + 1} \] Multiplying both sides by \(3a + 1\): \[ 3a + 1 = - (a - 9) \] Expanding and collecting terms: \[ 3a + 1 = -a + 9 \implies 4a = 8 \implies a = 2 \] Thus, the value of \(a\) is **2**. Therefore, the answer is: **c. 2**

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy