f(x) = − 5x + e^x− 2x^2 solve using bisection method

To solve \( f(x) = -5x + e^x - 2x^2 \) using the bisection method, we first need to find an interval \([a, b]\) where the function changes sign, indicating a root exists in that interval.

1. **Identify an Interval**: Evaluate \( f(x) \) at different points to find \( a \) and \( b \):
   - \( f(0) = -5(0) + e^0 - 2(0)^2 = 1 \) (positive)
   - \( f(2) = -5(2) + e^2 - 2(2)^2 \approx -10 + 7.39 - 8 = -10.61 \) (negative)

   Thus, \( f(0) > 0 \) and \( f(2) < 0 \), so we can choose \( a = 0 \) and \( b = 2 \).

2. **Bisection Steps**:
   - Calculate \( c = \frac{a + b}{2} \).
   - Check the sign of \( f(c) \):
     - If \( f(c) = 0 \), \( c \) is the root.
     - If \( f(a)f(c) < 0 \), set \( b = c \).
     - If \( f(b)f(c) < 0 \), set \( a = c \).
   - Repeat until \( |b - a| \) is less than a chosen tolerance (e.g., \( 0.01 \)).

### Example Iteration:
1. \( c = 1 \): \( f(1) = -5(1) + e^1 - 2(1)^2 \approx -5 + 2.71 - 2 = -4.29 \) (negative)
   - New interval: \([0, 1]\)

2. \( c = 0.5 \): \( f(0.5) = -5(0.5) + e^{0.5} - 2(0.5)^2 \approx -2.5 + 1.65 - 0.5 = -1.35 \) (negative)
   - New interval: \([0, 0.5]\)

Continue this process until convergence.

Would you like detailed steps for each iteration, or do you have any questions? 

### Follow-up Questions:
1. What tolerance level do you want for the bisection method?
2. Would you like to see a specific number of iterations?
3. Are you familiar with the concept of function continuity?
4. Would you like to know more about the bisection method's convergence?
5. Do you want to visualize the function graphically? 

### Tip:
Always check that \( f(a) \) and \( f(b) \) have opposite signs before applying the bisection method.

f(x) = − 5x + e^x − 2x^2 solve using bisection method

Explore how to solve the function f(x) = −5x + e^x − 2x^2 using the bisection method. This guide includes step-by-step instructions, example iterations, and crucial mathematical concepts. Ideal for high school students (Grades 11-12) learning about numerical methods.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation