Math Problem Statement

Find the average value of the function f(x) = 4x^2 - 3x + 5 on the interval [-3, 5].

Solution

To find the average value of a function f(x)f(x) on a given interval, we use the following formula:

Average Value=1baabf(x)dx\text{Average Value} = \frac{1}{b - a} \int_a^b f(x) \, dx

Where:

  • aa and bb are the bounds of the interval,
  • f(x)f(x) is the function being evaluated.

Given the function f(x)=4x23x+5f(x) = 4x^2 - 3x + 5 and the interval [3,5][-3, 5], we can now compute the average value.

Steps:

  1. Find the integral: We need to compute the integral of f(x)=4x23x+5f(x) = 4x^2 - 3x + 5 from x=3x = -3 to x=5x = 5.
  2. Divide by the length of the interval: After finding the integral, divide the result by ba=5(3)=8b - a = 5 - (-3) = 8.

Let me calculate this now. sympy import symbols, integrate

Define the variable and the function

x = symbols('x') f = 4x**2 - 3x + 5

Define the limits of integration

a, b = -3, 5

Calculate the integral

integral_value = integrate(f, (x, a, b))

Find the average value

average_value = integral_value / (b - a) average_valueThe average value of the function f(x)=4x23x+5f(x) = 4x^2 - 3x + 5 on the interval [3,5][-3, 5] is 823\frac{82}{3}.

Would you like more details on this solution, or do you have any other questions?

Here are some related questions you might find helpful:

  1. How do you evaluate the integral of a polynomial function?
  2. What does the average value of a function represent in the context of calculus?
  3. How do you compute the definite integral of a quadratic function?
  4. What is the significance of the interval in finding the average value?
  5. Can the average value formula be applied to other types of functions, like trigonometric ones?

Tip: When finding the average value, remember that it's related to the area under the curve, normalized by the width of the interval.

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Math Problem Analysis

Mathematical Concepts

Calculus
Definite Integration
Average Value of a Function

Formulas

Average Value = (1/(b - a)) ∫[a, b] f(x) dx

Theorems

Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12