Question

1. average value of a function

problem:
find the average value of
$f(x) = 3x^2 + 2$
on the interval $1, 4$.

Explanation:

Step1: Recall the formula for the average value of a function

The average value of a function \( f(x) \) on the interval \([a, b]\) is given by \( f_{\text{avg}}=\frac{1}{b - a}\int_{a}^{b}f(x)dx \). Here, \( a = 1 \), \( b = 4 \), and \( f(x)=3x^{2}+2 \).

Step2: Compute the integral of \( f(x) \)

First, find the antiderivative of \( f(x)=3x^{2}+2 \). The antiderivative of \( 3x^{2} \) is \( x^{3} \) (using the power rule \( \int x^{n}dx=\frac{x^{n + 1}}{n+1}+C \), for \( n = 2 \), \( \int 3x^{2}dx=3\times\frac{x^{3}}{3}=x^{3} \)) and the antiderivative of \( 2 \) is \( 2x \). So the antiderivative \( F(x)=x^{3}+2x \).

Step3: Evaluate the definite integral

Using the Fundamental Theorem of Calculus, \( \int_{1}^{4}(3x^{2}+2)dx=F(4)-F(1) \).
Compute \( F(4) \): \( 4^{3}+2\times4=64 + 8=72 \).
Compute \( F(1) \): \( 1^{3}+2\times1=1 + 2=3 \).
So \( \int_{1}^{4}(3x^{2}+2)dx=72-3 = 69 \).

Step4: Compute the average value

Now, use the formula for the average value: \( f_{\text{avg}}=\frac{1}{4 - 1}\times69 \). Since \( 4-1 = 3 \), then \( f_{\text{avg}}=\frac{69}{3}=23 \).

Answer:

The average value of the function \( f(x) = 3x^{2}+2 \) on the interval \([1,4]\) is \( 23 \).