for the equation f(x)=3x2+2x-5 find the average rate of change of f on [x,x+h]

To find the average rate of change of the function \( f(x) = 3x^2 + 2x - 5 \) on the interval \([x, x+h]\), we use the formula:

\[
\text{Average Rate of Change} = \frac{f(x+h) - f(x)}{h}
\]

1. First, calculate \( f(x+h) \). Substitute \( x+h \) into the function:

\[
f(x+h) = 3(x+h)^2 + 2(x+h) - 5
\]

Expand this expression:

\[
f(x+h) = 3(x^2 + 2xh + h^2) + 2(x+h) - 5
\]
\[
= 3x^2 + 6xh + 3h^2 + 2x + 2h - 5
\]

2. Now, subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = (3x^2 + 6xh + 3h^2 + 2x + 2h - 5) - (3x^2 + 2x - 5)
\]
\[
= 6xh + 3h^2 + 2h
\]

3. Finally, divide the result by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{6xh + 3h^2 + 2h}{h}
\]
\[
= 6x + 3h + 2
\]

Thus, the average rate of change of \( f(x) \) on the interval \([x, x+h]\) is:

\[
6x + 3h + 2
\]

Would you like more details or have any questions? Here are 5 related questions:

1. How do you interpret the average rate of change geometrically?
2. What is the significance of the term \( 3h \) as \( h \to 0 \)?
3. How does this expression simplify when \( h \to 0 \), representing the instantaneous rate of change?
4. Can you apply the same method to find the average rate of change for a linear function?
5. What are the steps to compute the derivative of \( f(x) \) using the limit definition?

**Tip:** The average rate of change is the slope of the secant line between two points on the function.

For the equation f(x)=3x^2+2x-5, find the average rate of change of f on [x, x+h].

Discover how to find the average rate of change for the quadratic function f(x)=3x^2+2x-5 on the interval [x, x+h]. This problem demonstrates key calculus concepts using the average rate of change formula, ideal for students in Grades 10-12.

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