https://file.aiquickdraw.com/mathbot/file/172689091559178a449gv.jpg

The problem in the image asks to calculate the **difference quotient** for the function \( f(x) = -3x^2 + 4x - 5 \), which is given by the formula:

\[
\frac{f(x+h) - f(x)}{h}
\]

### Step 1: Find \( f(x+h) \)

Substitute \( x+h \) into the given function \( f(x) = -3x^2 + 4x - 5 \):

\[
f(x+h) = -3(x+h)^2 + 4(x+h) - 5
\]

Now expand the terms:

\[
f(x+h) = -3(x^2 + 2xh + h^2) + 4(x+h) - 5
\]
\[
f(x+h) = -3x^2 - 6xh - 3h^2 + 4x + 4h - 5
\]

### Step 2: Compute \( f(x+h) - f(x) \)

Now subtract \( f(x) \) from \( f(x+h) \):

\[
f(x+h) - f(x) = \left(-3x^2 - 6xh - 3h^2 + 4x + 4h - 5\right) - \left(-3x^2 + 4x - 5\right)
\]

Simplify the expression:

\[
f(x+h) - f(x) = -6xh - 3h^2 + 4h
\]

### Step 3: Divide by \( h \)

Now, divide the result by \( h \):

\[
\frac{f(x+h) - f(x)}{h} = \frac{-6xh - 3h^2 + 4h}{h}
\]

Simplify by canceling out \( h \) from all terms:

\[
\frac{f(x+h) - f(x)}{h} = -6x - 3h + 4
\]

### Final Answer:

\[
\frac{f(x+h) - f(x)}{h} = -6x + 4 - 3h
\]

This is the difference quotient for the function \( f(x) = -3x^2 + 4x - 5 \).

---

Do you want any further details on this, or do you have any questions? Here are five related questions you can explore:

1. How is the difference quotient related to the derivative of a function?
2. What happens to the difference quotient as \( h \) approaches 0?
3. Can you calculate the derivative of the given function \( f(x) = -3x^2 + 4x - 5 \)?
4. What is the geometric interpretation of the difference quotient?
5. How would the difference quotient change for a cubic function?

**Tip:** The difference quotient is an important tool in calculus because it's used to define the derivative of a function, representing the slope of the tangent line at a point.

Calculate the difference quotient, (f(x+h) - f(x)) / h, for the function f(x) = -3x^2 + 4x - 5.

This problem involves calculating the difference quotient for the quadratic function f(x) = -3x^2 + 4x - 5. Learn the step-by-step process for simplifying and applying the difference quotient formula.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation