Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. (Simplify your answers completely.)
f(x) = −x2 + 7x
Step 1:	
f(x + h)

Let's start by finding \( f(x + h) \) for the given function \( f(x) = -x^2 + 7x \).

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

Given:
\[
f(x) = -x^2 + 7x
\]

Now, substitute \( x + h \) into the function:
\[
f(x + h) = -(x + h)^2 + 7(x + h)
\]

First, expand the terms:
\[
-(x + h)^2 = -(x^2 + 2xh + h^2) = -x^2 - 2xh - h^2
\]
\[
7(x + h) = 7x + 7h
\]

Now, combine these results:
\[
f(x + h) = -x^2 - 2xh - h^2 + 7x + 7h
\]

So, the simplified expression for \( f(x + h) \) is:
\[
f(x + h) = -x^2 - 2xh - h^2 + 7x + 7h
\]

This is the result for Step 1.

Would you like to proceed with Step 2? Let me know if you need any clarification or further details.

---

Here are 5 related questions you might explore:
1. What is the purpose of finding the difference quotient?
2. How does the limit process affect the calculation of the slope of a tangent line?
3. What is the difference between a secant line and a tangent line in terms of slope?
4. How do you simplify difference quotients with polynomials?
5. What does the derivative of a function represent geometrically?

**Tip:** Always expand and simplify algebraic expressions fully before proceeding to limits in calculus.

Use the four-step process to find the slope of the tangent line to the graph of the given function at any point. (Simplify your answers completely.) f(x) = −x^2 + 7x Step 1: f(x + h)

Learn how to use the four-step process to find the slope of the tangent line to the graph of the quadratic function f(x) = -x^2 + 7x. This step-by-step solution simplifies the difference quotient and covers key calculus concepts like tangent lines and the derivative. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation