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*) = 4*x*2 + 7*x*

Step 1:

*f*(*x* + *h*)

=
 
Step 2:

*f*(*x* + *h*) − *f*(*x*)

=
 
Step 3:

*f*(*x* + *h*) − *f*(*x*)

*h*
 
=
 
Step 4:

*f* *'*(*x*) = lim *h*→0 

*f*(*x* + *h*) − *f*(*x*)

*h*
 
=


To find the slope of the tangent line to the graph of the function \( f(x) = 4x^2 + 7x \) using the four-step process, we will follow the steps you've outlined.

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

To find \( f(x + h) \):

\[
f(x + h) = 4(x + h)^2 + 7(x + h)
\]

Expanding this:

\[
f(x + h) = 4(x^2 + 2xh + h^2) + 7x + 7h
\]
\[
= 4x^2 + 8xh + 4h^2 + 7x + 7h
\]
\[
= 4x^2 + 7x + 8xh + 7h + 4h^2
\]

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

Now we calculate:

\[
f(x + h) - f(x) = (4x^2 + 7x + 8xh + 7h + 4h^2) - (4x^2 + 7x)
\]

This simplifies to:

\[
= 8xh + 7h + 4h^2
\]

### Step 3: Find \( \frac{f(x + h) - f(x)}{h} \)

Now, divide the result from Step 2 by \( h \):

\[
\frac{f(x + h) - f(x)}{h} = \frac{8xh + 7h + 4h^2}{h}
\]

Simplifying gives:

\[
= 8x + 7 + 4h
\]

### Step 4: Find the limit as \( h \to 0 \)

Now, we find the limit:

\[
f'(x) = \lim_{h \to 0} (8x + 7 + 4h)
\]

As \( h \) approaches 0, the \( 4h \) term goes to 0:

\[
f'(x) = 8x + 7
\]

### Final Answer

The slope of the tangent line to the graph of \( f(x) \) at any point \( x \) is given by:

\[
f'(x) = 8x + 7
\]

If you have any questions or need further details, feel free to ask! Here are some related questions:

1. What does the slope represent in the context of the function?
2. How would the slope change if we modified the function?
3. Can you find the slope at a specific point, say \( x = 1 \)?
4. What is the significance of the tangent line in calculus?
5. How would you graph the function and its tangent line at a specific point?

**Tip:** Always check your calculations by substituting values back into the original function to ensure accuracy.

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) = 4x^2 + 7x

Step-by-step guide to finding the slope of the tangent line to the graph of f(x) = 4x^2 + 7x using the four-step derivative process. This calculus problem involves key concepts in limits and derivatives, ideal for high school students in Grade 11-12.

Math Problem Statement

h

h

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation