https://file.aiquickdraw.com/mathbot/file/1727886872892tmip366x.jpg

The image shows a question with two parts involving the function \( f(x) = 6x + 8x^2 \).

- In **part (a)**, the derivative \( f'(x) \) has been correctly computed as \( f'(x) = 6 + 16x \).
  
- In **part (b)**, you are asked to find the equation of the tangent line at \( x = 5 \).

To solve part (b), follow these steps:

### 1. Find \( f'(5) \) (the slope of the tangent line at \( x = 5 \)):
Using the derivative \( f'(x) = 6 + 16x \):
\[
f'(5) = 6 + 16(5) = 6 + 80 = 86
\]
So, the slope of the tangent line at \( x = 5 \) is 86.

### 2. Find \( f(5) \) (the value of the function at \( x = 5 \)):
Using the original function \( f(x) = 6x + 8x^2 \):
\[
f(5) = 6(5) + 8(5^2) = 30 + 8(25) = 30 + 200 = 230
\]
So, the point on the curve is \( (5, 230) \).

### 3. Use the point-slope form of the line equation:
The point-slope form is given by:
\[
y - y_1 = m(x - x_1)
\]
where \( m \) is the slope, and \( (x_1, y_1) \) is the point on the line. Substituting \( m = 86 \), \( x_1 = 5 \), and \( y_1 = 230 \):
\[
y - 230 = 86(x - 5)
\]
Simplifying:
\[
y - 230 = 86x - 430
\]
\[
y = 86x - 430 + 230
\]
\[
y = 86x - 200
\]
Thus, the equation of the tangent line at \( x = 5 \) is:
\[
y = 86x - 200
\]

Would you like further details or clarifications on any part of the process?

---

Here are some related questions for further practice:
1. How do you compute the equation of the tangent line at \( x = 0 \) for the same function?
2. What is the tangent line equation for \( x = 3 \) using the same steps?
3. How does the tangent line slope change as \( x \) increases?
4. What happens to the value of \( f'(x) \) as \( x \to \infty \)?
5. Can you find where the tangent line is horizontal (i.e., where \( f'(x) = 0 \))?

**Tip**: Always verify both the derivative and the point-slope form steps to ensure accuracy in tangent line problems!

Use the function f(x) = 6x + 8x^2 to find the following:
(a) Use the limit definition of the derivative to find f'(x).
(b) Find the equation of the tangent line at x = 5.

Learn how to calculate the derivative of f(x) = 6x + 8x^2 and find the equation of the tangent line at x = 5. This example covers key concepts in derivatives and the use of the point-slope form for tangent lines. Suitable for high school calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation