https://file.aiquickdraw.com/mathbot/file/172779822998493c7e2ox.jpg

We are tasked with finding the equation of the tangent line to the function \( f(x) = 5x^4 + 7x + 1 \) at \( x = 1 \) in point-slope form.

The point-slope form of a line is:
\[
y - y_0 = m(x - x_0)
\]
where \( m \) is the slope, and \( (x_0, y_0) \) is the point of tangency. Now, let's go step by step:

### Step 1: Identify \( x_0 \)
Since we are asked for the tangent line at \( x = 1 \), we have:
\[
x_0 = 1
\]

### Step 2: Calculate \( y_0 = f(x_0) \)
Now, we substitute \( x = 1 \) into the function \( f(x) \) to find \( y_0 \):
\[
f(1) = 5(1)^4 + 7(1) + 1 = 5 + 7 + 1 = 13
\]
So, \( y_0 = 13 \).

### Step 3: Find the slope \( m = f'(x_0) \)
To find the slope of the tangent line, we need to differentiate \( f(x) \):
\[
f'(x) = \frac{d}{dx} (5x^4 + 7x + 1) = 20x^3 + 7
\]
Now, substitute \( x = 1 \) to find the slope at that point:
\[
f'(1) = 20(1)^3 + 7 = 20 + 7 = 27
\]
Thus, \( m = 27 \).

### Summary of values:
- \( x_0 = 1 \)
- \( y_0 = 13 \)
- \( m = 27 \)

Would you like further clarification or details on these steps?

### Follow-up questions:
1. How do you differentiate polynomials step-by-step?
2. What is the geometric interpretation of the slope of a tangent line?
3. How can we use limits to find the slope of a tangent line?
4. How would the equation of the tangent line change if \( f(x) \) was a trigonometric function?
5. What is the significance of finding tangent lines in calculus?

**Tip**: When solving problems involving tangent lines, always ensure that both the point of tangency and the slope are calculated accurately from the derivative.

Suppose f(x) = 5x^4 + 7x + 1. If we were to express the equation of the tangent line to f(x) at x = 1 in point-slope form, what would be the following values?

Learn how to find the equation of the tangent line to the function f(x) = 5x^4 + 7x + 1 at x = 1 using the point-slope form. This problem involves differentiation and understanding how to apply the point-slope formula. 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