https://r2.erweima.ai/mathbot/image/17268022968878u0jg45f.jpg

The image contains a set of problems where the goal is to find the equation of the tangent line to the given function \( f(x) \) at the specified value of \( x \). This involves two main steps:

1. **Finding the derivative \( f'(x) \)** to get the slope of the tangent line at the given point \( x \).
2. **Using the point-slope form** of the equation of a line to determine the equation of the tangent line.

I'll help with one example, and you can follow the same steps for the others.

### Example Problem 7:
\( f(x) = x^2 - 6 \), and \( x = 3 \)

1. **Step 1: Find the derivative of \( f(x) \)**:
   \[
   f'(x) = \frac{d}{dx}(x^2 - 6) = 2x
   \]
2. **Step 2: Evaluate the derivative at \( x = 3 \)** to find the slope:
   \[
   f'(3) = 2(3) = 6
   \]
   So, the slope of the tangent line at \( x = 3 \) is 6.

3. **Step 3: Find the point on the function at \( x = 3 \)**:
   \[
   f(3) = 3^2 - 6 = 9 - 6 = 3
   \]
   So, the point of tangency is \( (3, 3) \).

4. **Step 4: Use the point-slope form of the line equation** \( y - y_1 = m(x - x_1) \), where \( m = 6 \), \( x_1 = 3 \), and \( y_1 = 3 \):
   \[
   y - 3 = 6(x - 3)
   \]
   Simplifying:
   \[
   y - 3 = 6x - 18
   \]
   \[
   y = 6x - 15
   \]
   So, the equation of the tangent line is \( y = 6x - 15 \).

You can apply this procedure for the remaining problems.

Would you like help with another specific problem from the list? 

### Here are 5 related questions to consider:
1. What is the general method to compute the tangent line of a curve at a point?
2. How can you apply the chain rule when differentiating more complex functions?
3. What does the derivative tell you about the behavior of the function at a given point?
4. How would you handle functions involving fractions when calculating the derivative?
5. How can the second derivative give additional insights into the curvature of the graph?

### Tip:
Always double-check the derivative, especially when dealing with polynomials or fractions, to ensure accurate slope calculation for the tangent line!

In the problems 7-18, use the derivative to find the slope of the tangent line to the graph of the given function at the value of x. Then find the equation of the tangent line at the corresponding point.

Learn how to find the slope of the tangent line to a function at a given point using derivatives. Problems include functions like polynomials and rational expressions. This is ideal for students in Grades 10-12, covering key calculus concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation