Find the point where the tangent line to
y
=
x
4
at
x
=
0.4
crosses the x-axis.

To find the point where the tangent line to \( y = x^4 \) at \( x = 0.4 \) crosses the x-axis, we need to follow these steps:

### Step 1: Find the derivative of \( y = x^4 \)
The derivative \( y' \) gives us the slope of the tangent line at any point \( x \).

\[
y' = \frac{d}{dx}(x^4) = 4x^3
\]

### Step 2: Evaluate the slope at \( x = 0.4 \)
Substitute \( x = 0.4 \) into the derivative to find the slope at that point:

\[
y'(0.4) = 4(0.4)^3 = 4(0.064) = 0.256
\]

So, the slope of the tangent line at \( x = 0.4 \) is 0.256.

### Step 3: Find the point on the curve at \( x = 0.4 \)
Now, substitute \( x = 0.4 \) into the original equation \( y = x^4 \) to find the corresponding \( y \)-coordinate:

\[
y(0.4) = (0.4)^4 = 0.0256
\]

Thus, the point of tangency is \( (0.4, 0.0256) \).

### Step 4: Use the point-slope form of the line
The equation of the tangent line is given by the point-slope form:

\[
y - y_1 = m(x - x_1)
\]

where \( m \) is the slope, and \( (x_1, y_1) \) is the point of tangency. Substituting the values we have:

\[
y - 0.0256 = 0.256(x - 0.4)
\]

Simplifying:

\[
y - 0.0256 = 0.256x - 0.1024
\]

\[
y = 0.256x - 0.1024 + 0.0256
\]

\[
y = 0.256x - 0.0768
\]

### Step 5: Find the x-intercept
To find where the tangent line crosses the x-axis, set \( y = 0 \) and solve for \( x \):

\[
0 = 0.256x - 0.0768
\]

\[
0.256x = 0.0768
\]

\[
x = \frac{0.0768}{0.256} = 0.3
\]

Thus, the tangent line crosses the x-axis at \( x = 0.3 \). The point is \( (0.3, 0) \).

---

Would you like any more details or clarifications? Here are five related questions to deepen your understanding:

1. How do you find the equation of a tangent line at a given point?
2. How can you verify if a point is on the tangent line?
3. What is the geometric significance of the x-intercept of a tangent line?
4. Can a tangent line have no x-intercept? Under what conditions?
5. How does the behavior of the tangent line change for different powers of \( x \) (e.g., \( x^2 \), \( x^3 \), \( x^5 \))?

**Tip:** When finding the tangent line at a point, always first differentiate the function to get the slope at that point.

Find the point where the tangent line to y = x^4 at x = 0.4 crosses the x-axis.

Discover how to calculate the point where the tangent line to the function y = x^4 at x = 0.4 crosses the x-axis. This solution includes the derivative of x^4, finding the slope at x = 0.4, and using the point-slope form of the line. Perfect for calculus students in Grades 11-12.

Math Problem Statement

Find the point where the tangent line to y

x 4 at x

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation