https://file.aiquickdraw.com/mathbot/file/17278609978203ustodfe.jpg

Let's solve the problem step by step from the image.

### Problem:
We are asked to calculate the tangent and normal for the function \( y = x\sqrt{x} + 2 \) at the point \( (x_0 = 4, y_0 = x_0\sqrt{x_0} + 2) \).

### Step 1: Find the equation of the tangent line
To find the equation of the tangent line at the given point, we need the derivative of the function, which gives us the slope of the tangent line.

The function is:
\[
y = x\sqrt{x} + 2 = x \cdot x^{1/2} + 2 = x^{3/2} + 2
\]

#### Derivative of the function \( y = x^{3/2} + 2 \):
\[
\frac{dy}{dx} = \frac{d}{dx}\left( x^{3/2} + 2 \right) = \frac{3}{2} x^{1/2}
\]

#### Slope of the tangent line at \( x_0 = 4 \):
Substitute \( x_0 = 4 \) into the derivative:
\[
\frac{dy}{dx}\bigg|_{x=4} = \frac{3}{2} \cdot 4^{1/2} = \frac{3}{2} \cdot 2 = 3
\]
So, the slope of the tangent line is \( m_t = 3 \).

#### Point \( (x_0, y_0) \):
Now, we calculate \( y_0 \) using the original function at \( x_0 = 4 \):
\[
y_0 = 4\sqrt{4} + 2 = 4 \cdot 2 + 2 = 8 + 2 = 10
\]
Thus, the point is \( (4, 10) \).

#### Tangent line equation:
Using the point-slope form \( y - y_1 = m(x - x_1) \), with \( m = 3 \) and \( (x_1, y_1) = (4, 10) \):
\[
y - 10 = 3(x - 4)
\]
Simplifying:
\[
y_t = 3x - 12 + 10 \quad \Rightarrow \quad y_t = 3x - 2
\]

### Step 2: Find the equation of the normal line
The slope of the normal line is the negative reciprocal of the tangent slope. Since the slope of the tangent line is \( 3 \), the slope of the normal line is:
\[
m_n = -\frac{1}{3}
\]

#### Normal line equation:
Using the point-slope form for the normal line:
\[
y - 10 = -\frac{1}{3}(x - 4)
\]
Simplifying:
\[
y - 10 = -\frac{1}{3}x + \frac{4}{3}
\]
\[
y_n = -\frac{1}{3}x + \frac{4}{3} + 10 = -\frac{1}{3}x + \frac{34}{3}
\]

### Final Answers:
- The algebraic expression for the vertical coordinate in the tangent equation in terms of \( x \) is:
\[
y_t = 3x - 2
\]
- The algebraic expression for the vertical coordinate in the normal equation in terms of \( x \) is:
\[
y_n = -\frac{1}{3}x + \frac{34}{3}
\]

Would you like further details or clarifications on any part of this? Here are 5 questions that build upon this problem:

1. How do we differentiate a composite function like \( y = x\sqrt{x} \)?
2. What is the geometric significance of the tangent line at a point on a curve?
3. How would you calculate the normal line if the function was more complex?
4. Why is the slope of the normal line the negative reciprocal of the tangent line’s slope?
5. What are some common uses of tangent and normal lines in physics or engineering?

**Tip:** When working with functions involving roots, rewrite them using exponents for easier differentiation.

Calculate the tangent and normal for the function y = x√x + 2 at point (x0 = 4, y0 = x0√x0 + 2). What are the algebraic expressions for the vertical coordinates in the tangent and normal equations in terms of x?

Learn how to calculate the tangent and normal lines for the function y = x√x + 2 at the point (4, 10). This problem involves differentiation and uses the point-slope form for both tangent and normal lines. Suitable for advanced 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