https://file.aiquickdraw.com/mathbot/file/1725485684172rgjo6inw.jpg

The image contains a differential equation, some provided values, and a set of problems for evaluation.

The equation given is:

\[ y(x) = x^2 + 3x \]

The task is to evaluate various expressions based on this function \( y(x) \), where \( x = 3 \).

The expressions to evaluate are:

a. \( (y + y') |_{x=3} \)

b. \( (xy) |_{x=3} \)

c. \( (y - y') |_{x=3} \)

d. \( \left( \frac{y}{x} \right) |_{x=3} \)

Let’s proceed step by step.

### 1. \( y(x) = x^2 + 3x \)
First, we will evaluate \( y(x) \) at \( x = 3 \):
\[
y(3) = 3^2 + 3(3) = 9 + 9 = 18
\]

### 2. \( y'(x) \)
Differentiate \( y(x) \) with respect to \( x \):
\[
y'(x) = \frac{d}{dx}(x^2 + 3x) = 2x + 3
\]
Now, evaluate \( y'(x) \) at \( x = 3 \):
\[
y'(3) = 2(3) + 3 = 6 + 3 = 9
\]

### Now evaluate each expression at \( x = 3 \):

**a.** \( (y + y') |_{x=3} = y(3) + y'(3) = 18 + 9 = 27 \)

**b.** \( (xy) |_{x=3} = 3 \cdot y(3) = 3 \cdot 18 = 54 \)

**c.** \( (y - y') |_{x=3} = y(3) - y'(3) = 18 - 9 = 9 \)

**d.** \( \left( \frac{y}{x} \right) |_{x=3} = \frac{y(3)}{3} = \frac{18}{3} = 6 \)

### Final answers:

a. \( 27 \)

b. \( 54 \)

c. \( 9 \)

d. \( 6 \)

Would you like more details or have any questions? 

Here are 5 related questions you might find helpful:
1. How do we differentiate composite functions like \( y(x) = f(g(x)) \)?
2. What is the product rule for differentiation, and when do we use it?
3. How do you compute higher-order derivatives of a function?
4. What is the significance of finding the derivative in real-life applications?
5. How does the chain rule help in differentiation?

**Tip:** Always verify that you correctly substitute the value of \( x \) after differentiation to avoid any errors in calculation!

Given y(x) = x^2 + 3x and y'(x) = 2x + 3, evaluate the following expressions at x = 3: (y + y'), (xy), (y - y'), and (y/x).

Learn how to evaluate the function y(x) = x^2 + 3x and its derivative y'(x) at x = 3. This example covers basic differentiation rules and algebraic function evaluation, suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation