.At what points of the curve y=x2+x−2is it tangent parallel to the line y=4x−1.

We are tasked with finding the points on the curve \( y = x^2 + x - 2 \) where the tangent line is parallel to the line \( y = 4x - 1 \).

### Step 1: Find the slope of the given line
The line \( y = 4x - 1 \) is in slope-intercept form \( y = mx + b \), where the slope \( m \) is 4.

### Step 2: Find the derivative of the curve
To find the points where the tangent line to the curve is parallel to \( y = 4x - 1 \), we need to compute the derivative of the curve \( y = x^2 + x - 2 \), which will give us the slope of the tangent line at any point \( x \).

\[
\frac{dy}{dx} = 2x + 1
\]

### Step 3: Set the derivative equal to the slope of the line
For the tangent to be parallel to the line \( y = 4x - 1 \), the slope of the tangent must be equal to 4. So we set the derivative equal to 4:

\[
2x + 1 = 4
\]

### Step 4: Solve for \( x \)
\[
2x = 4 - 1
\]
\[
2x = 3
\]
\[
x = \frac{3}{2}
\]

### Step 5: Find the corresponding \( y \)-coordinate
Now that we have \( x = \frac{3}{2} \), we substitute this back into the original curve equation \( y = x^2 + x - 2 \) to find the corresponding \( y \)-coordinate.

\[
y = \left( \frac{3}{2} \right)^2 + \frac{3}{2} - 2
\]
\[
y = \frac{9}{4} + \frac{3}{2} - 2
\]
\[
y = \frac{9}{4} + \frac{6}{4} - \frac{8}{4}
\]
\[
y = \frac{7}{4}
\]

### Final answer:
The point on the curve where the tangent line is parallel to the line \( y = 4x - 1 \) is \( \left( \frac{3}{2}, \frac{7}{4} \right) \).

Would you like more details on any specific step?

Here are 5 related questions:
1. How do you find the slope of a tangent line at any point on a curve?
2. Can you explain the significance of the derivative in finding tangent lines?
3. What happens when a tangent line has the same slope as a given line?
4. How can we apply this method to other curves, such as \( y = x^3 - x \)?
5. How would the problem change if the given line were \( y = -4x + 1 \)?

**Tip**: Always check the derivative to ensure the slope matches the given condition when solving problems involving tangents.

At what points of the curve y = x^2 + x - 2 is it tangent parallel to the line y = 4x - 1?

Discover how to find the points on the curve y = x^2 + x - 2 where the tangent line is parallel to y = 4x - 1. This step-by-step solution involves differentiation, solving for slopes, and applying algebraic methods. Ideal 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