Math Problem Statement
Find the acute angles between the curves at their points of intersection. (The angle between two curves is the angle between their tangent lines at the point of intersection. Give your answers in degrees, rounding to one decimal place. Enter your answers as a comma-separated list.) y = 7x2, y = 7x3
Solution
To find the acute angle between the curves and at their points of intersection, we follow these steps:
1. Find the points of intersection
To find the points where the two curves intersect, set the equations equal to each other:
Divide both sides by 7:
Factor the equation:
This gives two solutions:
Thus, the points of intersection are and .
2. Find the slopes of the tangent lines at each point
The slope of the tangent line to a curve at a given point is found by differentiating the equation of the curve.
For , the derivative is:
For , the derivative is:
3. Calculate the angle between the tangent lines
The formula for the angle between two curves at their point of intersection, based on their slopes and , is given by:
For each point of intersection, we calculate the slopes and apply the formula.
At :
- The slope of the tangent to is .
- The slope of the tangent to is .
So, , and the angle between them is:
Thus, the angle between the curves at is .
At :
- The slope of the tangent to is .
- The slope of the tangent to is .
Now, apply the formula for the angle:
Now, find :
Final Answer:
The acute angles between the curves at their points of intersection are approximately:
Would you like a detailed explanation of any of the steps, or have any additional questions?
Here are some related questions:
- How do we calculate the angle between two curves when their tangents are not perpendicular?
- How can we determine whether two curves are tangent at a point of intersection?
- How does the angle between the curves change if we have cubic vs quadratic functions?
- What is the significance of the angle between two curves in real-world applications?
- How do the slopes of tangents relate to the concavity of curves?
Tip: When finding the angle between curves, ensure you first find their points of intersection before calculating the slopes of their tangent lines.
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Math Problem Analysis
Mathematical Concepts
Calculus
Differentiation
Tangents
Intersection of Curves
Formulas
tan(θ) = |(m1 - m2) / (1 + m1 * m2)|
Theorems
Angle between two curves at a point of intersection
Suitable Grade Level
Grades 11-12
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