Math Problem Statement
Solution
Let's analyze the problem in the image you've uploaded step by step:
Problem Overview:
- We are given a point on the curve .
- The tangent line at point intersects the curve again at another point .
- We're tasked with finding certain information about this intersection.
Part (a): Equation of the Tangent Line at
Given that has an -coordinate of , we want to find the equation of the tangent line at .
- The function is .
- The slope of the tangent line at any point is given by the derivative of with respect to , so: At , where , the slope is: The equation of a tangent line is generally given by: At point , we plug in the values to get: Simplifying: So, the equation of the tangent line is .
Part (b): Intersection of the Tangent Line with the Curve
We are told that the tangent line intersects the curve at two points: one is , and we need to find the coordinates of the other point .
- Set the equation of the tangent line equal to the curve's equation: Rearrange this into a cubic equation: We know is a solution (since is on both the tangent line and the curve). So, we can factor out of the cubic equation: Now solve for the remaining factor . Using the quadratic formula: Simplifying: So, (which we already know) or .
Thus, the coordinates of the second intersection point are .
Part (c): Ratio of the Slopes at and
We are asked to find the ratio of the slope of the curve at to the slope at .
- The slope at any point on the curve is given by .
- At (where ), the slope is .
- At (where ), the slope is:
- The ratio of the slope at to the slope at is:
Thus, the ratio is .
Let me know if you would like any further clarifications!
5 Related Questions:
- How would the result change if the curve were instead of ?
- What happens to the tangent line equation if is at ?
- How can you solve higher-degree polynomial intersections like this problem more efficiently?
- What is the significance of the ratio of the slopes at different points of a curve geometrically?
- How would this problem differ if we worked with an implicit curve equation like ?
Tip:
Always check if the equation you're dealing with is factorable by known solutions to simplify your work. In this case, knowing as a root helped us quickly factor the cubic equation.
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Math Problem Analysis
Mathematical Concepts
Calculus
Derivatives
Polynomial Functions
Algebra
Formulas
Slope of tangent line: dy/dx = 3x^2
Tangent line equation: y - y1 = m(x - x1)
Quadratic formula: x = [-b ± sqrt(b^2 - 4ac)] / 2a
Theorems
Derivative of a Polynomial
Factor Theorem
Suitable Grade Level
Grades 11-12
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