Question:
For the curve \(y = 4x^3 − 2x^5\) , find all the points at which the tangents passes through the origin.
For the curve \(y = 4x^3 − 2x^5\) , find all the points at which the tangents passes through the origin.
Updated On: Sep 14, 2023
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Solution and Explanation
The equation of the given curve is y = 4x3 − 2x5
\(\frac {dy}{dx}\) = 12x2 - 10x4
Therefore, the slope of the tangent at a point (x, y) is 12x2−10x4 .
The equation of the tangent at (x, y) is given by
When x = 0, y = 4(0)3 - 2(0)5 = 0.
When x = 1, y = 4(1)3 − 2 (1)5 = 2.
When x = −1, y = 4(−1)3 − 2(−1)5 = −2.
Hence, the required points are (0, 0), (1, 2), and (−1, −2).
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Notes on Applications of Derivatives
Concepts Used:
Tangents and Normals
- A tangent at a degree on the curve could be a straight line that touches the curve at that time and whose slope is up to the derivative of the curve at that point. From the definition, you'll be able to deduce the way to realize the equation of the tangent to the curve at any point.
- Given a function y = f(x), the equation of the tangent for this curve at x = x0
- Slope of tangent (at x=x0) m=dy/dx||x=x0
- A normal at a degree on the curve is a line that intersects the curve at that time and is perpendicular to the tangent at that point. If its slope is given by n, and also the slope of the tangent at that point or the value of the derivative at that point is given by m. then we got
m×n = -1
- The normal to a given curve y = f(x) at a point x = x0
- The slope ‘n’ of the normal: As the normal is perpendicular to the tangent, we have: n=-1/m
Diagram Explaining Tangents and Normal:
