Find the points on the curve y = x^3 at which the slope of the tangent is equal to the y-coordinate of the point.
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given curve , y = x³
we know,slope of tangent is 1st order derivatives of the curve at the given point.
so, differentiate y with respect to x,
dy/dx = 3x²
Let the required point on the curve is (a, b).
slope of tangent at (a, b) = 3a² ------(1)
A/C to question,
slope of tangent is equal to the y - co - ordinate of the point. so, slope of tangent , dy/dx = b ----(2)
from equations (1) and (2),
b = 3a² ------(3)
put (a, b) on the curve, y = x³
e.g., b = a³ -----(4)
so, from equation (3) and (4),
a³ = 3a² => a²(a - 3) = 0
=> a = 0, 3
put a = 0 in the curve b = a³ => b = 0
put a = 3 in the curve , b = 3³ => b = 27
hence, required points (0,0) and (3,27)
we know,slope of tangent is 1st order derivatives of the curve at the given point.
so, differentiate y with respect to x,
dy/dx = 3x²
Let the required point on the curve is (a, b).
slope of tangent at (a, b) = 3a² ------(1)
A/C to question,
slope of tangent is equal to the y - co - ordinate of the point. so, slope of tangent , dy/dx = b ----(2)
from equations (1) and (2),
b = 3a² ------(3)
put (a, b) on the curve, y = x³
e.g., b = a³ -----(4)
so, from equation (3) and (4),
a³ = 3a² => a²(a - 3) = 0
=> a = 0, 3
put a = 0 in the curve b = a³ => b = 0
put a = 3 in the curve , b = 3³ => b = 27
hence, required points (0,0) and (3,27)
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