Find the points on the curve y = x3 at which the slope of the tangent is equal to the y-coordinate of the point.

Asked by Topperlearning User | 7th Aug, 2014, 10:14: AM

Expert Answer:

Let the required point on the curve y = x3 be P(x1, y1).

We have,
straight y equals straight x cubed rightwards double arrow dy over dx equals 3 straight x squared rightwards double arrow open parentheses dy over dx close parentheses subscript left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis end subscript equals 3 straight x subscript 1 squared
It is given that
Slope of the tangent at P(x1, y1) = Ordinate of P(x1, y1)
rightwards double arrow open parentheses dy over dx close parentheses subscript left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis end subscript equals straight y subscript 1
rightwards double arrow 3 straight x subscript 1 squared equals straight y subscript 1 space end subscript open square brackets because left parenthesis straight x subscript 1 comma straight y subscript 1 right parenthesis space lies space on space straight y equals straight x cubed rightwards double arrow straight y subscript 1 equals straight x subscript 1 cubed close square brackets
rightwards double arrow 3 straight x subscript 1 squared equals straight x subscript 1 cubed
rightwards double arrow straight x subscript 1 cubed minus 3 straight x subscript 1 squared equals 0
rightwards double arrow straight x subscript 1 squared open parentheses straight x subscript 1 minus 3 close parentheses equals 0
rightwards double arrow straight x subscript 1 equals 0 space comma straight x subscript 1 equals 3
Since (x1, y1) lies on y = x3. Therefore, y subscript 1 equals x subscript 1 cubed
Now, x1 = 0 Þ y1 = 0 and x1 = 3 Þ y1 = 33 = 27
Hence, required points are (0, 0) and (3, 27).

Answered by  | 7th Aug, 2014, 12:14: PM