Question
the tangent to the curve y = ax^3 at the point where x = 1 passes through the point( 2 , 8). find the value of a
Answers
GPT 3.5
To find the value of a, we first need to find the derivative of the curve y = ax^3.
Taking the derivative of y = ax^3 with respect to x, we get:
dy/dx = 3ax^2
The derivative represents the slope of the tangent line at any given point. We are given that the tangent line passes through the point (2, 8). Let's call the slope of the tangent line m.
m = dy/dx at x = 1
m = 3a(1)^2
m = 3a
Since the tangent line passes through (2, 8), we can use the point-slope form of a line to write the equation of the tangent line:
y - y1 = m(x - x1)
y - 8 = 3a(x - 1)
y = 3ax + (8 - 3a)
Since the tangent line passes through (2, 8), we can substitute these values into the equation and solve for a:
8 = 3a(2) + (8 - 3a)
8 = 6a + (8 - 3a)
8 = 6a + 8 - 3a
8 = 8 + 3a - 3a
8 = 8
We get 8 = 8, which is true for any value of a. Therefore, there is no unique value of a that satisfies the given conditions.
Taking the derivative of y = ax^3 with respect to x, we get:
dy/dx = 3ax^2
The derivative represents the slope of the tangent line at any given point. We are given that the tangent line passes through the point (2, 8). Let's call the slope of the tangent line m.
m = dy/dx at x = 1
m = 3a(1)^2
m = 3a
Since the tangent line passes through (2, 8), we can use the point-slope form of a line to write the equation of the tangent line:
y - y1 = m(x - x1)
y - 8 = 3a(x - 1)
y = 3ax + (8 - 3a)
Since the tangent line passes through (2, 8), we can substitute these values into the equation and solve for a:
8 = 3a(2) + (8 - 3a)
8 = 6a + (8 - 3a)
8 = 6a + 8 - 3a
8 = 8 + 3a - 3a
8 = 8
We get 8 = 8, which is true for any value of a. Therefore, there is no unique value of a that satisfies the given conditions.