find the equation of the tangent line to the curve at the given point by eliminating the parameter:
x=1+sqrt(t) y=e^(t^2) parameter: (2,e)
x = 1+√t
t = (x-1)^2
y = e^(x-1)^4
y' = e^(x-1)^4 * 4(x-1)^3
y'(2) = 4e
Check. Using the parameters, at t=1,
dy/dt = 2t e^t^2 = 2e
dx/dt = 1/(2√t) = 1/2
dy/dx = 2e/(1/2) = 4e
To find the equation of the tangent line to the curve at the given point, we need to eliminate the parameter "t" from the equations of the curve.
The given equations are:
x = 1 + √t
y = e^(t^2)
First, we need to find the value of "t" that corresponds to the given point, (2, e).
Given that x = 2, we can solve the equation "2 = 1 + √t" to find the corresponding value of "t".
2 = 1 + √t
√t = 2 - 1
√t = 1
t = 1 (squaring both sides)
Now, we have the value of "t" as 1. We will substitute this value of "t" into the equation for "y" to find the corresponding value of "y".
y = e^(t^2)
y = e^(1^2)
y = e
So, the given point (2, e) corresponds to t = 1.
To find the equation of the tangent line, we need to find the derivative of the parametric equations with respect to "t".
Differentiating x with respect to t:
dx/dt = d/dt (1 + √t)
dx/dt = (1/2) * (t)^(-1/2)
dx/dt = (1/2√t)
Differentiating y with respect to t:
dy/dt = d/dt (e^(t^2))
dy/dt = 2t * e^(t^2)
Now, we have the derivatives of x and y with respect to t. To find the slope of the tangent line at t = 1, we substitute t = 1 into the derivatives.
dx/dt = (1/2√t)
dx/dt = (1/2√1)
dx/dt = 1/2
dy/dt = 2t * e^(t^2)
dy/dt = 2(1) * e^(1^2)
dy/dt = 2e
The slope of the tangent line at t = 1 is dy/dt divided by dx/dt:
slope = (dy/dt) / (dx/dt)
slope = (2e) / (1/2)
slope = 4e
Now, we have the slope of the tangent line. To find the equation of the tangent line, we need a point on the line. Since we are given the point (2, e), we can use the point-slope form of a line to find the equation.
Using the point-slope form: y - y1 = m(x - x1)
Substituting the values:
y - e = 4e(x - 2)
Expanding and rearranging:
y = 4ex - 7e
Hence, the equation of the tangent line to the given curve at the point (2, e) is y = 4ex - 7e.