Question

For the following exercises, write the equation of the tangent line in Cartesian coordinates for the given parameter t

x=et ,y=(t-1)2 ,at(1,1)

Answer 1

y+2x = 3

Explanation:

The formula for finding the equation of a tangent line is expressed using the point slope equation y-y0 = m(x-x0) where:

m is the slope of the line and (x0, y0) is a point on the line.

m = dy/dx = dy/dt ÷ dx/dt

dy/dt = 2(t-1)

dx/dt = e^t

m = 2(t-1)/e^t

If x = e^t, and x = 1

1 = e^t

Taking ln of both sides

ln1 = lne^t

t = ln1

t = 0

Substituting t = 0 into the slope formula:

m = 2(t-1)/e^t

m = 2(0-1)/e^0

m = 2(-1)/1

m = -2/1

m = -2

Substituting the slope m= -2 and the given point (1,1) into the formula for calculating the equation of tangent line we will have;

y-1 = -2(x-1)

y-1 = -2x+2

y+2x = 2+1

y+2x = 3

Hence the equation of the tangent line in Cartesian coordinates for the given parameter t is y+2x = 3