Question

Find the equation of the tangent line to the curve f(x) = 1+e^-2x where it cuts the line y=2 is

Answer 1

Please see the attachment

Answer 2

2x + y = 2 is the equation of the tangent line to the curve f(x) = 1+e⁻²ˣ where it cuts the line y=2 is

Given:

• f(x) = 1 + e⁻²ˣ

To Find:

• Equation of the tangent line to the curve f(x) = 1+e⁻²ˣ where it cuts the line y=2 is

Solution:

Step 1:

Equate  1 + e⁻²ˣ with 2 and solve for x and find intersection point

1 + e⁻²ˣ = 2

=> e⁻²ˣ = 1

=> -2x = 0

=> x = 0

Hence intersection point is ( 0 , 2)

Step 2:

Find first derivative of f(x) = 1 + e⁻²ˣ

f'(x) =  -2e⁻²ˣ

Step 3:

Substitute x= 0 to find slope

Slope = -2e⁰   = - 2

Step 4:

Equation of line y - y₁ = m(x - x₁)   where x₍ = 0 and y₁ = 2

y - 2 = -2(x - 0)

=> y = -2x + 2

=> 2x + y = 2

2x + y = 2 is the equation of the tangent line to the curve f(x) = 1+e⁻²ˣ where it cuts the line y=2 is