FINDING THE EQUATION OF A TANGENT LINE

For each problem, find the equation of the tangent line to the function at the given point.

Problem 1 :

f(x) = x² + 1; (1, 2)

Solution :

Given, f(x) = x² + 1

f^'(x) at (1, 2)

x = 1 and y = 2

f^'(x) = m = 2x 

m = 2(1)

Slope m = 2

Equation of a tangent line is y - y₁ = m(x - x₁)

x₁ = 1 and y₁ = 2

y - 2 = 2(x - 1)

y - 2 = 2x - 2

y = 2x - 2 + 2

y = 2x 

Equation of a tangent line is y = 2x.

So, option A) is correct.

Problem 2 :

f(x) = 2x² + 2x + 2; (-1, 2)

Solution :

Given, f(x) = 2x² + 2x + 2

f^'(x) at (-1, 2)

x = -1 and y = 2

f^'(x) = m = 4x + 2 

m = 4(-1) + 2

m = -4 + 2

Slope m = -2

Equation of a tangent line is y - y₁ = m(x - x₁)

x₁ = -1 and y₁ = 2

y - 2 = -2(x + 1)

y - 2 = -2x - 2

y = -2x - 2 + 2

y = -2x 

Equation of a tangent line is y = -2x.

So, option B) is correct.

Problem 3 :

y = x² + x - 2; (1, 0)

Solution :

Given, y = x² + x - 2

y^' at (1, 0)

x = 1 and y = 0

y^' = m = 2x + 1 

m = 2(1) + 1

m = 2 + 1

Slope m = 3

Equation of a tangent line is y - y₁ = m(x - x₁)

x₁ = 1 and y₁ = 0

y - 0 = 3(x - 1)

y - 0 = 3x - 3

y = 3x - 3

Equation of a tangent line is y = 3x - 3.

So, option C) is correct.

Problem 4 :

y = 2x² + 1; (-1, 3)

A)  y = 2x + 5        B)  y = -4x – 1     C)  y = 12x + 15

Solution :

Given, y = 2x² + 1

y^' at (-1, 3)

x = -1 and y = 3

y^' = m = 4x 

m = 4(-1)

Slope m = -4

Equation of a tangent line is y - y₁ = m(x - x₁)

x₁ = -1 and y₁ = 3

y - 3 = -4(x + 1)

y - 3 = -4x - 4

y = -4x - 4 + 3

y = -4x - 1

Equation of a tangent line is y = -4x - 1.

So, option B) is correct.

Problem 5 :

Solution :

Equation of a tangent line is y - y₁ = m(x - x₁)

x₁ = 0 and y₁ = 1

y - 1 = -1(x - 0)

y - 1 = -1x

y = -x + 1

Equation of a tangent line is y = -x + 1.

So, option C) is correct.

Problem 6 :

Solution :

Hence, option B) is correct.

Problem 7 :

Solution :

Equation of a tangent line is y - y₁ = m(x - x₁)

x₁ = -2 and y₁ = -1

y  + 1 = 1(x + 2)

y = x + 2 - 1

y = x + 1

Equation of a tangent line is y = x + 1.

So, option B) is correct.

Problem 8 :

Solution :

So, option A) is correct.