In this page tangent and normal question2 we are going to see solution of first question.

(1) Find the equation of the tangent and normal of the following curves

(iii) y = 2 sin ² 3 x, at x = π/6

To find the equation of the tangent line we need to have two information that is a point on the tangent and the slope of the tangent line.

Here we have a point, to find the slope of the tangent line at that particular point we have to differentiate the given equation

y = 2 sin ² 3 x

dy/dx = 2 (2 sin 3 x) (cos 3 x) 3

= 6 (2 sin 3 x cos 3 x)

= 6 sin 2 (3 x)

= 6 sin 6 x

slope at x = π/6

= 6 sin 6 (π/6)

= 6 sin π

= 6 (0)

= 0

Substitute x = π/6 in the given equation to get y-coordinate value

y = 2 sin² 3 x

= 2 sin² 3 (π/6)

= 2 sin² (π/2)

= 2 (1)

= 2

So the required point is (π/6,2)

**Equation of tangent :**

(y - y₁) = m (x - x₁)

(y - 2) = 0 [x - (π/6)]

y - 2 = 0

**Equation of normal :**

(y - y₁) = (-1/m) (x - x₁)

(y - y₁) = (-1/0) (x - x₁)

0 (y - 2) = -1 [x - (π/6)]

0 = -1 [x - (π/6)]

x - (π/6) = 0

(iv) y = (1+sin x)/cos x,at x = π/4

To find the equation of the tangent line we need to have two information that is a point on the tangent and the slope of the tangent line.

Here we have a point, to find the slope of the tangent line at that particular point we have to differentiate the given equation

y = (1+sin x)/cos x

dy/dx = [cos x (cos x) - (1 + sin x) (-sin x)]/cos² x

= [cos² x + sin x + sin² x]/cos² x

= [1 + sin x ]/cos² x

slope at x = π/4

= [1 + sin π/4]/cos² π/4

= [1 + (1/√2)]/(1/√2)²

= [(√2 + 1)/√2]/(1/2)

= [(√2 + 1)/√2] x (2/1)

= (√2 + 1)√2

= 2 + 2√2

Substitute x = π/4 in the given equation to get y-coordinate value

y = (1+sin x)/cos x

= (1 + sin π/4)/cos π/4

= [1 + (1/√2)]/(1/√2)

= [(√2 + 1)/√2]/(√2/1)

= (√2 + 1)

So the required point is (π/4,(√2 + 1))

**Equation of tangent :**

(y - y₁) = m (x - x₁)

[y - (√2 + 1)] = (2 + 2√2) [x - (π/4)]

(y - y₁) = (-1/m) (x - x₁)

[y - (√2 + 1)] = [-1/(2 + 2√2)] [x - (π/4)] tangent and normal question2

tangent and normal question2

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- Rolle's theorem
- Lagrange's theorem
- Finding increasing or decreasing interval
- Increasing function
- Decreasing function
- Monotonic function
- Maximum and minimum
- Examples of maximum and minimum