Question

Find the equations of the tangent and normal to the given curves at the indicated points:

(i) y = x⁴ − 6x³ + 13x² − 10x + 5 at (0, 5)
(ii) y = x⁴ − 6x³ + 13x² − 10x + 5 at (1, 3)
(iii) y = x³ at (1, 1)
(iv) y = x² at (0, 0)
(v) x = cos t, y = sin t at t=π/4

Answer 1

(i) The equation of the curve is y = x⁴ − 6x³ + 13x² − 10x + 5. On differentiating with respect to x, we get:

Thus, the slope of the tangent at (0, 5) is −10. The equation of the tangent is given as:

y − 5 = − 10(x − 0)
⇒ y − 5 = − 10x
⇒ 10x + y = 5

(ii) The equation of the curve is y = x⁴ − 6x³ + 13x² − 10x + 5. On differentiating with respect to x, we get:

Thus, the slope of the tangent at (1, 3) is 2. The equation of the tangent is given as:

(iv) The equation of the curve is y = x². On differentiating with respect to x, we get: