TANGENTS AND NORMAL TO CURVES

FURTHER MATHEMATICS SSS2 FIRST TERM

WEEK 2

TANGENTS AND NORMAL TO CURVES

Contents

For any curve, dydx is the gradient function. At any point on the curve, dydxat that point, gives the gradient of the tangent at the point. The derivation of y with respect to x at x = x₁ is denoted.

Solution

x²y + y²x + 3x – 13 = 0

Sketch the curve y = x³ – 6x² + 9x.

Find:

(i)the equation of the tangent to the curve y = x³ + x2 – 8x + 2 at the point A(1, -4);

(ii) the coordinate of the point where the tangent meets the x–axis.

Evaluation

Find the equation of the normal to each of the following curves

(4) y = (2x – 3) (x + 2) at x = 1

 

General Evaluation

Find the equation of the tangent to each of the following curves at the given points

(1) y = x² – 3x – 4 at x = 1

(2) y = x³ + 2x² – 3x + 1 at x = -1

(3) y = 1 – 2x + 5x² – x³ at x = 3

Find the equation of the normal to each of the following curves

(5) y = 6 – 2x + 3x² – 2x³ at x = 0

 

Reading Assignment

New Further Maths Project 2 pages 143, 144, 169

Weekend Assignment

A curve y = 4x³ – 2x² + 7x + 5 at point x = 3, find the

(1) Gradient of its tangent

(a) 12                                       (b) 108                            (c) 103

(d) 115

(2) Gradient of its normal

(a) -112b-1108c-1103d-1115

(3) Equation of the tangent

(a)y = 108x – 116                (b) y = 103x – 193           (c) y = 115x - 90

(d) y = 12x - 309

(4) Equation of the normal

(a) 103y + x – 11 = 0            (b) 103x + y – 116 = 0     (c) 116x + x – 119 = 0

(d) 116x + y – 103 = 0

(5) Find the gradient of this curve y = 2x² – 5x + 8 at point x = 1

(a) 1                      (b) -1           (c) 2            (d) 3

Theory

(1) Find the gradient of the tangent and the normal of the curve y = x³ – 6x² – 15x – 1 at point x = 1

(2) A curve passes through the point (1, 0) and its gradient at any point p(x, y) is 3x² – 1, find the equation of the curve.