1. Gradient of a Curve and
Equation of a Tangent to a
Curve
Math Studies 2 – Topic 7
2. Find the gradient of the tangent to the curve whose
4
equation is y = + 2 at the point where x=2.
x
−1
y = 4x + 2 substitute
f ' ( x ) = −4 x + 0
−2 −4
f ( x) = 2 = −1
2
4
f ' ( x ) = −4 x −2
= 2
x
3. A power boat moves in a straight line such that at time t
seconds its distance s from a fixed point O on that line is
given by s = 2t − 3t + 1
2
Find the speed after 3 seconds.
ds substitute
= 4t − 3 t =3
dt
speed = 4(3) − 3 = 9m / s
4. Find the value of the gradient of the curve whose
equation is y=(x-3)(x+2) at the point where it crosses the
positive x-axis (x=3).
multiply
substitute
y = x − x−6
2
x=3
f ' ( x) = 2(3) − 1 = 5
f ' ( x) = 2 x − 1
6. Steps to finding the gradient
of any curve
• Differentiate to find the gradient
• Substitute the particular value of x
to find the gradient of the curve at
that point
7. Find the equation of the tangent to the curve whose
2
equation is y = 3x 2 + − 5 at the point P where x=2.
x
dy 2 2
−2
= 6x − 2x = 6x − 2 y = 3(2) 2 + −5 = 8
dx x 2
y =8
P(2,8)
substitute
dy 2 1 y = mx + c
= 12 − = 11
dx 4 2 1
1 8 = 11 (2) + c
m = 11 2
2
c = 15
10. Equation of the Tangent at a Given Point
• Find the gradient at the point P and call it m
• Find the y-coordinate of P as well as the x-
coordinate (sometimes it will be given)
• Use y=mx+c and substitute for m, x, and y at the
point P
• Write the equation with values of m and c
12. 1) Find the equation of the tangent to the curve whose
4
equation is y=x²-2x +3 at the point where x=1. Show
all working.
2) Find the equation of a tangent to the curve whose
equation is y=2x³-4x at the point where x=2.
3) The curve whose equation is y=x²-4x+15 has gradient
6 when x=a. Find the value of a.
4) The tangent to the parabola given by y=x²+3x-8 has
gradient 7 at the point P. Find the coordinates of P.
