Gradient of a Curve and Equation of a Tangent

to a Curve

Math Studies 2 – Topic 7

Find the gradient of the tangent to the curve whose

equation is at the point where x=2.2

4x

y

24 1 xy

04' 2 xxf

2

2 44'

xxxf

12

4)(

2

xf

substitute

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 Find the speed after 3 seconds.

132 2 tts

34 tdt

ds

smspeed

t

substitute

/93)3(4

3

Find the value of the gradient of the curve whose equation is y=(x-3)(x+2) at the point where it crosses thepositive x-axis (x=3).

62 xxy

multiply

12)(' xxf51)3(2)('

3

xf

x

substitute

Think Back

What are the steps I used to

solve the problems?

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

Find the equation of the tangent to the curve whose

equation is at the point P where x=2.5

23 2

xxy

22 2626

xxxx

dx

dy

2

111

2

111

4

212

m

dx

dy

substitute)8,2(

8

852

2)2(3 2

P

y

y

15

)2(2

1118

c

c

cmxy

152

111 xy

equation

Think Back

What are the steps I used to

solve the problems?

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

IndependentPractice

1) Find the equation of the tangent to the curve whose 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.

4

Solution1)y=2x+42)y=20x-323)54)(2,2)

Resources

www.info.oxford-consulting.comwww.wclipart.comwww.clipartspot.netMathematical Studies Course Companion (Oxford)