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)
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