Tangents and normals
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The teaching part (1 min) is free to watch. A free account opens the whole lecture: 2 worked examples, 5 practice questions, 1 real past-paper question with the answer explained, the mistakes that lose marks, the full-marks tip, a one-card summary and a quick quiz.
What this lecture covers
- A straight line that fits the curve
- Tangent and normal
- The method
Read the transcript of the teaching part
Last time
Before we start: last time, in Differentiation, you learned these points. Here they are again on one card. Keep them in mind, because today builds on them.
A straight line that fits the curve
Zoom in on any smooth curve far enough, and it looks like a straight line. That line is the tangent. Its gradient is the derivative. And the normal is the line at right angles to it, which you meet in optics, engineering and mechanics.
Tangent and normal
The gradient of the tangent at a point is the value of dy by dx there. The normal is perpendicular to the tangent. Its gradient is minus one over the tangent gradient. Both lines go through the point on the curve. Use y minus y one equals m, x minus x one.
The method
One: find the y coordinate of the point, if only x is given. Two: differentiate, and put in x to get the gradient. Three: for a normal, take the negative reciprocal. Four: write the equation of the line through the point.
Chapter 14: Differentiation and integration
- Differentiation: standard functions, chain, product and quotient rules 5 min
- Tangents and normals 5 min
- Stationary points and the derivative tests 5 min
- Practical maxima and minima 5 min
- Connected rates of change and small increments 5 min
- Integration: powers of x and the reverse of differentiation 5 min
- Integrating (ax + b)ⁿ, exponentials and trigonometric functions 5 min
- Definite integrals and plane areas 5 min
- Kinematics: displacement, velocity, acceleration and graphs 5 min
Also for O Level Additional Maths 4037
Which topics come up most in O Level Additional Maths 4037 past papers, counted from every paper since 2004.