Tangents, Normals and Approximations

Name: Tangents, Normals and Approximations
Author: StemPy Lab

The derivative is the gradient (also called the slope) of a curve at a point. Write the equations of the tangent and the normal, find where a tangent is horizontal, parallel or perpendicular to a given line, and measure the angle between two curves. Then use differentials to estimate small changes, roots and percentage errors. It covers the Indian syllabus (NCERT, JEE Main) and the matching topics of A-Level, IB and AP.

3 lessons, about 1 h 15 min3 simulations10 problems3 practice questionsPython in every lesson

Created by StemPy Lab

Course outline (3 lessons)

Lesson 1about 25 min
Tangents and normals
The tangent to a curve at a point is the straight line that touches the curve there, and its gradient is the derivative at that point. The normal is the line through the same point at a right angle to the tangent, so its gradient is minus the reciprocal. The tangent is $y - y_0 = f'(x_0)(x - x_0)$ . For an implicit or a parametric curve, find $\dfrac{dy}{dx}$ first.
1 simulation1 practice question1 Python program2 problems1 Python lab
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Lesson 2about 25 min
Tangents in a given direction and the angle between curves
To find where a tangent is parallel to a given line, set the derivative equal to the gradient of that line. For a perpendicular tangent, use the negative reciprocal. A tangent through a point outside the curve is found with the point of contact as the unknown. The angle between two curves at a common point is the angle between their tangents, with $\tan\varphi = \left|\dfrac{m_1 - m_2}{1 + m_1m_2}\right|$ .
1 simulation1 practice question1 Python program5 problems
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Lesson 3about 25 min
Small changes, approximations and errors
A small change $dx$ in $x$ changes $y = f(x)$ by about $dy = f'(x)\,dx$ . This gives the estimate $f(x + dx) \approx f(x) + f'(x)\,dx$ , which is the tangent line used as a copy of the curve. It estimates roots and trigonometric values, and it gives the error in a calculated quantity from the error in a measurement. The relative error and the percentage error compare the error with the size of the quantity.
1 simulation1 practice question1 Python program3 problems1 Python lab1 question set1 note
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Tangents, Normals and Approximations: Timed Test

10 questions · 25 min

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Tangents, Normals and Approximations

Course outline (3 lessons)

Tangents and normals

Tangents in a given direction and the angle between curves

Small changes, approximations and errors

Test Your Learning

Your teacher

Discussion (0)