tangents & normals

Quick links:
downloadable teaching materials for tangents & normals
► syllabus content for the Calculus Topic: SL syllabus (see syllabus section 6.1); HL syllabus (see syllabus sections 6.1).

Course planning / teaching notes:

Finding the equation of a line that is tangent or normal (perpendicular) to a given function at a particular point is a very common question on exams - both as a short question on its own (Section A) or as part of a longer question (Section B). The accessibility level for students on such a question can vary widely - both for SL and HL. One difference to clearly illustrate to students is whether or not the given point through which the tangent line or normal line passes is on the graph of the function or is not. This difference can be demonstrated with a comparison of the solution approach (and answer) for the following two examples.

Example 1: Find equations for the tangent and normal to the graph of at the point where .

The first thing to do is to determine that . Thus the point where the tangent line and the normal line both intersect the graph of the function is at , which, of course, is a point on the graph of the given function. Clearly there will be just one tangent line at this point (answer: ) and just one normal line at this point (answer: ).
Compare this to the following example.

Example 2: Find the equation(s) of the line(s) that pass through and are tangent to the graph of .

The solution and answer for this question will differ significantly from the one in Example 1 because the given point is not on the given function. As is hinted in the wording of the question, it is possible that there will be more than one answer. In fact, there are two lines that pass through and are tangent to the function (of course, the points of tangency are different). Students should be strongly advised to make a sketch to assist them in visualizing this question - which then helps understand why there will be two answers. Answers: and

4 questions - ‘accessible’ to ‘discriminating’

download: 4_Qs_tangents_normals_1_with_answers_v3

accessible SL question

moderate SL / accessible HL question

discriminating SL / moderate HL question

discriminating HL question


Click 'eye' to see graph showing solution to Question 2 above

Given function is in blue, and the answer - the tangent line passing through function's x-intercept - is in red.

Click 'eye' to see solution (and graph) for Question 4 above

♦ teaching materials

6 exercises (some with multiple parts) on finding equations of tangents & normals. Various differentiation techniques including the chain rule and derivatives of logarithmic and exponential functions are necessary.  Answers included.

click here to go to calculus unit tests

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