Work through the list without notes or a calculator. If you cannot do a statement, follow its link, then come back. Guide = study guide; Practice = practice questions. This topic is shared by Calculus AB and Calculus BC.
Understanding
- I can explain why sine, cosine and tangent must be restricted before they have inverses, and state the range of arcsin, arccos and arctan. (Guide: “Why the ranges are restricted”; Practice Q7(d))
- I can explain the difference between sin⁻¹ x and 1/sin x. (Guide: “A note on notation”)
- I can explain why arcsin and arccos are not differentiable at x = ±1, while arctan is differentiable everywhere. (Guide: “Where the derivatives fail”; Practice Q4, Q6(c))
Skills
- I can state the derivatives of arcsin x, arccos x and arctan x from memory. (Revision notes; Practice Q1, Q2)
- I can derive the derivative of arcsin x or arctan x by implicit differentiation, using a reference triangle or an identity. (Guide: Figure 1; Practice Q6(a))
- I can use the chain rule with an inner function, squaring the whole inner function. (Guide: Worked example 1; Practice Q1, Q2, Q7)
- I can combine an inverse trigonometric derivative with the product rule and evaluate it exactly. (Guide: Worked example 2)
- I can write the tangent line to an inverse trigonometric graph at a given point. (Guide: Worked example 1; Practice Q5(b), Q7(c))
- I can find a derivative at a point with the inverse-function rule from Topic 3.3. (Guide: Worked example 3; Practice Q3)
Reasoning
- I can justify the sign of the square root using the range of the inverse function. (Guide: arcsin derivation; Practice Q6(a))
- I can decide where an expression such as arcsin(x − 2) is defined and where it is differentiable. (Practice Q4)
- I can use the sign of a derivative to say whether an inverse trigonometric function is increasing or decreasing. (Practice Q5(c))
All ticked? Move on to Topic 3.5, Selecting Procedures for Calculating Derivatives, or return to the Calculus AB roadmap or the Calculus BC roadmap.