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 local linearity: near a point where f is differentiable, the graph looks almost like its tangent line. (Guide: “Zoom in and the curve looks straight”)
- I can explain why a linearization does not exist at a corner. (Guide: “The linearization formula”)
- I can explain, with a sketch, why a graph that bends upward gives underestimates and one that bends downward gives overestimates. (Guide: Figure 1 and “Overestimate or underestimate?”; Practice Q3)
Skills
- I can write L(x) = f(a) + f′(a)(x − a) for a given function, including one that needs the chain rule. (Guide: Worked example 1; Practice Q2, Q5(a))
- I can choose a sensible point a and estimate a value such as ∛8.3 or 1/4.9 by hand. (Guide: Worked example 1; Practice Q5(b))
- I can use a value and a derivative from a context or a table to estimate a nearby value, with a positive or negative step. (Guide: Worked example 2; Practice Q1, Q4)
- I can estimate f(x) when I am given f(a) and a formula for f′ only. (Guide: Worked example 3; Practice Q6(b))
Reasoning
- I can use the sign of f″ between a and x to justify whether an estimate is too high or too low. (Guide: Worked example 3; Practice Q5(c), Q6(c))
- I can handle a point of tangency where f″ changes sign, treating each side separately. (Practice Q7(b))
- I can explain in context why an estimate gets less reliable further from a. (Guide: error table and Worked example 2(d); Practice Q6(d))
- I can state an estimate with “≈” and units, and say how it relates to the actual value. (Guide: Worked example 2; Practice Q5(d), Q6(b), Q7(c))
All ticked? Move on to Topic 4.7, Using L’Hospital’s Rule for Determining Limits of Indeterminate Forms, or return to the Calculus AB roadmap or the Calculus BC roadmap.