Linear approximation practice question 1
Use a linear approximation of at to estimate .
Why is 25 a good base point for ? The tangent line there needs two facts about at 25: which two?
The tangent line at is , with .
A decimal.
Theory UBC Math 100 §9.2CLP-1 §3.4Key results
Worked solution
Strategy
Near a point where the function is easy to evaluate, replace it by its tangent line. 25 is a perfect square next door to 26.
Working
Value and slope at the base point.
The true value is 5.0990…, so the estimate is slightly high, as expected for a concave function.
Answer
The answer is 5.1.
Check
5.1² = 26.01, just 0.01 above 26, so 5.1 is a slight overestimate of √26.
Where this goes wrong
Using the derivative at 26 rather than at 25, which defeats the purpose: the whole point is that everything is evaluated at the easy point 25.
The idea behind it
The linear approximation lesson, a figure to move, in a few steps
Leans on
The derivative of √x, from Differentiation rules
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