- Topic
- Linear approximation
- Difficulty
- Introductory
- Suggested time
- 2 min
- Marks
- 2
Linear approximation practice question 4
Find the linear approximation of at , where is the natural logarithm.
Why is a convenient base point for ? The tangent line there is built from one value and one slope.
. Here and .
A formula in x. Use ^ for powers and brackets for every input: sin(2x), e^(x^2), log(x+1). Give exact values: 1/6, not 0.167.
Theory UBC Math 100 §9.2CLP-1 §3.4Key results
Worked solution
Strategy
The linear approximation at a is the tangent line there: the value f(a) plus the slope f′(a) times the step x − a. At e both are exact.
Working
Value and slope at the base point.
The slope multiplies the step from the base point.
It simplifies: this tangent line passes through the origin.
Answer
The answer is L of x equals 1 plus (x minus e) over e, which is x over e.
Check
L(e) = 1 = log e, and L(3) = 3/e ≈ 1.104 is close to log 3 ≈ 1.099.
Where this goes wrong
Writing L(x) = 1 + x/e, with the slope times x. The slope multiplies the step from the base point, x − e, so that L(e) = 1 = f(e).
Leans on
The derivative of log x, from The chain rule and logarithms
Try a similar question
Use a linear approximation of at to estimate .Saved in this browser only.