- Topic
- Linear approximation
- Difficulty
- Standard
- Suggested time
- 4 min
- Marks
- 2
Linear approximation practice question 9
The speed of sound in air at a temperature of degrees Celsius is modelled by metres per second. Find the linear approximation of at .
The tangent line at needs the speed and its slope there. Which rule does the square root of call for when you differentiate?
The base point is 0, so .
Write . The chain rule brings in the derivative of the inside, .
A formula in x, with exact fractions rather than decimals. 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
Linearise at x = 0, where the square root is 1. The slope needs the chain rule, and the inner derivative 1/273 is what makes it small.
Working
The speed at freezing, in m/s.
Power rule on the outside, times the derivative of the inside, 1/273.
Sound travels about 0.6 m/s faster for each degree warmer.
The base point is 0, so the step is x itself.
Answer
The answer is L of x equals 331 plus 331 over 546 times x, about 331 plus 0.606 x.
Check
At 20 °C the line gives about 343.1 m/s, against 342.9 m/s from the model itself.
Where this goes wrong
Leaving out the inner derivative 1/273, which gives a slope of 331/2: sound would speed up by 165.5 m/s for each degree. The chain rule multiplies by the derivative of 1 + x/273.
Leans on
The chain rule, from The chain rule and logarithms
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