- Topic
- Curve sketching
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Curve sketching practice question 8
The concavity of the graph of is read from its second derivative. Find .
Concavity needs , and is a product, so this takes the product rule twice. Watch the sign the chain rule gives the derivative of .
Product rule: . The chain rule gives the in the derivative of .
Differentiate with the product rule again, then factor out .
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 §7CLP-1 §3.6Key results
Worked solution
Strategy
Differentiate twice with the product rule, factoring out the exponential each time so that the sign of what remains is easy to read.
Working
Product rule; the chain rule makes the derivative of e−x equal to −e−x.
Product rule again, on (1 − x) times e−x.
e−x > 0, so f is concave down for x < 2 and concave up for x > 2: an inflection point at x = 2.
Answer
The answer is f double prime of x equals x minus 2, times e to the minus x.
Check
At x = 0 the formula gives −2, and the second difference (f(0.1) − 2f(0) + f(−0.1))/0.1² ≈ −2.00 agrees.
Where this goes wrong
Losing the minus sign from the derivative of e−x, which turns every −e−x into +e−x and gives (x + 2)e−x, with the inflection point at −2 instead of 2.
Leans on
The product rule, from Differentiation rules
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