- Topic
- Curve sketching
- Difficulty
- Introductory
- Suggested time
- 3 min
- Marks
- 2
Curve sketching practice question 1
Find the largest open interval on which is increasing.
Being positive and being increasing are different things. On each interval, which sign tells you that is rising?
Increasing where . Use the quotient rule and factor the numerator; the denominator squared and are both positive.
Theory UBC Math 100 §7CLP-1 §3.6Key results
Worked solution
Strategy
Differentiate, throw away the factors that are always positive, and read the sign of what remains.
Working
Quotient rule, then factor eˣ.
eˣ > 0 and (x + 2)² > 0 wherever f is defined.
Answer
The answer is x greater than negative 1.
Check
f(−1.5) ≈ 0.45, f(−1) ≈ 0.37 and f(0) = 0.5, so f is still falling just left of −1 and rising after it.
Where this goes wrong
Answering (−2, ∞) because that is where f is defined and positive. Being positive and being increasing are different; between −2 and −1 the function is positive but falling.
Leans on
The quotient rule, from Differentiation rules
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