- Topic
- Antiderivatives
- Difficulty
- Challenging
- Suggested time
- 5 min
- Marks
- 3
- Theory
- CLP-1 §4.1Key results
Antiderivatives practice question 9
A function satisfies , and . Find .
Two antidifferentiations, two constants. The condition on fixes the first constant; the condition on fixes the second.
with gives .
A whole number.
How sure are you?
Theory CLP-1 §4.1Key results
Worked solution
Strategy
Work down one derivative at a time, using the condition on f′ as soon as f′ is known and the condition on f as soon as f is known.
Working
First antiderivative and its constant.
Second antiderivative and its constant.
Evaluate.
Answer
The answer is 13.
Check
f = 2x³ − 3x² + 2x + 5 meets all three conditions: f(0) = 5, f′(1) = 6 − 6 + 2 = 2 and f″ = 12x − 6.
Where this goes wrong
Using f(0) = 5 to set the constant in f′, or applying f′(1) = 2 to f. Each condition belongs to the function it names.
Leans on
Higher-order derivatives, from Differentiation rules
Try a similar question
A function satisfies and . Find .Saved in this browser only.