- Difficulty
- Standard
- Suggested time
- 3 min
- Marks
- 2
The definition of the derivative practice question 14
For a base , the slope of at is the limit below.
For a = 2 this limit is about 0.69, and for a = 3 it is about 1.10. Which statement about the number e is correct?
Every exponential satisfies a⁰ = 1. Is that a fact about the value at 0 or about the slope at 0?
The slope at 0 grows with the base: about for 2 and about for 3. Somewhere between them is a base whose slope at 0 is exactly 1.
Theory UBC Math 100 §3.5Key results
Worked solution
Strategy
Separate what every exponential shares at x = 0 from what depends on the base, then use the two quoted slopes to bracket the base you want.
Working
The definition of the derivative at 0, with a⁰ = 1.
A larger base gives a steeper graph at 0, so the base with slope exactly 1 lies between 2 and 3.
That base is e. Every exponential passes through (0, 1); only y = eˣ does so with slope 1.
Answer
The answer is e is the base whose exponential has slope exactly 1 at x = 0, so e lies between 2 and 3.
Check
At h = 0.001 the quotient is about 0.693 for a = 2, 1.0005 for a = e and 1.099 for a = 3.
Where this goes wrong
Confusing the value of aˣ at 0, which is 1 for every base, with its slope at 0, which depends on the base. e is singled out by the slope.
Try a similar question
Use to evaluate the limit below.Saved in this browser only.