Maths
Sketch y = x e⁻ˣ
Sketch the graph of y = x e⁻ˣ. Talk me through what you check as you go.
- Hint 1Where does it cross the axes, and what is its sign either side?
- Hint 2What happens as x grows large, and as x becomes very negative? Then find where the gradient is zero.
- It passes through the origin; y < 0 for x < 0 and y > 0 for x > 0.
- As x → ∞, e⁻ˣ shrinks faster than x grows, so y → 0 from above. As x → −∞, y → −∞.
- dy/dx = (1 − x)e⁻ˣ, zero at x = 1: a maximum at (1, 1/e).
- d²y/dx² = (x − 2)e⁻ˣ, so the curve changes from concave to convex at x = 2.
- Draw it rising steeply through the origin, peaking at (1, 1/e) and then decaying towards the x-axis.
Where it lands: Maximum at (1, 1/e), point of inflection at x = 2, y → 0 as x → ∞ and y → −∞ as x → −∞.
The trap: Drawing the curve crossing the x-axis again for large x. It approaches the axis but stays positive.
What a tutor might ask next: How many solutions does x e⁻ˣ = k have, depending on k?
Now do one out loud. In the interview you think aloud with a tutor. Try a Maths problem with hints, follow-ups and Mia's feedback on how you think.