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Math Analysis 1

Studying the graph of a function: step-by-step procedure with a full example

Domain, sign, limits and asymptotes, first and second derivatives, maxima, minima and inflection points, with a full study of x times e to the minus x and maximum power transfer.

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Sketching the graph of a function brings the whole course together: domain, limits, derivatives, monotonicity, concavity. The goal is a correct qualitative graph, with asymptotes, maxima, minima and inflection points in the right place, without computing hundreds of points. Here is the procedure step by step, applied to a complete example.

The steps

  1. Domain and any symmetry (even, odd, periodic).
  2. Intercepts with the axes and sign of f.
  3. Limits at the ends of the domain and asymptotes (vertical, horizontal, oblique).
  4. First derivative: where it exists, where it vanishes, its sign. This gives intervals of increase, maxima and minima, corners or cusps.
  5. Second derivative: its sign, hence concavity and inflection points.
  6. Graph, checking that everything is consistent.

Concavity: where f'' > 0 the graph is convex (opening upwards, like x^2), where f'' < 0 it is concave. An inflection point is where the concavity changes.

A complete example: f(x) = x\,e^{-x}

1. Domain. All of \mathbb R. No symmetry: f(-x) = -x e^{x} is neither f(x) nor -f(x).

2. Intercepts and sign. f(x) = 0 only at x = 0. Since e^{-x} > 0, the sign is that of x: negative for x < 0, positive for x > 0.

3. Limits. As x \to -\infty: x \to -\infty and e^{-x} \to +\infty, so f \to -\infty. Oblique asymptote check: \frac{f(x)}{x} = e^{-x} \to +\infty, so there is none. As x \to +\infty: \frac{x}{e^x} \to 0 (the exponential beats the power), horizontal asymptote y = 0.

4. First derivative. f'(x) = e^{-x} - x e^{-x} = (1 - x)e^{-x}. It vanishes at x = 1, is positive for x < 1 and negative for x > 1. So f increases up to 1 and then decreases: absolute maximum at \left(1, \frac1e\right) \approx (1,\ 0.368).

5. Second derivative. f''(x) = -e^{-x} - (1 - x)e^{-x} = (x - 2)e^{-x}. Negative for x < 2 (concave), positive for x > 2 (convex): inflection point at \left(2, \frac{2}{e^2}\right) \approx (2,\ 0.271).

6. Graph. It comes up from -\infty on the left, crosses the origin with slope f'(0) = 1, rises to the maximum at x = 1, goes down, changes concavity at x = 2 and flattens towards the x axis without ever touching it. This shape, a quick rise and a slow return, is the response of a critically damped RLC circuit to an impulse.

Useful checks

  • The sign of f' must agree with the limits: if f \to +\infty on the right, eventually the function must increase.
  • Between two consecutive maxima there is always a minimum (for a continuous function on an interval).
  • A horizontal asymptote can be crossed by the graph: it is not a barrier, it only describes the behaviour at infinity.
  • Where f is not differentiable (roots, absolute values) compute the right and left derivatives to see whether there is a corner or a cusp.

Maxima and minima on a closed interval

For absolute extrema on [a, b] it is enough to compare the value of f at the interior stationary points, at the points where it is not differentiable and at the endpoints a and b. This is the “engineering” version of graph sketching: for example, the power delivered to a load R_L by a source with internal resistance R_s is P = \frac{V^2 R_L}{(R_s + R_L)^2}. Differentiating with respect to R_L gives the maximum at R_L = R_s: the maximum power transfer theorem.

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