The Mean Value Theorem and the shape of graphs

A derivative tells you the slope of a function at a single point. A great deal of calculus depends on going the other way: using information about the derivative everywhere on an interval to draw conclusions about the function itself. If the derivative is positive throughout an interval, is the function really increasing there? If two functions have the same derivative, how different can they be? The bridge between local slopes and global behavior is the Mean Value Theorem, one of the central results of calculus. This lesson states and interprets the theorem, uses it to justify the tests for increasing and decreasing functions and for concavity, and then combines these tests with limits and asymptotes into a systematic method for sketching graphs. Being able to predict the shape of a function from its formula, and to check a computer plot against that prediction, is a basic engineering skill.

Rolle's theorem

Suppose a function \(f\) is continuous on a closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and takes the same value at both ends, \(f(a) = f(b)\). Rolle's theorem says that there is at least one point \(c\) between \(a\) and \(b\) where \(f'(c) = 0\). The idea is visual: a smooth curve that starts and ends at the same height must turn around somewhere, and at the turning point its tangent is horizontal. The conditions matter. The function \(|x|\) on \([-1, 1]\) has equal endpoint values but no horizontal tangent, because it is not differentiable at 0.

The Mean Value Theorem

The Mean Value Theorem (MVT) removes the requirement of equal endpoint values. If \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\), then there is at least one point \(c\) in \((a, b)\) where:

The right side is the slope of the secant line joining the endpoints of the graph, the average rate of change over the interval. The theorem says that somewhere in between, the instantaneous rate equals the average rate: there is a point where the tangent is parallel to the secant. Rolle's theorem is the special case in which the secant is horizontal.

Consequences of the Mean Value Theorem

The MVT is rarely used to find numbers. Its importance lies in three consequences that justify much of calculus:

• If \(f'(x) = 0\) for every \(x\) in an interval, then \(f\) is constant on that interval. (Apply the MVT between any two points: the average slope is zero, so the values are equal.)
• If \(f'(x) = g'(x)\) on an interval, then \(f\) and \(g\) differ by a constant: \(f(x) = g(x) + C\). This is why antiderivatives in Lesson 8 always include a "+ C".
• If \(f'(x) > 0\) on an interval, then \(f\) is increasing there; if \(f'(x) < 0\), \(f\) is decreasing.

A further use is bounding errors. If \(|f'(x)| \le M\) on \([a, b]\), the MVT gives \(|f(b) - f(a)| \le M |b - a|\): a function whose slope is bounded cannot change faster than that bound allows.

Increasing, decreasing and the first derivative test

Since the sign of \(f'\) determines where \(f\) rises and falls, the intervals of increase and decrease are found by locating the critical points (where \(f' = 0\) or \(f'\) does not exist) and testing the sign of \(f'\) in each interval between them. A sign chart organizes the work.

At a critical point, the change in sign of \(f'\) classifies the point. If \(f'\) changes from positive to negative, \(f\) rises then falls, so the point is a local maximum; from negative to positive, a local minimum; if the sign does not change, neither. This is the first derivative test, which Lesson 7 applies to optimization.

Concavity and inflection points

The second derivative describes how the slope itself changes. Where \(f''(x) > 0\), the slope is increasing and the graph is concave up, bending upward like a cup; it lies above its tangent lines. Where \(f''(x) < 0\), the graph is concave down, like a cap, and lies below its tangent lines. A point where the concavity changes is an inflection point.

Concavity has physical meaning. If \(s(t)\) is position, concave up means positive acceleration. For a beam, the curvature of the deflected shape is proportional to the bending moment, so an inflection point in the deflection curve marks where the bending moment is zero. In economics, diminishing returns appear as a concave-down production curve.

At a critical point, the second derivative test classifies the point quickly: if \(f'(c) = 0\) and \(f''(c) > 0\), the graph is a cup and \(c\) is a local minimum; if \(f''(c) < 0\), it is a cap and \(c\) is a local maximum. If \(f''(c) = 0\), the test is inconclusive, as \(x^4\) (a minimum), \(-x^4\) (a maximum) and \(x^3\) (neither) all show at \(x = 0\).

Asymptotes and end behavior

Limits from Lesson 1 describe the behavior of a graph far from the origin and near points where it blows up:

• Vertical asymptotes occur where \(f(x) \to \pm\infty\) as \(x\) approaches a finite value, typically where a denominator is zero and the numerator is not.
• Horizontal asymptotes are the limits of \(f(x)\) as \(x \to \pm\infty\), when these are finite.
• Slant (oblique) asymptotes occur for rational functions whose numerator degree exceeds the denominator degree by one: polynomial division writes \(f(x) = mx + b + r(x)\) with \(r(x) \to 0\), and the line \(y = mx + b\) is the asymptote.

Exponential and logarithmic terms add their own behavior (Lesson 2): \(e^{-x}\) decays to zero, so functions such as \(x e^{-x}\) approach zero far to the right despite the growing factor \(x\).

A procedure for sketching graphs

Combining these tools gives a reliable procedure:

• Domain: where is the function defined? Note any excluded points.
• Intercepts: find \(f(0)\) and, if practical, the zeros.
• Symmetry: even functions (\(f(-x) = f(x)\)) are symmetric about the \(y\)-axis; odd functions (\(f(-x) = -f(x)\)) are symmetric about the origin.
• Asymptotes: vertical, horizontal and slant.
• First derivative: critical points, intervals of increase and decrease, local extrema.
• Second derivative: concavity and inflection points.
• Plot the key points and join them with a curve consistent with all of the above.

Graphing software can plot any function in an instant, so why learn this? Because software plots only the window you choose and can miss behavior outside it or between sample points, such as a narrow spike near an asymptote or a minimum just outside the window. Knowing what shape to expect lets you choose sensible windows and recognize when a plot is misleading.