Explore derivatives through limits and understand the Mean Value Theorem
The derivative is the slope you see when you zoom in until a curve looks straight. Shrink the gaps below and watch slopes, tangents, and polynomial approximations appear.
Choose a function and a point a, then shrink h. The secant line swings into the tangent, and its slope closes in on f'(a).
Error: |secant slope - f'(a)| = 1.000000
The derivative f'(a) is defined as:
f'(a) = limh→0 [f(a + h) - f(a)] / h
Watch as h shrinks: the secant line (connecting two points on the curve) approaches the tangent line (touching at exactly one point).
The derivative measures instantaneous rate of change. It's defined as a limit of difference quotients: f'(a) = limh→0 [f(a + h) - f(a)] / h. We'll explore when this limit exists, prove the Mean Value Theorem, and discover the surprising ways that differentiability relates to continuity.
Watch secant lines approach the tangent line as h → 0. The slope of the secant line converges to the derivative.
Key insight: The derivative exists only when the left and right difference quotients agree in the limit, making it a fundamentally two-sided concept.
Drag the interval endpoints and find c, where the tangent runs parallel to the secant from a to b.
c = 0.5000
f'(c) = 1.0000
= (f(b) - f(a)) / (b - a) = 1.0000
If f is continuous on [a, b] and differentiable on (a, b), then there exists c ∈ (a, b) such that:
f'(c) = [f(b) - f(a)] / (b - a)
Geometrically: there's a point where the tangent line is parallel to the secant connecting the endpoints.
For a differentiable function on [a, b], there's always a point c where the tangent line is parallel to the secant connecting the endpoints. This theorem is the workhorse of differential calculus, powering results from L'Hopital's rule to the Fundamental Theorem of Calculus.
Key insight: If f is continuous on [a, b] and differentiable on (a, b), then f'(c) = [f(b) - f(a)] / (b - a) for some c in (a, b).
Pick a function and set an interval where f(a) = f(b). A horizontal tangent always shows up in between.
Rolle's Theorem applies!
If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then there exists c ∈ (a, b) such that:
f'(c) = 0
This is a special case of MVT: when the secant slope is 0 (horizontal), the tangent must also be horizontal somewhere in between.
A special case of MVT: when f(a) = f(b), there must be a point c where f'(c) = 0 (horizontal tangent). Rolle's theorem is both the stepping stone to the MVT and a powerful tool for counting roots of derivatives.
Key insight: Rolle's Theorem is the geometric statement that a smooth curve returning to the same height must have a horizontal tangent somewhere in between.
Pick a function and shrink h with the left and right secants showing. Where their slopes disagree, there is no derivative.
Left and right derivatives exist but are different
Left derivative: -1, Right: 1
If a function is differentiable at a point, it must be continuous there. However, continuity does NOT guarantee differentiability.
• Corner: |x| is continuous but not differentiable at 0
• Cusp: x^(2/3) is continuous but has infinite one-sided derivatives
• Vertical tangent: ∛x is continuous but f'(0) = ∞
• Discontinuity: If not continuous, cannot be differentiable
Differentiability implies continuity, but not vice versa. Explore corners, cusps, and vertical tangents where continuous functions fail to be differentiable.
Key insight: If f is differentiable at a, then f is continuous at a. The converse is false — |x| at x = 0 is the classic counterexample.
Pick a function and raise the degree. The polynomial hugs the curve over a wider and wider stretch.
Taylor polynomial:
T3(x) = 1.000 +x +0.500x^2 +0.167x^3
The Taylor polynomial of degree n centered at a is:
Tn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)²/2! + ... + f(n)(a)(x-a)n/n!
As n increases, the polynomial approximates f better near the center. For analytic functions, Tn → f as n → ∞ (within the radius of convergence).
Higher-order derivatives give us polynomial approximations that match a function's behavior near a point. Watch convergence as degree increases.
Key insight: An n-times differentiable function can be approximated by a polynomial plus a remainder term whose size is controlled by the (n+1)-th derivative.
Spot a mistake or have an idea?