*Draw the secant line between any two points on the graph of a quadratic polynomial. The Mean Value Theorem tells us there is a point in that interval at which the derivative equals the slope of...*

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A Mean Value Property of the Derivative of Quadratic Polynomials-without Mean Values and Derivatives

*The authors use inequalities to solve optimization problems without resorting to calculus, as illustrated by four common examples in calculus.*

*The authors derive an upper bound of the expected range of random points chosen from the unit interval according to any distribution.*

*The author finds Riemann sums that equal exactly the definite integrals for polynomials and negative-integer power functions.*

*The authors model a real traffic problem by using the fundamental theorem of calculus.*

*The author proves visually four chain inequalities among five common means: harmonic, geometric, arithmetic, root square, and contraharmonic.*

*The authors show that the locus of the focus of a parabola rolling on the* \(x\)*-axis is a catenary.*

*The authors provide a condition for a function to have nested \(n\)-th degree Taylor polynomials with varying centers, which can approximate the function visually.*