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Plot of f(x) = sin(2x) from − π /4 to 5 π /4; the second derivative is f″(x) = –4sin(2x), and its sign is thus the opposite of the sign of f. Tangent is blue where the curve is convex (above its own tangent), green where concave (below its tangent), and red at inflection points: 0, π /2 and π
The tangent of half an angle is important in spherical trigonometry and was sometimes known in the 17th century as the half tangent or semi-tangent. [2] Leonhard Euler used it to evaluate the integral ∫ d x / ( a + b cos x ) {\textstyle \int dx/(a+b\cos x)} in his 1768 integral calculus textbook , [ 3 ] and Adrien-Marie Legendre described ...
An osculating curve from a given family of curves is a curve that has the highest possible order of contact with a given curve at a given point; for instance a tangent line is an osculating curve from the family of lines, and has first-order contact with the given curve; an osculating circle is an osculating curve from the family of circles ...
In calculus, a derivative test uses the derivatives of a function to locate the critical points of a function and determine whether each point is a local maximum, a local minimum, or a saddle point. Derivative tests can also give information about the concavity of a function.
Geometrically, the map from a conic to its dual is one-to-one (since no line is tangent to two points of a conic, as that requires degree 4), and the tangent line varies smoothly (as the curve is convex, so the slope of the tangent line changes monotonically: cusps in the dual require an inflection point in the original curve, which requires ...
The second derivative of a function f can be used to determine the concavity of the graph of f. [2] A function whose second derivative is positive is said to be concave up (also referred to as convex), meaning that the tangent line near the point where it touches the function will lie below the graph of the function.
inflection point In differential calculus, an inflection point, point of inflection, flex, or inflection (British English: inflexion) is a point on a continuous plane curve at which the curve changes from being concave (concave downward) to convex (concave upward), or vice versa. instantaneous rate of change
The blue squares are inflection points. In mathematics, particularly in calculus, a stationary point of a differentiable function of one variable is a point on the graph of the function where the function's derivative is zero. [1] [2] [3] Informally, it is a point where the function "stops" increasing or decreasing (hence the name).
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