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In mathematics, the vertical line test is a visual way to determine if a curve is a graph of a function or not. A function can only have one output, y , for each unique input, x . If a vertical line intersects a curve on an xy -plane more than once then for one value of x the curve has more than one value of y , and so, the curve does not ...
Variations of the horizontal line test can be used to determine whether a function is surjective or bijective: The function f is surjective (i.e., onto) if and only if its graph intersects any horizontal line at least once. f is bijective if and only if any horizontal line will intersect the graph exactly once.
In general, implicit curves fail the vertical line test (meaning that some values of x are associated with more than one value of y) and so are not necessarily graphs of functions. However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self ...
An example of a stationary point of inflection is the point (0, 0) on the graph of y = x 3. The tangent is the x-axis, which cuts the graph at this point. An example of a non-stationary point of inflection is the point (0, 0) on the graph of y = x 3 + ax, for any nonzero a. The tangent at the origin is the line y = ax, which cuts the graph at ...
The sine and tangent small-angle approximations are used in relation to the double-slit experiment or a diffraction grating to develop simplified equations like the following, where y is the distance of a fringe from the center of maximum light intensity, m is the order of the fringe, D is the distance between the slits and projection screen ...
The diagram at right shows a circle with centre O and radius r = 1. Let two radii OA and OB make an arc of θ radians. Since we are considering the limit as θ tends to zero, we may assume θ is a small positive number, say 0 < θ < 1 / 2 π in the first quadrant.
Signs of trigonometric functions in each quadrant. All Students Take Calculus is a mnemonic for the sign of each trigonometric functions in each quadrant of the plane. The letters ASTC signify which of the trigonometric functions are positive, starting in the top right 1st quadrant and moving counterclockwise through quadrants 2 to 4.
The tangent line to a point on a differentiable curve can also be thought of as a tangent line approximation, the graph of the affine function that best approximates the original function at the given point. [3] Similarly, the tangent plane to a surface at a given point is the plane that "just touches" the surface at that point.