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- Convex curves, or convex functions are curves that curve upwards. Graphically, if a curve is above the line segment connecting any two points on it, the curve is said to be convex. Conversely, concave functions curve downwards. A function is classified as concave if it lies beneath the line segment connecting any two points on it.
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Example 1: Characterising Graphs. Say we have a graph of the function f(x) = x(x^2 + 1). Find the parts of the graph where the function is convex or concave, and find the point(s) of inflexion. [3 marks] f(x) = x(x^2 + 1) = x^3 + x gives. f''(x) = 6x. f''(x) = 0, when x = 0. f''(x) \textcolor{red}{< 0} when x<0. Here we have a concave section.
Convex curves, or convex functions are curves that curve upwards. Graphically, if a curve is above the line segment connecting any two points on it, the curve is said to be convex. Conversely, concave functions curve downwards.
Sep 9, 2023 · Ways to Remember. Concave: Think “Con-cave”; it has a “cave” or an inward dip. Convex: Think of “Con-vex” as “convicts” trying to escape, bulging outwards. Concave and Convex Lenses. Concave lenses focus light inside the curve of the lens, while convex lenses focus light using the outer curve.
Graphically, a concave function opens downward, and water poured onto the curve would roll off. A function f is convex if f’’ is positive (f’’ > 0). A convex function opens upward, and water poured onto the curve would fill it. Of course, there is some interchangeable terminology at work here.
Dec 21, 2020 · The graph of \(f\) is concave up if \(f''>0\) on \(I\), and is concave down if \(f''<0\) on \(I\). Figure \(\PageIndex{3}\): Demonstrating the 4 ways that concavity interacts with increasing/decreasing, along with the relationships with the first and second derivatives.
The focal length f is positive for concave mirrors and negative for convex mirrors. The image distance di is positive for real images and negative for virtual images. Notice that rule 1 means that the radius of curvature of a spherical mirror can be positive or negative.
Jan 26, 2023 · focal lengths of concave mirrors are positive, \(f>0\). focal lengths of a convex mirrors are negative, \(f<0\). image distances for real images are positive, \(i>0\).
