concave

Examples of concave in the following topics:

• Concavity and the Second Derivative Test

  • A related but distinct use of second derivatives is to determine whether a function is concave up or concave down at a point.
  • Specifically, a twice-differentiable function $f$ is concave-up if $f''(x)$ is positive and concave-down if $f''(x)$ is negative.
  • If it is concave-up at the point, it is a minimum; if concave-down, it is a maximum.

• Image Formation by Spherical Mirrors: Reflection and Sign Conventions

  • Spherical mirrors can be either concave or convex.
  • A summary of the properties of concave mirrors is shown below:
  • The focal point is the same distance from the mirror as in a concave mirror.
  • This figure shows the difference between a concave and convex mirror.
  • This is a ray diagram of a concave mirror.

• Derivatives and the Shape of the Graph

  • The shape of a graph may be found by taking derivatives to tell you the slope and concavity.
  • At an inflection point, a function switches from being a convex function to being a concave function or vice versa.
  • Sketch the shape of a graph by using differentiation to find the slope and concavity

• Aging and the Skeleton

  • One example is Hyperkyphosis, colloquially called a "dowager's hump", which refers to an anteriorly concave, curvature of the thoracic region of the spine .
  • Hyperkyphosis, colloquially called a "dowager's hump,, refers to an anteriorly concave, curvature of the thoracic region of the spine.

• Linear Approximation

  • If $f$ is concave-down in the interval between $x$ and $a$, the approximation will be an overestimate (since the derivative is decreasing in that interval).
  • If $f$ is concave-up, the approximation will be an underestimate.

• Ilium

  • The body contributes to the formation of the acetabulum, a concave structure where the head of the femur articulates to form the hip joint.
  • The internal surface of the ala is smooth and concave and forms the iliac fossa.

• Capillary Action

  • When the lower end of a vertical glass tube is placed in a liquid, a concave meniscus forms.
  • It can be either convex or concave.
  • Conversely, a concave meniscus occurs when the molecules of the liquid are attracted to those of the container, causing the surface of the liquid to cave upwards.
  • Capillary action acts on concave menisci to pull the liquid up, increasing the favorable contact area between liquid and container, and on convex menisci to pull the liquid down, reducing the amount of contact area.

• Refraction and Magnification

  • In general, two types of lenses exist: convex lenses, which cause parallel light rays to converge, and concave lenses, which cause parallel light rays to diverge.

• The Spine

  • Kyphosis is an exaggerated concave (kyphotic) curvature of the thoracic vertebral column; it is commonly known as "humpback."
  • The pelvic curve begins at the sacrovertebral articulation, and ends at the point of the coccyx; its concavity is directed downward and forward.

• Types of Synovial Joints

  • The condyloid joint occurs where an egg-shaped suface of a bone fits into a concavity in another bone.
  • The surface of a saddle joint has both convex and concave areas which resemble a saddle and permit the same movements as the condyloid joints.