KhanAcademyVideos

A conversation between Dr. Lauren Kilroy-Ewbank and Dr. Beth Harris in front of Félix Parra, Fray Bartolomé de las Casas, 1875, oil on canvas, 263 x 356.5 cm (Museo Nacional de Arte, Mexico City)

Finding the limit of g(h(x)) at x=-1 when the limit of h(x) at x=-1 doesn't exist. Does it mean that the composite limit doesn't exist? Not necessarily! See how we analyze it.

Suppose we are looking for the limit of the composite function f(g(x)) at x=a. This limit would be equal to the value of f(L), where L is the limit of g(x) at x=a, under two conditions. First, that the limit of g(x) at x=a exists (and if so, let's say it equals L). Second, that f is continuous at x=L. If one of these conditions isn't met, we can't assume the limit is f(L).

Finding the limit of g(h(x)) at x=1 when the limit of h(x) at x=1 is 2 and the limit of g(x) at x=2 doesn't exist. Does it mean that the composite limit doesn't exist? Not necessarily! See how we analyze it.

Sal graphs y=-cos(π⋅x)+1.5 by thinking about the graph of y=cos(x) and analyzing how the graph (including the midline, amplitude, and period) changes as we perform function stransformations to get from y=cos(x) to y=-cos(π⋅x)+1.5.

Sal graphs y=3⋅sin(½⋅x)-2 by thinking about the graph of y=sin(x) and analyzing how the graph (including the midline, amplitude, and period) changes as we perform function stransformations to get from y=sin(x) to y=3⋅sin(½⋅x)-2.

A conversation with Dr. Lauren Kilroy-Ewbank and Dr. Steven Zucker below a mudéjar-style ceiling, 16th century, carved, painted, and gilded wood, 28 x 33 ft., Spain (The Metropolitan Museum of Art)

Evan Freeman and Anne McClanan, PhDs in Byzantine Art History, here discuss a Byzantine chalice, now at the Metropolitan Museum of Art (Acc. 1986.3.2). Video Editor: Anna Weltner This video is available CC BY 4.0 Here's info on the object from the Met's website: https://www.metmuseum.org/art/collection/search/466136 Date: 500–650 Geography: Made in Attarouthi, Syria Culture: Byzantine Medium: Silver and gilded silver Dimensions: Overall: 9 11/16 × 6 9/16 in., 16.7oz. (24.6 × 16.7 cm, 474g) Diam. of foot: 3 15/16 in. (10 cm) Diam. of knop: 1 5/8 in. (4.1 cm) Capacity of cup: 2000 ml With a youthful Christ with a cruciform halo, a deacon saint with censer (probably Saint Stephen), a youthful saint with staff, the Virgin Mary in orant pose, a military saint in armor killing a dragon (Saint George ?), and a long-haired Saint John the Forerunner, under arcades Inscribed in Greek: Of Saint Stephen of the village of Attaroutha An unusual aspect of these chalices is their repeated representation of military saints. The figures in armor killing a dragon may be the earliest surviving depictions of Saint George, who according to tradition was martyred in the eastern Mediterranean in the fourth century or earlier.

A conversation between Dr. Lauren Kilroy-Ewbank and Dr. Beth Harris in front of Nicola da Urbino, armorial plate (tondino), The Story of King Midas, c. 1520–25, tin-glazed earthenware, 27.5 cm in diameter (The Metropolitan Museum of Art)

Eugène Delacroix, Greece on the Ruins of Missolonghi, 1826, oil on canvas, 208 cm × 147 cm (Musée des Beaux-Arts de Bordeaux). Speakers: Dr. Steven Zucker and Dr. Beth Harris

At the resonant frequency, the L.C.R. circuit has a minimum impedance and maximum current. Impedance is a minimum when capacitive reactance equals inductive reactance.

Transformers step up (increase) or step down (decrease) AC voltage using the principle of electromagnetic induction - mutual induction. A changing current in the primary coil induces an e.m.f in the secondary. Since the e.m.f generated depends on the number of turns, the voltage induced in the secondary can be changed - stepped up or down - by altering the turn's ratio.

Huygen's principle states that every point on a wavefront behaves as a source for secondary waves, whose common tangent (envelop) becomes the new wavefront. Using this principle, let's prove the laws of reflection

The frequency response graph is a graph of current vs frequency, keeping all other variables a constant. For LCR circuits, the graph has a peak at the resonant frequency. This is used to tune radios and other devices to specific radio frequency channels. A high-quality LCR resonance has a narrow graph, while a low-quality has a flatter graph - the sharpness determines the quality.

This video visualizes the counter intuitive phenomenon of the transformers, when voltage is stepped up, the current gets stepped down. This is a consequence of energy conservation. Transformers work on the principle of electromagnetic induction - mutual induction. Power line transmission requires both step up and step down transformers.

We can use the mean and standard deviation and normal shape to calculate probability in a sampling distribution of the difference in sample proportions.

We can calculate the mean and standard deviation for the sampling distribution of the difference in sample proportions. Also, we can tell if the shape of that sampling distribution is approximately normal.

Distillation is the process of separating the components of a liquid mixture through selective evaporation and condensation. The basis of separation is the difference in the vapor pressures (volatilities) of the respective components. To improve the separation in a distillation, chemists often use a fractionating column, which allows for multiple cycles of evaporation and condensation (this is known as fractional distillation).

The progress of a distillation can be represented using a distillation curve, which plots temperature versus volume of distillate collected. In this video, we'll learn how to interpret the various regions of a distillation curve for the fractional distillation of a 1:1 mixture of methyl and propyl acetate.

Column chromatography applies the same principles as TLC on a larger scale. In column chromatography, a glass column is filled with a stationary phase (typically silica gel), and the mixture of compounds to be separated is placed on top. Then, a mobile phase (solvent) is passed through the column from top to bottom. As with TLC, the rate at which each compound in the mixture moves down the column depends on its relative attractions to the stationary and mobile phases.