Expansion Rate
Expansion Rate - Related Videos
09:42
Analysis of Thermal Expansion via Dilatometry
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2023
Source: J. Jacob Chavez, Ryan T. Davis, and Taylor D. Sparks, Department of Materials Science and Engineering, The University of Utah, Salt Lake City, UT Thermal expansion is extremely important when considering which materials will be used in systems that experience fluctuations in temperature. A high or low thermal expansion in a material may or may not be desirable, depending on the application. For instance, in a common liquid thermometer, a material with a high thermal expansion would be...
02:53
Reaction Rate
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2020
The rate of reaction is the change in the amount of a reactant or product per unit time. Reaction rates are therefore determined by measuring the time dependence of some property that can be related to reactant or product amounts. Rates of reactions that consume or produce gaseous substances, for example, are conveniently determined by measuring changes in volume or pressure. The mathematical representation of the change in the concentration of reactants and products, over time, is the rate...
03:14
Ex Vivo Expansion of Natural Killer Cells from Peripheral Blood Mononuclear Cells
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2025
This video demonstrates a technique for the expansion of natural killer (NK) cells directly from peripheral blood mononuclear cells (PBMCs) using a feeder cell line. PBMCs are co-cultured with genetically modified lymphoblastoid feeder cells expressing membrane-bound interleukin-21 (mIL-21), and the culture is supplemented with the cytokines interleukin-2 (IL-2) and interleukin-15 (IL-15). Together, this cocktail of ligands leads to the prolonged expansion of NK cells with high purity.
01:24
Heat and Free Expansion
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2023
The work done by a thermodynamic system depends not only on the initial and final states but also on the intermediate states—that is, on the path. Like work, when heat is added to a thermodynamic system, it undergoes a change of state, and the state attained depends on the path from the initial state to the final state. Consider an ideal gas cylinder fitted with a piston. When the cylinder is heated at a constant temperature, the gas molecules absorb energy and expand slowly in a controlled...
01:22
Thermal Expansion
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2023
The expansion of alcohol in a thermometer is one of many commonly encountered examples of thermal expansion, which is the change in size or volume of a given system as its temperature changes. The most visible example is the expansion of hot air. When air is heated, it expands and becomes less dense than the surrounding air, which then exerts an upward force on the hot air to, for example, make steam and smoke rise, and hot air balloons float. The same behavior happens in all liquids and gases,...
FAQs
The derivative provides a local reading of how a function responds at a particular input, rather than summarizing its behavior over an entire interval. Examining f'(t) at different values of t shows whether the function’s absolute change remains consistent or varies as the input changes. This makes the derivative useful for interpreting local behavior in a mathematical model.
Relative expansion rate becomes important when the same absolute change has different significance at different scales. The ratio f'(t)/f(t) compares the current change with the current value, provided f(t) is nonzero. Consequently, it supports proportional comparisons between quantities or between different stages of one model, whereas f'(t) reports change in the quantity’s own scale.
These growth patterns can be compared by inspecting both the derivative and, when appropriate, the relative rate across inputs. A model’s rate may be considered at one point or tracked as the input changes, revealing whether its behavior is consistent or variable. This comparison helps identify which mathematical description best represents the trend being modeled.
To compute an Expansion Rate, begin with the modeled quantity f(t) and differentiate it to obtain f'(t). Evaluate that derivative at the input of interest for the instantaneous value. If proportional change is the goal and f(t) is nonzero, also calculate f'(t)/f(t). This workflow separates absolute and relative assessments.
The relative calculation requires f(t) to be nonzero because the expression divides f'(t) by f(t). At an input where the modeled quantity equals zero, that ratio cannot be used as stated. In that situation, the derivative still provides the absolute rate, so analysis can continue without making an invalid relative comparison.
It can support models of populations, investments, algorithms, and dynamical systems. In each setting, examining the relevant rate helps compare behavior, estimate how a quantity may change, and assess longer-term trends. The same mathematical framework therefore connects local calculations, proportional comparisons, and broader interpretation without requiring every application to use the same underlying model.