Math Insight

Pages similar to: Basic idea and rules for logarithms

• The derivative of the natural logarithm
  Determining the derivative of the natural logarithm from the fact that it is the inverse of the exponential function.
• Basic rules for exponentiation
  A brief overview of the basic rules for exponents or powers.
• Exploring the derivative of the exponential function
  A guided tour into the reasons that the derivative of the exponential function with base e is the function itself.
• Developing an initial model to describe bacteria growth
  By analyzing some data and hypothesizing rules for cell division, we develop a discrete dynamical system for the growth of a population of bacteria.
• An introduction to ordinary differential equations
  An introduction using simple examples explaining what an ordinary differential equation is and how one might solve them.
• Developing a logistic model to describe bacteria growth
  Modeling the slowing growth rate caused by environmental carrying capacity in bacteria growth data.
• From discrete dynamical systems to continuous dynamical systems
  Illustration of how a discrete dynamical system approaches a continuous dynamical as the size of the time step shrinks.
• Initial dynamical systems exploration
  Exploring models of anteater population growth and lead decay in the bloodstream using interactive applets.
• The idea of the chain rule
  An illustration of the basic concept of the chain rule using interactive graphics to diagram the relevant points on the graphs and the corresponding slopes.
• Spruce budworm outbreak model
  A graphical analysis of a differential equation model reveals how the spruce budworm could devastate forests every 40 years.
• Introducing rabbit predators
  Exploring the interactions between fox and rabbit populations when foxes are introduced to an exploding population of rabbits.
• A model of chemical pollution in a lake
  A discrete dynamical system model of chemical pollution in a lake illustrates the concept of movement toward equilibrium.
• Approximating functions by quadratic polynomials
  Determining how to use information from the second derivative to form a quadratic approximation to a function.
• Using the Forward Euler algorithm to solve pure-time differential equations
  By pretending that the slope of a function is constant over small intervals, we following tangent lines to estimate the solution to pure-time differential equations.
• Discrete dynamical systems as function iteration
  The simplest versions of discrete dynamical systems can be viewed as evolving due to iterating a function repeatedly.
• The idea of a probability density function
  A probability density function captures the probability of being close to a number even when the probability of any single number is zero.
• Redundant parameters in the exponential function
  Calculation showing that in the exponential function, changing either the base or a constant in the exponent have the same effect on the function.
• Solutions to elementary discrete dynamical systems biology problems
  Solutions to sample problems illustrating discrete dynamical systems in the context of biology.
• A discrete SIR infectious disease model
  
• Exponential growth and decay: a differential equation
  Solving a differential equation to find an unknown exponential function.

Back to: Basic idea and rules for logarithms