##### Northwestern University

Math

##### June , 1990

Other

##### University of Minnesota

Master's
### Education

Northwestern University (Math)

June , 1990 (Other)

University of Minnesota (Master's)

##### About Ronald

Tutoring and Teaching Approach to Differential Calculus

In calculus the concepts of variables, functions and rates of change of functions are of fundamental importance. A variable often denoted by x, y, z or t, is a quantity that is able to vary: as opposed to a constant such as 1, 2, -3, etc., which are fixed numbers. So remember that a variable is able to vary. Numerically a function may be thought of as a relationship between two variables, the independent variable and the dependent variable, such that for every value of the independent variable there is one and only one value of the dependent variable. The term “dependent variable” may be used inter-changeably with the term “function”. We may state this briefly as follows: the value of a function depends upon the value of the independent variable.

A good practical example of a function, which may be denoted by x, is the distance (from a fixed point called the origin) that a car travels on the expressway in a given amount of time which we may denote by t. This fact can be concisely written as follows: x = x(t). This may be read as “x equals x of t” which means that x is a function of t: where in our example x is the distance traveled and it depends upon the elapsed time t.

The instantaneous rate of change of a function x with respect to an independent variable t is called the derivative of that function with respect to the independent variable. It is denoted by dx/dt and is read as the derivative of x with respect to t. In our example dx/dt would be the instantaneous velocity of the car (if it is moving in a straight line) or the speed of the vehicle. Acceleration is the derivative with respect to time of dx/dt and is called the second order derivative of x with respect to t. Acceleration is the instantaneous rate of change of velocity.

I also tutor integral calculus, and all levels of high school and college math through ordinary and partial differential equations. Most branches of physics and chemistry make extensive use of math and I tutor those subjects also.

I earned a BS degree in math from Northwestern University, Evanston, IL in 1990 and an MS degree in math from the University of Minnesota, Minneapolis, Minnesota in 1993.

I taught math in the Chicago Public Schools and at the City Colleges of Chicago. I was a graduate teaching assistant in math at the University of Minnesota.
Tutoring and Teaching Approach to Differential Calculus

In calculus the concepts of variables, functions and rates of change of functions are of fundamental importance. A variable often denoted by x, y, z or t, is a quantity that is able to vary: as opposed to a constant such as 1, 2, -3, etc., which are fixed numbers. So remember that…
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