# Mathematics for Technology II (Math 2131) Durham College, Mathematics
Free • 0 lessons
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• 14 week duration
• ##### Solving Systems of Equations

This unit introduces how to systematically solve a system of equations, namely linear equations. Examples of non-linear systems, including systems of 3 unknowns will be of emphasis.

• ##### Graphs of Trigonometric Functions

The unit focuses primarily on how to graph periodic sinusoidal functions, and how to identify features of a waveform to produce an equation by inspection.

• ##### Polar Coordinate Functions

An introduction to the polar coordinate system.

• ##### Complex Numbers

This unit is an extension of what was introduced in Math 1131. To learn how to work with radicals, knowing your exponent laws in crucial. Hence, this unit begins with a thorough review.

• ##### Logarithmic Functions

This chapter introduces you to exponential functions, and how they can be solved using logarithms.

• ##### Trigonometric Identities and Equations
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• ##### Analytic Geometry
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## Mathematics for Technology II (Math 2131)

### Exponential Growth to an Upper Limit

Sometimes you’ll be asked to find the final amount of a substance that’s decreasing exponentially relative to its initial amount. For instance, suppose that a bucket initially contains a x liters of water, and a hole is placed at the bottom where water leaks out at an exponential rate. Then assume that the amount remaining in the bucket, in turn, decreases exponentially. The formula shown below allows you find out how much water has poured out ‘y’ at any instant ‘t’ you substitute into the equation:

Upper limit formula:

$y=a–a{e}^{–nt}\phantom{\rule{0ex}{0ex}}$

Now factor the ‘a’ from both terms:

$y=a\left(1–{e}^{–nt}\right)$ Graphically, the upper limit resembles this graph when plotted. Substituting a very large number into ‘t’ should get you the initial amount ‘a’.

To reduce confusion, most questions seeking an upper limit will explicitly mention the phrase ‘upper limit’ in the question, and this is typically the case. Examples of these types of questions are provided in the videos below for clarity.

• A second example can be watched here. Interestly, this question does not mention “upper limit”, so it’s important that you recognize what’s being asked based on the wording.

Try one more:

A steel casting initially at 0°C is placed in a furnace at 801 °C. If it increases in temperature at the rate of 5.00% per minute, find its temperature after 20.0 min.