Exponential and Logarithmic Functions
To test this material, students can be given problems of exponential growth and logarithmic plots. For example, there are endless problems having to do with interest rates:
which implies that:
Differentiation and Integration of Logarithmic and Exponential Functions
In general it tends to be difficult for students to simply memorize all of these cumbersome formulas for derivatives. The easiest thing is to teach the simplest formulas:
This can be written as:
and so we can just use the Chain Rule:
can be written as:
and so we just get:
Exponential Growth and Decay
A nice problem to walk through with the students is the following:
so it would seem that our differential equation is solved by the function:
After completing this, it is worthwhile to erase the board and have students attempt to work either alone or in groups to solve a similar problem. This will force them to try recalling each step along the way.
Derivatives and Integrals Involving Inverse Trigonometric Functions
A quick glance at the plot of any trigonometric function:
shows that they all fail the horizontal line test miserably. However we can restrict their domains so that over these new functions defined only on the restricted domains do have inverses. Their plots over these restricted domains look like:
These functions, restricted to the smaller domains, clearly have no problems passing the horizontal-line test. Therefore on these domains the functions are invertible and the inverses are determined quite easily.
The other formulae can be similarly derived and leaving this out will only make the material seem more odd and difficult to swallow. Students will have an easier time using, manipulating, and recalling the ideas if they seem them presented fully.
Students now have the means to solve this problems and important ones like it using L’Hopital’s Rule:
To test this material it is sufficient to have students find some limits that are not obvious. However, it is always best to continuously remind them of how the results can often be predicted by making a sketch of the plot near the limiting point. Similarly, it is a good idea to be able to mentally plug in some values near the limit to see if any trend is clearly visible.