MTH 251 - Week 3 Notes

MTH 251 - Week 3 Notes

Use the graph shown in Figure 1 to estimate the value of for the function . Next, find the exact value of using the formal definition of the first derivative function at a point. Then find the equation of the tangent line to at the point .

Solution: Using the points and , I can see the slope is

Note that I did not write any limit laws or show any of the limit law steps. This was only a requirement up through test 1. Since this is material after test 1, I am no longer required to show these steps.

Since the slope is , I can use the point and the slope, , to find the equation of the tangent line as follows.

or

The equation of the tangent line is

Slope Units

Recalling the classic slope formula "rise over run", it naturally follows that if you are working with an applied function then the units on the slope of the function are the "rise units" over the "run units." Since the rise is associated with the output of the function and the run is associated with the input of the function, it follows that the units on the slope of an applied function are the function's "output units" over the function's "input units."

Let's find the slope units for each of the following functions; state the practical meaning of the slope.

is the population of Oregon (in million people) t years after 1900. Pretend that the slope value is constantly 0.025.

The slope units are and the slope means that the population of Oregon is increasing at a constant rate of

is the velocity of Newton's apple (in ft/sec) t seconds after the apple began its plunge towards earth. Pretend that the slope value is constantly -32.1.

The slope units are and the slope means that the velocity of Newton’s apple is decreasing at a constant rate of

Note that while the value of the slope is , I do not use the negative in my sentence. The fact that the velocity is decreasing tells me the slope is negative. Note also that you would probably write the slope units as in a physics class. However, we will never reduce the slope units in this class. The units are important in interpreting the slope.

Notes about Homework for Lab 3

There is an excellent example for Activity 1. You should follow this format when writing your solutions. In particular, make sure you use a complete sentence for the tangent line.

When grading problem set 3, I will be looking for the phrase “increasing at a rate of” or “decreasing at a rate of.” You will not receive full credit unless you interpret the slope as a rate of change. Also, you need to use complete sentences (and, of course, follow the directions). You could answer all of the parts in one sentence (if you want to). For example:

The slope is 0.30 dollars/donut which means that the total cost of the donuts is increasing at a rate of 30 cents per donut. (Note that I have given the slope, the units of the slope and a practical interpretation of the slope all in one sentence.)

Page 2 of 2