Prepared: Wed Apr 14 17:09:55 CDT 1999

MATH 1151

Quiz II

1) Solve the following equation:

(Note that and .)

Since exponential functions are one-to-one functions, so is . This means if , the exponents, and must be equal, in other words . In our case then we have

2) Graph ALL of the following functions on the SAME set of axes. Make sure that EACH graph (of each function) is CLEARLY distinguishable from the others and the axes. You DO NOT have to WORRY about PROPORTIONALITY.

Observe that being of the form for b>1 will have an increasing type exponential graph. Observe next that can be rewritten as hence it will be the reflection of the first graph with respect to the y-axis. The graph of should be obvious. The last graph will be the contraction of the second graph because the last function is obtained by replacing x be 100x in the second function.

3) Solve the following equation:

There are two ways of solving the rest. First we try to factor the polynomial on the left. 35 has factors 7 and 5, so trying all possible combinations we find that

has zeros

If you cannot see the factorization easily, the second method, which is more direct but rather requires more work, is using the quadratic formula (after making sure that you remember it correctly).

In our case a=2, b=9, c=-35. So zeros are

4) For the following question write down the formulas you use. At the end, circle your answer.

Is it better to invest at 8% compounded quarterly or 7.75% compounded continuously?

The investment formula for interest compounded n times with annual interest rate r is

So $100 in the first plan will become

after one year.

When an interest is compounded continuously, we use the growth function model to find the balance with interest rate r.

So $100 in the second plan will become

after one year.

Obviously the first plan is better since the balance is larger.

5) For the following question write down the formula you use.

In 1930 the population of the northeastern US was 34,427,000. In 1950 the population was 39,478,000. According to the model for the population growth, a) predict the population in the year 2000 and b) find the population predicted by this model in 1980 and compare to the actual population of 49,135,000.

In this problem we use the growth function model.

Taking 1930 as the initial time, the function becomes

To find out what r is we use the population in 1950. 1950 is 20 years after 1930, so we should have P(20)=39478000.

Divide both sides by 34427000:

Take ln of both sides:

Divide both sides by 20:

Now we know r, hence we can rewrite the growth function.

a) 2000 is 70 years after 1930, so to find out the predicted population just plug in t=70.

b) 1980 is 50 years after 1930.

Comparing the predicted population and the actual population should be in terms of percentage.

So the error is less than 1.3%.

Note: You do not need to worry if your quiz did not mention about the year 1980. I made a printing mistake and I did not realize it in the first section. Students in the first section will not be penalized if they did not work with 1980.