Solutions to Math 1411- Test 1
1. Estimate the limit by substituting smaller and smaller values of h. If the values start to cluster about a certain number, that number is the limit. Give answer to three decimal places.
a. 
Answer: 1.099
Solution:
|
|
|
|
0.1 |
1.16123 |
|
0.001 |
1.09921 |
|
0.00001 |
1.09862 |
|
0.0000001 |
1.09861 |
|
0.000000001 |
1.0986 |
Since the
instructions were to round to three decimals, then 
.
b.
Answer: 1.359
Solution:
|
|
|
|
0.1 |
1.42942 |
|
0.001 |
1.35982 |
|
0.00001 |
1.359148 |
|
0.0000001 |
1.359141 |
|
0.000000001 |
1.35915 |
Since the instructions were to round
to three decimals, them 
2. Use the following figure to give approximate values for the following limits (if they exist).
a. 
Answer: 3
Solution: As you evaluate the function visually from the graph, the function values get closer and closer to 3 as x approaches – 4 from the left and as x approaches – 4 from the right.
b.
Answer: The Limit Does Not Exist.
Solution:
As you approach – 1 from the left, the function values approach
2. As you approach – 1 from the right, the function values approach
5. Therefore, there is no limit at 
c.
Answer: The Limit Does Not Exist
Solution:
As you approach 2 from the left, the function values approach 3. As you
approach 2 from the right, the function values approach 2.
Therefore, there is no limit at 
d.
Answer: 0.5
Solution: As you evaluate the function visually from the graph, the function values get closer and closer to 0.5 as x approaches 6 from the left and as x approaches 6 from the right.
3.
If 
and
, evaluate the
following limits if they exit. If they do not exist, explain why.
a. 
Answer: -9
Solution:
Remember that polynomials are continuous everywhere.
b.
Answer: 0
Solution:
The function 
is only undefined at 
Therefore 
.
c. 
Answer: The Limit Does Not Exist
Solution:
The function 
is undefined at 
. Therefore, the limit does
not exist at .
d. 
Answer: 13
Solution:
Both of the functions are polynomials and 
is a polynomial.
Polynomials are continuous everywhere; therefore,
4. You have $500 invested in a bank account earning 8.2% compounded annually.
a. Write an equation for the money M in your account after t years.
Answer:
Solution: Since the money is increasing at an annual rate of 8.2%, the base is 1.082.
b. How long will it take to triple your money?
Answer: 13.94 Years
Solution:
c. Suppose the interest were compounded monthly instead, that is you earned interest each month. What interest would you then earn for 1 year?
Answer: 8.52%
Solution:
At the end of one year, one dollar grows to 
. So the annualized rate is
8.52%.
5. Find a possible formula for each of the following functions. Check that your formula fits the data points.
(a)
Answer: 
Solution:
This can either be represented by a sine function or a cosine function.
Since the highest point is on the y-axis, let’s use the cosine function. The
typical model for the cosine function is 
.
Since the highest point of the graph is on the y-axis, there is no
horizontal shift and thus 
.
Two full cycles occur in 8
units; therefore, the period is 4 units. Since 
, then 
.
A represents the amplitude which is half the distance between the highest point and the lower point or 2.5.
D represents the midpoint which is 5.
is the correct model.
Let’s check it with the point 
. 
(b)
Answer: 
Solution:
This is an exponential function of the form 
. Using the point 
yields 
. Using the other
two points, we get
Let’s check it with the
point 
.
(c)
Answer: 
Solution:
This is a polynomial of degree 3 with a zero at and a double zero at
. We can
write it as 
.
Use the point 
to find the value of a. 
and 
.
can be
written as
Let’s check it with the
point 
.
