Written assignment unit 1 (math 1211)
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Nov 24, 2024
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Uploaded by AngelRoseBenedict
Written Assignment Unit 1
1. What linear function,
has
and
?
2. If
, what is
?
3. Find all solutions to the equation
. Write your answer in radians in terms of
4. Sketch the graph of
. Find the domain, range, and horizontal asymptote. Include the horizontal asymptote in your graph.
5. Sketch the graph of
. Find the domain, range, and vertical asymptote. Include the vertical asymptote in your graph.
6. Find the domain fo the function
.
7. From the graph below, find what is
and for what numbers
is
.
8. Determine whether the graph is that of a function. If it is, use the graph to find its domain and range, the intercepts, if any, and any symmetry with respect to the x-axis, the y-axis, or the origin.
9. Determine whether the function is even, odd, or neither.
a.
b.
c.
10.
A cellular phone plan had the following schedule of charges: Basic service, including 100 minutes of calls is $20.00/month; 2nd 100 minutes of calls is $0.075/minute; additional minutes of calls is $0.10/minute.
a. What is the charge for 200 minutes of calls in one month?
b. What is the charge for 250 minutes of calls in one month?
c. Construct a function that relates the monthly charge C for x minutes of calls?
Solution:
1. We can use the slope-intercept form of a linear equation to find the linear function y = mx + b, where m is the slope and b is the y-intercept. We can use the given points to find the slope:
m = (f(7) - f(0)) / (7 - 0) = (14 - 8) / 7 = 6/7
Here we use the slope and one of the points to find the y-intercept:
8 = (6/7)(0) + b b = 8
Therefore, the linear function is:
f(x) = (6/7)x + 8
2.
To find f(t + h) - f(t) / h, we substitute the given values into the function and simplify:
Now we can simplify further by factoring out a common factor:
Therefore, 3.
To solve the equation , we can use the identity to rewrite it as:
Simplifying, we get:
Dividing both sides by Cos(x) (which is not zero), we get:
Therefore, is not a real number, and there are no solutions to the equation.
4.
To sketch the graph of y = log base , we can start by finding the domain, range, and horizontal asymptote.
The domain is all positive real numbers greater than -2, since the argument of the logarithm must be positive.
The range is all real numbers, since the logarithm can take on any real value.
The horizontal asymptote is y = 5, because as x approaches infinity or negative infinity, the logarithm approaches infinity or negative infinity at a slower rate than the constant term.
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To sketch the graph, we can plot the vertical asymptote at x = -2 and the horizontal asymptote at y = 5. Then we can plot a few points and draw a smooth curve that approaches the asymptotes.
5.
To sketch the graph of y = , we can start by finding the domain, range, and vertical asymptote.
The domain is all real numbers, since any real number can be raised to a power.
The range is all positive real numbers greater than 5, since the base of the exponential function is positive, and the constant term is 5.
There is no vertical asymptote, since the exponential function can take on any positive value.
To sketch the graph, we can plot the point (-2, 5) and a few other points, and
then draw a smooth curve that increases rapidly as x approaches negative infinity and approaches the horizontal asymptote y = 5 as x approaches infinity.
6.
To find the domain of the function , we need to find all values of x that make the denominator zero, since division by zero is undefined.
x = ±9i
Therefore, the domain of g(x) is all real numbers except ±9i.
7.
To find f(-15), we need to look at the graph and find the point where x = -15. From the graph, we can see that the point where x = -15 is on the y-axis and has a y-coordinate of 2.
Therefore, f(-15) = 2.
To find the values of x where f(x) = 0, look for x-intercepts of the graph. From the graph, we can see that there are two x-intercepts, one at x = -3 and another at x = 4. Therefore, f(x) = 0 when x = -3 or x = 4.
8.
The graph is that of a function because it passes the vertical line test. The domain of the function is all real numbers since there are no restrictions on the input values. The range of the function is from negative infinity to positive infinity since the graph extends indefinitely
in both the positive and negative y-directions.
The y-intercept of the graph is at (0, 1). There are no x-intercepts other than the ones already mentioned in question 7. The graph is symmetric with respect to the origin since it is an odd function.
9. a. To determine whether is even, odd, or neither, we need to check whether , or neither.
Since
, the function is odd.
b. To determine whether is even, odd, or neither, we need to check whether
or neither.
Since , the function is even.
c. To determine whether is even, odd, or neither, we need to check whether
, or neither.
Since , the function is odd.
10. a. For 200 minutes of calls, the first 100 minutes are included in the basic
service, so we only need to calculate the charge for the additional 100 minutes. The charge for the second 100 minutes is 100 x $0.075 = $7.50. Therefore, the total charge for 200 minutes of calls is $20.00 + $7.50 = $27.50.
b. For 250 minutes of calls, the first 100 minutes are included in the basic service, so we only need to calculate the charge for the additional 150 minutes. The charge for the second 100 minutes is $7.50, and the charge for
the remaining 50 minutes is 50 x $0.10 = $5.00. Therefore, the total charge for 250 minutes of calls is $20.00 + $7.50 + $5.00 = $32.50.
c. Let C(x) be the monthly charge for x minutes of calls.
For 0 ≤ x ≤ 100, C(x) = $20.00
For 100 < x ≤ 200, C(x) = $20.00 + (x - 100) x $0.075
For x > 200, C(x) = $20.00 + $7.50 + (x - 200) x $0.10
Therefore, the function that relates the monthly charge C for x minutes of calls is:
C(x) = $20.00 if 0 ≤ x ≤ 100 $20.00 + (x - 100) x $0.075 if 100 < x ≤ 200 $20.00 + $7.50 + (x - 200) x $0.10 if x > 200
Reference:
Herman, E. & Strang, G. (2020).
Calculus volume 1
. OpenStacks. Rice University.
https://openstax.org/books/calculus-volume-1/pages/2-introduction
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