For each of the 6 problems linked below, make a table that describes the position, velocity, and acceleration indicated by the graph. Make sure you notice what kind of graph you are given – is it a position vs. time graph, or a velocity vs. time graph?
In all of these problems, the up direction is positive, the down direction is negative. Let’s use a rounded value for the acceleration due to gravity: -10 m/s2.
- An astronaut in outer space throws a rock. The rock leaves her hand and travels toward the Milky Way at 10 m/s. Where is the rock 2 seconds later?
- The same astronaut is now on earth and throws the same rock toward the Milky Way – assume she throws the rock straight up and it leaves her hand with a speed of 10 m/s. Where is the rock 2 seconds later? What is the difference between situation 1 and situation 2?
- Imagine a shelf full of dishes. The shelf is above a counter. A person grabs a dish and lowers the dish to the dish to the counter. Describe the velocity and acceleration of the dish as it is moved from shelf to counter.
- At the same shelf of dishes and counter, a cat knocks a dish down and the dish falls to the counter and breaks. Describe the velocity and acceleration of the dish as it is moved from shelf to counter. Why does it break? What is the difference between situation 3 and situation 4?
Which of these objects is at mechanical equilibrium:
- A seagull standing on top of a fence post.
- A car moving due south at 40 kilometers per hour.
- A water drop from a kitchen tap, just before it hits the surface of the sink.
- A rain drop from the sky, just before it hits the ground.
- A car as it enters the freeway, traveling from a surface road where the speed limit is 60 km/h to the freeway where the speed limit is 100 km/h. The driver’s foot is pressing the accelerator.
- A can in a factory on a conveyor belt moving at 0.1 m/s.
- A bicycle wheel rolling in a straight line at 50 revolutions per second.
- A coin after it is dropped from someone’s hand and before it hits the ground.
- A rock thrown up in the air, at the very top of its journey through the air.
- The earth as it travels around the sun. (This is a trick question, something to think about. Assume that the earth travels around the sun in a circle, and assume that the earth’s speed is constant. The earth’s orbit is approximately circular and its speed is approximately constant – after all, it goes around pretty steadily once a year.)
Horizontal Asymptotes (end behavior, symmetry)
If the degree of the numerator of a rational function is less than the degree of the denominator, the function has a horizontal asymptote. Use division by the highest power of x to determine the equation of the horizontal asymptote.
If the degree of the numerator is one more than the denominator, there is a slant asymptote. Use long division to find the equation of the slant asymptote.
If the degree of the numerator is two or more than the denominator, the end behavior of the function does not follow a linear asymptote. Instead, it is similar to that of a polynomial of the same degree as the difference between numerator and denominator.
What is the end behavior of each of these functions? If there is a horizontal or slant asymptote, what is its equation?
1. f(x) = (2x + 1) / (x – 4)
2. f(x) = (x2 + 3) / (x – 1)
3. f(x) = (3x4 – x3 + 5) / (x2 + 7)
4. f(x) = 17 / (x2 + 4x – 5)
5. f(x) = (6x2 + 1) / (3x2 – 1)
Sign (intervals of definition, increasing, decreasing)
The graph of a rational function can be broken into disconnected segments by the presence of vertical asymptotes. Remember, at a vertical asymptote the function approaches either positive or negative infinity (sometimes both). To graph a rational function correctly it helps to determine where it is positive or negative.
Here is an organized way to do this.
A. Find the zeros and undefined x-values of the function (places where numerator or denominator are zero).
B. Arrange these values in increasing order.
C. Divide the x-axis into intervals around these values. For example, if the values are -1 and 0, the intervals are:
-1 < x, -1 < x < 0, and 0 < x. If there are N values, there will be N+1 intervals.
D. Test the sign of the function on each interval. You don’t have to compute exact values of the function, just check to see if it is positive or negative.
Here is a set of problems that guide through the process:
1. f(x) = (x2 + x) / (x2 + 6x + 5)
Factor the numerator and denominator of f(x).
2. Where does f(x) have zero value(s)?
3. Where does f(x) have vertical asymptote(s)?
4. Where does f(x) have a hole?
5. Using the information from problems 1-4, test the sign of f(x) over the appropriate intervals.
Asymptotes are lines that the graph of a function approaches. A vertical line in the x-y plane is a line that goes to plus and minus infinity. So vertical asymptotes indicate that the values of a function approach infinity (or minus infinity) as x approaches a finite value.
Example notes: The function f(x) = 1/(x – 1) has a vertical asymptote at x = 1. But the function g(x) = (x2 – 1)/(x – 1) does not have a vertical asymptote at x = 1. As long as x is not equal to 1, the function g(x) is equivalent to x + 1, which has no vertical asymptote. (Factor and simplify g(x) to make sure that this is true.)
Which functions below have vertical asymptotes? If there are any vertical asymptotes, what are their equations?
1. f(x) = 3 / (x – 2)
2. f(x) = (x3 + x)/x
3. f(x) = x / (x2 + 5)
4. f(x) = (2x3 – 32) / (x2 – 6x + 5)
5. f(x) = 1 – 1/x + 2/(x – 1)
The range is the set of all values taken by the function. State the range of the following rational functions, in interval notation.
1. f(x) = 1/(x – 1)
2. f(x) = 3
3. f(x) = 3/(x2 + 1)
4. f(x) = 1/x2
5. f(x) = (x3 + 1)/x
The definition of a rational function is: it can be written as a ratio of two polynomials.
1. Suppose f(x) = 1/x and g(x) = 1/(x + 1). Are f(x) and g(x) both rational functions?
2. Is h(x) = f(x) + g(x) from problem 1 above a rational function? Explain.
3. What is the domain of f(x) = 1/x?
4. What is the domain of the rational function f(x) = (x – 1) / (x2 + 1) ?
5. What is the domain of h(x) = 1/x + 1/(x + 1)?
There are connections between complex numbers and geometry. You can graph complex numbers in the complex plane, where the horizontal axis is the real part, and the vertical coordinate is the complex part. The complex number 2 + 3i, for example, would occupy a point similar to the coordinates (2, 3) in the real plane.
1. Graph the number 1 in the complex plane. (Hint, 1 as a complex number is 1 + 0i, and it occupies a point similar to (1,0) in the real plane). Multiply the number 1 by i and graph the result. Do it again and again. What is the geometrical effect of multiplying by i?
2. Graph the complex number 1 + i in the complex plane. What is the effect of multiplying this number by i? Graph the result.
3. Graph the complex number 2 + i in the complex plane. Multiply this number by the real number 2 and graph the result. What is the effect of multiplying by 2?
4. Graph the complex number -i in the complex plane. Multiply -i by 2 and graph the result. What is the effect of multiplying by 2?
5. Graph the complex number 1 + 2i in the complex plane. Multiply this number by -1 and graph the result. What is the effect of multiplying by -1?
Roots of Unity
1. The non-real number i raised to what power is one? (Of course, when raised to the zeroth power it is one, but there are other powers as well.)
2. Compute the square of:2-1/2(1 + i)
3. Compute the cube of: 2-1/2(1 + i)
4. What power of 2-1/2(1 + i) is one? In other words, 2-1/2(1 + i) raised to what power is one?
5. Compute the square of: 2-1/2(1 – i)