IMPORTANT! The hints for each problem are in white text, so you need to highlight the text to be able to see it.
Email me if you're stuck, and maybe I can give you some more hints.
Homework 3 - Due Wednesday 11/7
Problem 3, #59
Look at page 136 for the limit of e^x as x->negative infinity.
Problem 4 - all
Use the formula given in class to find the slope of the tangent. Then use point-slope form: y - y1 = m(x - x1) to write the equation of the tangent line.
Problem 6
Use a table to find the limits... some values for x could be 10, 100, 1000, 10000. That should be enough to get you going for each one. You should find that the values in d match up with the values from a-c. If they don't, you messed up somewhere.
Problem 7
Part c - use the formula I gave you in class for the secant line (average velocity). In part d, find the limit to get the slope of the tangent (instantaneous velocity).
Problem 8
This will be a graded problem, and you will need to show your work to get credit. So, some hints: First of all, make sure that your values in a match up with the graph. You might question yourself because the change at the beginning of the time is slow and the change in population at the end of the ten years is fast.You don't need to show every step of the calculation, but show enough so that I can tell you didn't just copy the answers.
For parts b and c, you can get the maximum wolf population from the graph, then use the function to solve for the time. Once you know the time, plug it into the caribou function to get their population.
For pard d, it is asking for the limit as t->infinity (beyond ten years). As if the population would follow this model forever.
Problem 9
You won't get a numerical answer in this problem because you don't know the values of a and m.
Homework 2 - Due Wednesday 10/10
Problem 7
At the end, you are asked to find a simple formula for the average velocity between 1 second and 1 + delta t sec. Use your average velocity formula and simplify [f(1 + delta t) - f(1)]/[(1 + delta t) - 1].
Problem 9
[0,2] is interval notation, not a coordinate pair. It means find the average velocity from 0 to 2 seconds. As you learned in the previous worksheet, average velocity is the same as slope. In this case, it would be [f(2)-f(0)]/[2-0]. Repeat for each interval. If you have a graphing calculuater, the tip I gave you in the Module 2 Student Guide will come in handy here. :-)
Problem 10
I cannot stress how important it is to read the parametric equations handout if you are struggling with this question. Once you have...
Let the origin be the ground under the axis of the ferris wheel (so that the center is at the point (0, 103). Remember that the fancy "w" is the greek letter omega and represents angular speed. You need to find the angular speed using the formula omega=theta/t. Once you have omega, you can represent the angle as a funcion of time.
From here, I found the funtions two different ways. 1: sketch a triangle that goes the the center of the wheel, through point Q, and a vertical line. Use that to develop sin and cos functions in terms of theta (which is then in terms of time). 2: sketch your triangle along the radius that has been drawn for you, and a horizonal line under that radius. The benefit of this point of view is that you will get the same answer as given in the online answer key. However, if you use this method, you have to keep in mind that the angle in this diagram is actually pi/2 less than the angle than the rider has traveled.
To find the time at 80 feet: once you have the function, it is easy to find the first time (on the way up). To find what time she hits 80 on the way down, you could find the total time of revolution and go back from there.
Homework 1 - Due Thursday 9/20
Problem 4
There are two unknowns (the x- and y-coordinate of the point of tangency). If you can find these, you can write the equation in point-slope form.
To find two unknowns, you'll need two equations.
One equation is that of the unit circle.
I found two possible second equations:
1. Write an expression for the slope of the tangent line and an expression for the slope of the radius to the point of tangency. Use what you know about the relationship between these slopes to write an equation. Solve the system.
2. Think of the circle with center (4, 2) that passes through those two points of tangency. What is the radius? (it helps to draw a picture). Find the equation of that circle. Solve the system.
Problem 5
First of all, the tree is not twirling the apple. You are standing at (20, 15) and you are twirling the apple.
For part c, the problem involves distance (10 feet) so you are probably going to be needing to use the distance formula here. Since you are given the distance in the problem, you'll need to work backward to find what you're looking for.
Problem 6
Part C: think of the formula d=rt. You have the rate, the question asks you to find t. What do you need to know in order to use that equation to find t?