Math 1500: Tips for Assignment 5

Published: Mon, 03/07/16

Don't have my book or audio?  You can download a free sample of my book and audio lectures containing Lessons 1 and  2:
Did you read my tips on how to study and learn Math 1500?  If not, here is a link to those important suggestions:
Did you read my tips I sent earlier on making sign diagrams and finding range? 
Here are the links:
Did you see my tips for Assignment 1? Click here.
Did you see my tips for Assignment 2? Click here.
Did you see my tips for Assignment 3? Click here.
Did you see my tips for Assignment 4? Click here.
Tips for Assignment 5
These are tips for the first assignment in the Distance/Online Math 1500 course, but I strongly recommend that you do this assignment as homework even if you are taking the classroom lecture section of the course.  These assignments are very good (and challenging) practice.  The first assignment is a great way to build and review key skills that will be helpful for this course.

Here is a link to the actual assignment, in case you don't have it:
Remember, it is almost a certainty that, for every hour you spend studying my lessons, that will save you at least one hour, if not more, in time spent struggling on the assignment.  You are not saving time to start working on an assignment unprepared!  You are really increasing the time you will need to do an assignment.

Study Lesson 10 (Max/Min Word Problems)  and Lesson 11 (Antiderivatives and Integrals) from my Intro Calculus book to prepare for this assignment.
Question 1
First, make sure you have read my previous email about finding the domain in Max/Min Word Problems.  Be sure you do state the domain as part of your answer to the first two questions:

Similar to my Lesson 10, Lecture Problem 9 and Practise Problem 15. 
  1. Sketch the ellipse by finding the x-intercepts (y=0) and the y-intercepts (x=0).  Then, label an arbitrary point on the ellipse as (x,y).  Choose a spot on the curve itself, stay away from the x- and y-axis, so that you don't lose a general sense of the situation.
  2. Plot the given point (1,0).
  3. Draw a line from the (x,y) point to the (1,0) point.  Label the length of that line D (or z, if you want).  That is the distance D or z that you want to Maximize.
  4. Use the distance between two points formula as given in my problem 15 above.
  5. You can isolate either x or y, your preference in the ellipse formula.  I strongly suggest that you isolate y and sub the result into the formula for D.  The result will be a formula entirely in terms of x.
  6. Proceed to find your maximum.
Question 2
First, make sure you have read my previous email about finding the domain in Max/Min Word Problems.  Be sure you do state the domain as part of your answer to the first two questions:

Very similar to my Lesson 10, Lecture Problem 6.
Question 3
This is just standard antiderivative and indefinite integral stuff like what I am doing in my Lecture Problem 1 in Lesson 11. 
Question 4
This is just standard antiderivative and integral stuff applied to velocity and acceleration, like what I am doing in my Lecture Problem 9 in Lesson 11. Note that they are using s to represent the position of the particle, so v = s' and a = s".
Question 5
This question is NOT to be solved using antiderivative formulas! You couldn't solve it that way anyway. 

First, split the problem up into two separate definite integrals using properties of antiderivatives.  You can say it is the definite integral from -4 to 0 of x dx MINUS the definite integral from -4 to 0 of the square root expression.  Don't forget to put dx at the end of each integral!

For the integral of x, you have to draw a graph of f(x) = x from x=-4 to x=0, then find the area between the vertical lines x= -4 and x=0, between the graph of f(x) and the x-axis using geometric methods.  You should notice the shape is just a triangle, so you can find the area by using the area of a triangle. Remember! Area below the x-axis is negative.

For the other integral, draw the graph of f(x) = square root of (16 - x^2) from x=-4 to x=0 and find the area between the curve and the x-axis.  Hint: the graph is a quarter circle.

Don't forget that the original integral is the area of the first graph MINUS the area of the second graph.
Question 6
Use antiderivative properties again to split this up into three separate integrals.  Note that you can factor the coefficients out of the integrals after they have been separated.  Which is to say, the definite integral of 2 f(x) dx is 2 times the definite integral of f(x) dx.
Question 7
This is classic Fundamental Theorem stuff like what I am doing in my Lecture Problem 10 in Lesson 11 (especially part (d)).

Make sure that you write out part 2 of the Fundamental Theorem as given at the start of Lesson 11, and say that you are combining chain rule with the Fundamental Theorem of Calculus.  I have given you a chain rule version of the Fundamental Theorem, but they may take marks off if you don't explain that chain rule is also being used. 
Question 8
This is just standard antiderivative and indefinite integral stuff like what I am doing in my Lecture Problem 1 in Lesson 11.   It is just a definite integral this time. 
Question 9
This is again exploiting the fact that a definite integral is computing the area between a curve and the x-axis (but regions below the x-axis have negative areas).  Note, since g(x) is the definite integral between 0 and x, that means g(1) is the definite integral between 0 and 1, while g(2) is the definite integral between 0 and 2, etc.

Part (a)
Visualize the area each value is describing and solve it using the fact the shapes are just rectangles and triangles.  Remember! Areas below the x-axis are negative.

Part (b)
Almost everyone is going to get this question wrong, because they won't read the graph properly. Don't be guilty of jumping to conclusions.  The graph is showing you f, not g!

One way to answer this question is to observe your answers to part (a).  That should give you a pretty good idea where g is increasing or decreasing.  You may want to throw in a few more for clarity.  I would make sure I plotted every value of g where the graph of f changed.  For example, I would also compute g(0), the starting point, g(1), where f stopped increasing, g(6), etc.

A more sophisticated way is to use the Fundamental Theorem of Calculus (see my Lesson 11, question 10).  Note, you have been given that g(x) is a definite integral.  You can compute g'(x) using the Fundamental Theorem.  Now that you have computed g'(x), you need to establish where g'(x) is positive (since that is where g(x) is increasing).  The key is to establish how does g'(x) relate to the graph of f(x) you have been given.  Therefore, how can you establish where g'(x) is positive?

Part (c)
There are several ways you can solve this.  The easiest is probably by looking at your answers to part (a) and (b).  You can also make a sign diagram for g'(x) using the Fundamental Theorem of Calculus again.

Part (d)
Again, the easiest way is to plot the points you found in part (a) and (b).  You may want to throw in a few more for clarity.  I would make sure I plotted every value of g where the graph of f changed.  For example, I would also compute g(0), the starting point, g(1), where f stopped increasing, g(6), etc.
Question 10
This is just area word problem applications like what I am doing in my Lecture Problems 6 to 8 in Lesson 11.