I fixed a typo in my tips for #4 below.  I also added a little more help.

Did you read my tips on how to study and learn this course?  If not, here is a link to those important suggestions:

Here is a link to the actual assignment, in case you don't have it:

Study Lesson 10 (Max/Min Word Problems)  and Lesson 11 (Antiderivatives and Integrals) from my Intro Calculus book to prepare for this assignment.  Make sure you have done all my lecture problems in Lesson 11 (OMIT #13 now, and forever!) before attempting the Lesson 11 questions in the assignment.  You need to make sure you have had ample practice and solving antiderivatives.

Very similar to my Lesson 10, Lecture Problem 6.  Except the base is not a square it is a rectangle with length three times the width (x by 3x by y).  Also, this box has a top (which costs more than the sides and bottom).

Similar to my Lesson 10, Lecture Problem 9 and Practise Problem 15.  Do note that in both of my examples I choose to keep y as the variable and eliminate x.  I strongly recommend you do the opposite.  In your problem, it is much more sensible to keep x and eliminate y.

Good luck on this one!  Similar to my Lesson 10, Lecture Problem 12 and Practise Problem 18, but even more challenging.  To be honest, this question is so hard, I doubt anyone could do it without copying the answer from someone else.  And, since he is quite paranoid about plagiarism, I think the best thing to do is just skip this one if you can't do it yourself.  It isn't worth the time and effort for three-tenths of 1 percent of your grade.

Some hints:

I have given you as much hints as I dare for this one.

This is just standard antiderivative and integral stuff applied to velocity and acceleration, like what I am doing in my Lesson 11, Lecture Problem 9 and Practise Problems 45-47.

Note that, if we are using h to represent the height of the rock at any time t, then v = h' and a = h".  Make sure you watch your signs during the calculations for h, v and a. Up is positive and down is negative here, so you are given positive values for h and v when t=0.  Note that a = -16 always! It is constant.  The antiderivative of that constant (don't forget to put dt in your integral!) will tell you v.

Remember, the velocity = 0 when the rock reaches maximum height.

This is just standard antiderivative and indefinite integral word problems like what I am doing in my Lesson 11, Lecture Problem 2. 

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 0 to 5 of 5dx MINUS the definite integral from 0 to 5 of the square root expression.  Don't forget to put dx at the end of each integral!

For the integral of 5, you have to draw a graph of f(x) = 5 from x=0 to x=5, then find the area between the vertical lines x=0 and x=5, between the graph of f(x) and the x-axis using geometric methods.  You should notice the shape is just a rectangle, so you can find the area by using the area of a rectangle.

For the other integral, draw the graph of f(x) = square root of (25- x^2) from x=0 to x=5 and find the area between the curve and the x-axis.  Hint: the graph is a quarter of a circle.

Don't forget that the original integral is the area of the first graph MINUS the area of the second graph.

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.

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.

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(3), g(5), and g(10).

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?

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.

This is just standard antiderivative and indefinite integral stuff like what I am doing in my Lecture Problem 1 in Lesson 11.   They are just definite integrals this time. 

This is just an area word problem applications like what I am doing in my Lecture Problems 6 to 8 in Lesson 11.