You should thoroughly study Lesson 10: Max/Min Word Problems and Lesson 11: Antiderivatives (Integrals) before attempting this lesson.

 

Make sure you have read my earlier email about finding the domain in word problems.  Here is a link to that email:

Finding the Domain in Max/Min Word Problems

 

Question 1 is similar to my Practise Problem 1 in Lesson 10 except you have two nonnegative numbers instead of negative numbers.  Don't forget to include the domain of x (assuming you use x as your variable in the derivative you eventually compute).

 

Question 2 is very similar to my Lecture Problem 3 in Lesson 10.  Don't forget to include the domain of x (assuming you use x as your variable in the derivative you eventually compute).

Question 3 is very complicated and messy.  It is similar to my Practise Problems 22 and 23.  Note though that x is the perimeter of the equilateral triangle and 1-x is the perimeter of the square, so the actual sides of the triangle are x/3 and the sides of the square are (1-x)/4 (making the problem even messier, in my opinion).  Note, the area of an equilateral triangle is (the square root of 3) divided by 4 all times s-squared where s is the length of its side.  Also note that the domain of x in parts (a), (b) and (c) will all be the same.

 

Question 4 is just standard antiderivative and indefinite integral stuff like what I am doing in my Lecture Problem 1 in Lesson 11.

 

Question 5 is just like my Lecture Problem 2 in Lesson 11.

 

Question 6 requires the method I use in Lecture Problem 13 in Lesson 11.  Your question is actually more pleasant than either of my two examples.  I suggest you do this question open book.  Note, you can check your answer by doing the definite integral of x dx from x=0 to x=1.

 

Question 7 is NOT to be solved using antiderivative formulas! You couldn't solve part (b) that way anyway.  You can solve part (a) that way and you can use that answer as a check, but you will get no credit for that method.  You have to draw a graph of f(x) = x+2 from x=1 to x=5 in part (a), then find the area of between the vertical lines x=1 and x=5 below the graph of f(x) and above the x-axis using geometric methods.  You should notice the shape is a trapezoid, so you can find the area by cutting it into a rectangle and triangle.  Again, in part (b), draw the graph of f(x) = square root of 9 - x^2 from x=-3 to x=3 and find the area between the curve and the x-axis.  Hint: the graph is a semicircle.