Here are some tips to help you with Assignment 1 if you are taking Math 1500 via distance/online.

 

You should thoroughly study Lesson 9: Curve-Sketching before attempting this lesson.

 

Be sure to compute and simplify the necessary derivatives and double-check you are right before you proceed to answer the questions.

 

Question 1 is asking for the critical points and singular points.  Make sure you give both the x and y coordinates.  The top and bottom zeros of the first derivative are the critical numbers.

 

Question 2.  Similar to question 5 in my Lesson 9.

Question 3 is a Mean Value Theorem question.  I give you the Mean Value Theorem in the Proofs section of Lesson 9.  You are given a and b, so you can compute f(b) - f(a) all divided by b - a.  You can also compute, the derivative of f(x), to get f ' (x).  Literally sub c in place of x to get f ' (c).  Set f ' (c) equal to your answer for f(b) - f(a) over b - a, and solve for c, keeping in mind that the answer for c must be between a and b.

 

Question 4.  The key here is to first pull everything over to the left side to make the inequality > 0.  Let f(x) equal the left side of this inequality and use the Mean Value Theorem, where a = 0 and b = x.

 

Question 5.  Consider the conditions f(x) must meet in order for Mean Value Theorem to apply.

 

Question 6 is analyzing f ' (x).

 

Question 7 is analyzing f '' (x).

 

Question 8 is a classic curve-sketch.