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 numbers. That means they want the critical points and
singular points. The
top and bottom zeros of the first derivative are the critical numbers. Make sure you give both the x and y coordinates of your critical numbers (i.e. make a table of values for each). Recall, as I say in Lesson 9, eu has no zeros.
Question 2. Similar to question 5 in my Lesson 9.
Question 3
is a Mean Value Theorem question. Click the link below for the procedure to follow to "verify" the Mean Value Theorem:
Question 4. Consider the conditions f(x) must meet in order for Mean Value Theorem to apply. Is the function continuous on [a, b]? Is it differentiable on (a, b)? Which is to say, is the derivative defined for all values between a and b? Look for bottom zeros. A function or derivative is undefined if the bottom is zero. If the function is not continuous, or not differentiable, then it has nothing to do with the Mean Value Theorem, and so it is no surprise if the Mean Value Theorem does not work.
Question 5 is analyzing f ' (x). The first derivative tells you where a function is increasing or decreasing and if the critical points are local maximums or minimums. Be sure to give the (x, y) coordinates of all the local extremes you identify.
Question 6 is analyzing f '' (x). The second derivative tells you where a function is concave up or concave down and if you have inflection points.
Be sure to give the (x, y) coordinates of all the inflection points you identify.
Question 7 is a classic curve-sketch. Just like my questions 3 and 4 in the Lecture.
