Tentative Problem Sets for Math Xa

Problem Set to be assigned on Wednesday December 15

Read Chapter on ``e" - to be given as a handout

Do: >From Chapter on ``e" # 6, 7, 8, 9, 10 a-d, 11, 13

plus

So far although we have said that critical points of a function are the endpoints and points at which the derivative is either zero or undefined, we have had very few, if any, examples in which f' is undefined at a point in the domain of f. The first problem below (along with the ones to be assigned on Friday) are an attempt to rectify this!

1.

Let h(x) = |g(x)|, where g(x) = x² -1.

(a)

Graph g(x) and h(x) on the same set of axes.

(b)

What are the critical points of g?

What are the critical points of h?

(c)

Identify all local extrema of h.

(d)

Does h have an absolute maximum value? If so, what is it, and where is it attained?

(e)

Does h have an absolute minimum value? If so, what is it, and where is it attained?

2.

Let $f(x) = \displaystyle\frac{2x^2 + 3}{5x-5}$. While you should feel free to use your graphing calculator, the work for this question must stand on its own, completely independent of the machine with the exception of the very last part of the question.

(a)

Identify any x-intercepts, y-intercepts, vertical and horizontal asymptotes of f.

(b)

Find f '(x). Show your work.

(c)

What are the critical points of f? Give exact answers, not numerical approximations from your calculator.

(d)

Graph f(x), labelling the x coordinates of any turning points.

(Note: If there is a critical point at x=a and you know it is a local minimum due to your knowledge of the behavior of a rational function, you need not prove that it is a local minimum. )

(e)

What is the domain of f?

(f)

Approximately what is the range of f? (Here you can use your graphing calculator, in either its graphing or numerical capabilities.)

Problem Set to be assigned on Friday December 17

Read §14.2

Do:

1.

Let f(x) = e^x(x³ - 3x² + 5x -5).

(a)

Find all the critical points of f. Classify each one as either a local maximum, local minimum, or point of inflection.

(b)

Does f have an absolute maximimun value? If so, what is it and where is it attained?

(c)

Does f have an absolute minimimun value? If so, what is it and where is it attained?

2.

Let $f(x) =\sqrt{(-x^2 + 4)^2}$.

Please note that $f(x) \ne -x^2 + 4$. For instance, if x = 10 f(x) is positive, but -x² + 4 is negative.

(a)

Differentiate f using the Chain Rule. Do not try to simplify first.

(b)

What are the critical points of f? Classify them.

3.

You are in a dune buggy 40 km north of the nearest point (we'll call it P) on a straight paved road. 50 km down the road from P is the nearest gas station. The dune buggy is low on gas and it's late in the afternoon, so you want to get to the gas station in the minimum amount of time.

(a)

Suppose your dune buggy can travel 45 kmin the desert and 75 kmon a paved road. If you take the route that minimizes time, how long will it take you to get to the gas station?

(b)

Suppose your dune buggy can travel 45 kmin the desert and 55 kmon a paved road. If you take the route that minimizes time, how long will it take you to get to the gas station?

Although you can use your calculator to help check your work, your work should stand on its own, independent of the calculator.

Chapter 14 #6, 11, 13

Chapter 15 #14

Extra credit:

(a)

Let $f(x) =\sqrt{(x^2-1)^2} + x$.

Please note that $f(x) \ne x^2 - 1 + x$. For instance, if x = 0 f(x) is 1, not -1.

i.

Differentiate f using the Chain Rule. Do not try to simplify first.

ii.

What are the critical points of f? Classify them.

Problem Set to be assigned on Monday December 20

Do an exam from the Review Packet: January 24, 1996. Due, (self-corrected), on Monday, January 10, 200

Note: Your final exam is on Friday, January 21st at 2:15 - 5:15.

Let f(x) = e^x(x³ - 3x² + 5x -5).

1.

Find all the critical points of f. Classify each one as either a local maximum, local minimum, or point of inflection.

2.

Does f have an absolute maximimun value? If so, what is it and where is it attained?

3.

Does f have an absolute minimimun value? If so, what is it and where is it attained?

Let $f(x) =\sqrt{(-x^2 + 4)^2}$.

Please note that $f(x) \ne -x^2 + 4$. For instance, if x = 10 f(x) is positive, but -x² + 4 is negative.

1.

Differentiate f using the Chain Rule. Do not try to simplify first.

2.

What are the critical points of f? Classify them.

You are in a dune buggy 40 km north of the nearest point (we'll call it P) on a straight paved road. 50 km down the road from P is the nearest gas station. The dune buggy is low on gas and it's late in the afternoon, so you want to get to the gas station in the minimum amount of time.

1.

Suppose your dune buggy can travel 45 kmin the desert and 75 kmon a paved road. If you take the route that minimizes time, how long will it take you to get to the gas station?

2.

Suppose your dune buggy can travel 45 kmin the desert and 55 kmon a paved road. If you take the route that minimizes time, how long will it take you to get to the gas station?

Although you can use your calculator to help check your work, your work should stand on its own, independent of the calculator.

Chapter 14 #6, 11, 13

Chapter 15 #14

Extra credit:

1.

Let $f(x) =\sqrt{(x^2-1)^2} + x$.

Please note that $f(x) \ne x^2 - 1 + x$. For instance, if x = 0 f(x) is 1, not -1.

(a)

Differentiate f using the Chain Rule. Do not try to simplify first.

(b)

What are the critical points of f? Classify them.

Problem Set to be assigned on Monday December 20

Do an exam from the Review Packet: January 24, 1996. Due, (self-corrected), on Monday, January 10, 200

Note: Your final exam is on Friday, January 21st at 2:15 - 5:15.