Has Stewart Calculus (8th edition) bungled 14.6 #29?

Question

(From Calculus by Stewart, 8th edition (Cengage Learning)):

14.6 29 Find all points at which the direction of fastest change of the function $f(x,y)=x^2+y^2-2x-4y$ is $\bf{i+j}$.

Roughly in order of least serious to most serious we have:

1) The question casually conflates f with the rule for evaluating f. (This is a minor and common trangression).

2) If we agree that at $(x,y)$ the direction of fastest change of $f$ is the direction of the vector $\bigtriangledown f(x,y)$, then the question conflates the notion "direction of a vector" with the notion "vector".

Comment. Can we agree that two vectors "have the same direction" if each is a positive scalar multiple of the other, and two vectors "have opposite directions" if each is a negative scalar multiple of the other?

I infer Stewart would say "yes", since in section 12.2 vectors are treated as arrows, and Stewart says "...the arrow points in the direction of the vector".

3) The question is now at least ambiguous. Do we wish to solve for $(x,y)$

3a) $\bigtriangledown f(x,y)=\bf{i+j}$?

3b) $\bigtriangledown f(x,y)=\lambda (\bf{i+j}$) for some $\lambda>0$?

3c) $\bigtriangledown f(x,y)=\lambda (\bf{i+j}$) for some real number $\lambda$?

4) The answer in the back of the book is "all points on the line $y=x+1$". This answers interpretation 3c.

If Stewart intended 3c) as the question then it is now true that "Two vectors have the same direction if they have opposite directions".

5) Both the question at hand and its answer have appeared in the 5th,6th,7th, and 8th editions of Stewart.

Summary: The following is a guess as to the intent of the original question.

If the function $f$ is defined by the rule $f(x,y)=x^2+y^2-2x-4y$, find all locations $(x,y)$ so that the direction of fastest change of $f$ is also the direction of the vector $\bf i+j$.

The answer depends on which definition of "direction of a vector" is employed.

If we infer the definition from Stewart's prose, the answer in the back is incorrect.

Answer 1

Given a vector $v$ we usually say that the vector $\frac{v}{|v|}$ is the direction of $v$.

Recall that $\nabla f(x,y)$ gives the direction one should move to increase $f$ the fastest. The question just asks for the fastest change of $f$. Therefore, when thinking about this question, we should make no distinction between $- \frac{\nabla f(x,y)}{|\nabla f(x,y)|} = \frac{\vec{i} + \vec{j}}{|\vec{i} + \vec{j}|}$ and $\frac{\nabla f(x,y)}{|\nabla f(x,y)|} = \frac{\vec{i} + \vec{j}}{|\vec{i} + \vec{j}|}$.

So, your interpretation of two vectors $\vec{u}, \vec{v}$ having the same direction if and only if there exists $\lambda > 0$ such that $\vec{u} = \lambda \vec{v}$ is correct. However, due to only caring about $f$ changing and not $f$ increasin, we should still find $\lambda \neq 0$ so that $\nabla f(x,y) = \lambda(\vec{i} + \vec{j})$.

In short, I think Stewart's interpretation of the problem is correct. Your confusion sounds like it may have come from equating "fastest change in $f$" with "direction in which $f$ increases fastest".

Answer 2

The "conflation" of $f$ with $f(x,y)$ is prevalent throughout the text, and in the context of the problem, it's not likely to be misinterpreted, so I wouldn't change that.

The answer in the back is wrong -- Stewart surely means "in the same direction" as the vector $\bf{i+j}$ (i.e., a positive multiple of $\bf{i+j}$).

Update: Fastest change doesn't necessarily mean fastest increase, hence the book answer is OK. Note -- the answer key is often done by someone other than the author, so the person working out the answers may have taken "fastest change" literally. But Stewart probably intended fastest increase.

I would change the wording only minimally:

Find all points at which the direction of fastest increase of the function $f(x,y)=x^2+y^2-2x-4y$ is in the direction of $\bf{i+j}$.

And with that suggested change, the answer would be: the ray $y=x+1$, $\;x > 1$.

Answer 3

Employing the insight in mlg480 and quasi's answers, it looks like replacing the word "the" by the word "a" can create a new question with a simple interpretation, yielding the answer in the back of the book.

However, this comes at the cost of arguably contradicting Stewart's intended question.

Suppose we replace Stewart's original question by the following:

Find all points at which a direction of fastest change of the function $f(x,y)=x^2+y^2-2x-4y$ is $\bf i+j$.

If $|\bigtriangledown f(x,y) \cdot \bf \frac{v}{|v|} |$ means the rate at which $f$ is $changing$ at $(x,y$) in the direction of the nonzero vector $\bf v$, then the adjusted question is clear, and the answer in the back of the book is correct.

Comments:

1) Changing "the direction of fastest change" to "a direction of fastest change" suggests to the reader that at a given location $(x,y)$ there may/must be at least two directions of fastest change, thus helping the reader clarify the potential meaning of the question.

2) In the new wording of the question, we need not modify $\bf i+j$.

3) $ \bf i+j $ is a direction of fastest change at $(1,2)$, even though $\bigtriangledown f(1,2)=\bf0$.

4) However, both the text and examples in 14.6 suggest Stewart may intend "direction of fastest change" to mean "a positive scalar multiple of $\bf i+j$".

4a) From 14.6 p992. "...in which of (all possible) directions does $f$ change fastest...?"

"...the same direction as the gradient vector $\bigtriangledown f(\bf x)$".

There is no suggestion that this answer is not unique. Moreover Stewart's notion of "direction of a vector" is implied to mean the direction in which the corresponding arrow is pointing.

4b) 14.6 Example 6b) "In what direction does $f$ have the maximum rate of change?". Notice Stewart does not say "directions". Does he mean the direction in which $f$ changes fasted? Stewart's answer is the direction of the gradient vector.