David B.

# Finding Where the slope is Undefined

So I was helping a student with a problem today. The answer we tried to submit was wrong and I'm not sure I understand why.

The question was:

Find where the slope is undefined for x = 4sec(y)

At first I thought this was fairly straightforward:

We implicitly differentiated, getting:

1 = 4y'sec(y)tan(y)

Then solved for y':

y' = 1/(4y'sec(y)tan(y))

Now I assumed that the slope should be undefined wherever the denominator was zero:

4sec(y)tan(y) = 0
sec(y)tan(y) = 0

Both sec(y) and tan(y) should be zero for the same values of y, namely:

y = pi/2 + k*pi

But this was the wrong answer! I'm worried it might have something to do with the fact that we're dealing with y instead of x, but I was stumped as to how to proceed.

Howard J.

In this case, slope depends on how you define it. I think it would any change in the dependent variable (x) divided by the commensurate change in the independent variable (y). In that case, the slope is undefined at any one of the horizontal asymptotes.
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10/08/19

## 2 Answers By Expert Tutors

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Tutor
4.8 (5)

Principal Mechanical Engineer with >30 years' math coaching experience

David B.

When you say "when cos(y) is some multiple of π/2" Did you actually mean when y is some multiple of π/2?
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10/08/19

Howard J.

See the updated solution.
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10/08/19

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