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Calculus/Slope of the tangent line.

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Expert: - 10/15/2008


Question
A function is defined by f(x)= x(x+3)^(3/5).  For what values of x does the function have one or more vertical tangents?

Answer

Vertical Tangent
Questioner:   Lindsay
Category:  Calculus
Private:  No
 
Subject:  Calculus
Question:  A function is defined by f(x)= x(x+3)^(3/5).  For what values of x does the function have one or more vertical tangents?
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Hi, Lindsay,

Let's put a few facts together:

1. The slope of the tangent line at a point is the derivative of the function at that point.
2a. A horizontal line has a slope of zero. (Not needed here, but some day...)
2b. A vertical line has a slope that is undefined.
3. A fraction is undefined if its denominator is zero.

OK. Now we know what to look for.  A point where the:

slope is undefined, so
the derivative is undefined, so
some denominator is zero.


f(x)= x(x+3)^(3/5)

f'(x) = [product rule.] (x)( 3/5(x + 3)^(-2/5) ) + (1)(x + 3)^(3/5)

f'(x) =  3x/5(x + 3)^(2/5) + (x + 3)^(3/5)

The only fraction there is the first term, and the denominator there would be zero if  x = -3.  That's your answer.

(See attached picture.)

Paul Klarreich

Expertise

All topics in first-year calculus including infinite series, max-min and related rate problems. Also trigonometry and complex numbers, theory of equations, exponential and logarithmic functions. I can also try (but not guarantee) to answer questions on Analysis -- sequences, limits, continuity.

Experience

I taught all mathematics subjects from elementary algebra to differential equations at a two-year college in New York City for 25 years.

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(See above.)

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