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Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 3.1 - Problem 61ae
Get Full Access to Calculus: Early Transcendentals - 1 Edition - Chapter 3.1 - Problem 61ae

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# Solved: Vertical tangent lines If a function f is ISBN: 9780321570567 2

## Solution for problem 61AE Chapter 3.1

Calculus: Early Transcendentals | 1st Edition

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Problem 61AE

Vertical tangent lines? If a function f is continuous at a and limx?a (x)| = ? , then the curve y= f(x) has a vertical tangent line at a and the equation of the tangent line is x = a. If a is an endpoint of a domain, then the appropriate one-sided derivative (Exercises 59-60) is used. Use this definition to answer the following questions. Graph the following functions and determine the location of the vertical tangent lines. a.? f(x) = (x ? 2)1/3 b.? f(x) = (x + 1)2/3 c.? f(x) = ?|x?4| d.? f(x) = x5/3 ? 2x1/3

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SOLUTION If a function f is continuous at a and lim|xa (x)| = , then the curve y= f(x) has a vertical tangent line at a and the equation of the tangent line is x = a A function is said to be continuous if it satisfies the condition limf(x) = f(a) xa STEP 1 3 (a).Given function f(x) = (x + 1) By definition of continuity the given function is continuous. 2 21 2 1 f (x) = (3 + 1) 3 = (3 + 1) 3 Then, 2 3 x1|f (x)| = x1(x3+ 1) = Thus,the graph of f(x) will be having a vertical tangent line at x=-1. The graph is STEP 2 3 (b).Given function f(x) = (x 2) By definition of continuity the given function is continuous. 1 2 f (x) = (x 2) 31 = (x 2) 3 3 3 Then, 2 lim|f (x)| = lim (x 2) 3 = x2 x2 3 Thus,the graph of f(x) will be having a vertical tangent line at x=2. The graph is STEP 3 f(x) = |x 4| (c). iven function By definition of continuity the given function is continuous. f (x) = |x4| 2 x4 |x4| lim|f x)| = lim = x4 x4 2 x4 Thus,the graph of f(x) will be having a vertical tangent line at x=4. The graph is STEP 4 (d).Given function f(x) = x 2x 3 By definition of continuity the given function is continuous. 5 51 2 11 5 2 2 2 f (x) = x3 3 3 3 = x3 x 3 3 5 3 2 3 x0|f (x)| = x0 x3 x 3 = Thus,the graph of f(x) will be having a vertical tangent line at x=0. The graph is

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##### ISBN: 9780321570567

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