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Solved: Vertical tangent lines If a function f is
Chapter 4, Problem 61AE(choose chapter or problem)
If a function f is continuous at a and \(\lim _{x \rightarrow a}\ |f'(x)|=\infty\), 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. \(f(x)=\sqrt{|x-4|}\) d. \(f(x)=x^{5/3}-2x^{1/3}\)
Questions & Answers
QUESTION:
If a function f is continuous at a and \(\lim _{x \rightarrow a}\ |f'(x)|=\infty\), 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. \(f(x)=\sqrt{|x-4|}\) d. \(f(x)=x^{5/3}-2x^{1/3}\)
ANSWER: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)