if a function f is continuous at a and \lim_{x \to 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 is used. graph the following functions and determine the location of the vertical tangent lines.

a. f(x) = (x + 2)^{1/7}
b. f(x) = (x - 3)^{4/5}
c. f(x) = \sqrt{|x - 3|}
d. f(x) = x^{5/3} - 5x^{1/3}

c. choose the correct graph of f(x) = \sqrt{|x - 3|} below.

Question

if a function f is continuous at a and \lim_{x \to 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 is used. graph the following functions and determine the location of the vertical tangent lines.

a. f(x) = (x + 2)^{1/7}
b. f(x) = (x - 3)^{4/5}
c. f(x) = \sqrt{|x - 3|}
d. f(x) = x^{5/3} - 5x^{1/3}

c. choose the correct graph of f(x) = \sqrt{|x - 3|} below.

a. f(x) = (x + 2)^{1/7}
b. f(x) = (x - 3)^{4/5}
c. f(x) = \sqrt{|x - 3|}
d. f(x) = x^{5/3} - 5x^{1/3}

c. choose the correct graph of f(x) = \sqrt{|x - 3|} below.

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [], "new_concepts": [ "Vertical Tangent Lines", "Cusp and Corner Analysis", "Absolute Value Radical Graphs" ], "current_concepts": [ "Vertical Tangent Lines", "Cusp and Corner Analysis", "Absolute Value Radical Graphs" ] } ## Analyze the function and its derivative $$ \begin{aligned} f(x) &= \sqrt{|x-3|} = |x-3|^{1/2} \ f'(x) &= \frac{1}{2}|x-3|^{-1/2} \cdot \frac{d}{dx}(|x-3|) \ f'(x) &= \frac{x-3}{2|x-3|^{3/2}} = \frac{\operatorname{sgn}(x-3)}{2|x-3|^{1/2}}, \quad x eq 3 \end{aligned} $$ ## Evaluate the limit of the derivative as x approaches 3 $$ \begin{aligned} \lim_{x \to 3^+} f'(x) &= \lim_{x \to 3^+} \frac{1}{2\sqrt{x-3}} = \infty \ \lim_{x \to 3^-} f'(x) &= \lim_{x \to 3^-} \frac{-1}{2\sqrt{3-x}} = -\infty \ \lim_{x \to 3} |f'(x)| &= \infty \end{aligned} $$ Since the one-sided limits of the derivative have opposite signs ($\infty$ and $-\infty$), the graph has a cusp at $x = 3$, not a vertical tangent line. ## Identify the correct graph of the function $$ \begin{aligned} &f(3) = 0 \ &f(4) = 1, \quad f(2) = 1 \ &f(7) = 2, \quad f(-1) = 2 \end{aligned} $$ The graph is symmetric about $x = 3$, has a V-shaped cusp pointing down at $(3,0)$, and opens upwards. Looking at the options: - Option A: A very flat curve. - Option B: A cubic-like curve. - Option C: A symmetric V-like curve with a cusp at $(3,0)$. - Option D: A shifted radical curve. Thus, Option C is the correct graph.

A. (A)

B. (B)

C. (C)

D. (D)

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Calculus", "Differentiability from Graph" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Analyze the function and its derivative

Evaluate the limit of the derivative as x approaches 3

Identify the correct graph of the function

Answer: