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Questions and Answers
How do you find vertical asymptotes algebraically?
How do you find vertical asymptotes algebraically?
- Simplify the function 2. Set the denominator equal to 0 and solve for x. 3. If x is undefined, then there is no vertical asymptote.
What is the shortcut to finding horizontal asymptotes?
What is the shortcut to finding horizontal asymptotes?
If degN < degD, horizontal asymptote = 0; if degN = degD, ratio of leading terms; if degN > degD, no horizontal asymptotes.
How do you find horizontal asymptotes using limits?
How do you find horizontal asymptotes using limits?
- Factor the rational function by the coefficient with the largest degree. 2. Check if terms become 0 or 1 when factoring. 3. Horizontal Asymptote = 0.
A function will have a vertical tangent at a point x = c in its domain if...
A function will have a vertical tangent at a point x = c in its domain if...
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At a vertical tangent, there is...
At a vertical tangent, there is...
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A function will have a vertical cusp at a point x = c in its domain if...
A function will have a vertical cusp at a point x = c in its domain if...
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At a vertical cusp, there is...
At a vertical cusp, there is...
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What are the steps using calculus to graph a function?
What are the steps using calculus to graph a function?
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Study Notes
Vertical Asymptotes
- Vertical asymptotes occur where a function is undefined; identify them by setting the denominator of a simplified function equal to zero.
- If a variable factor remains in the denominator after simplification, solving for zero gives x-values for vertical asymptotes; if x is undefined, no asymptote exists.
Horizontal Asymptotes
- Determine horizontal asymptotes using degree comparisons of numerator (degN) and denominator (degD):
- If degN < degD, the horizontal asymptote is y = 0.
- If degN = degD, y is the ratio of the leading coefficients.
- If degN > degD, there are no horizontal asymptotes.
Finding Horizontal Asymptotes with Limits
- Factor the rational function by its largest degree term to analyze behavior at infinity.
- Evaluate limits by checking if the remaining terms approach 0 or 1 after factoring out.
- In cases where the limits lead to 0, the horizontal asymptote is y = 0.
Vertical Tangents
- A function has a vertical tangent at x = c if both left-hand and right-hand limits of the first derivative approach infinity.
- There is no sign change in f'(x) at vertical tangents.
Vertical Cusps
- A function has a vertical cusp at x = c if the left-hand limit of the first derivative approaches infinity while the right-hand limit approaches negative infinity.
- There is a sign change in f'(x) at vertical cusps.
Steps for Graphing Functions Using Calculus
- Determine the Domain: Odd roots cover all real numbers; for even roots, set the content (radicand) ≥ 0.
- Find Asymptotes: Identify vertical and horizontal asymptotes for rational functions.
- Identify Intercepts: Solve f(x) = 0 for x-intercepts; evaluate f(0) for y-intercepts, considering domain restrictions.
- First Derivative (f') Analysis: Find critical points to determine intervals of increase/decrease and identify local extremes as well as vertical tangents and cusps.
- Second Derivative (f') Analysis: Assess concavity and locate points of inflection through f''.
- Plot Points of Interest: Mark intercepts, extremes, and inflection points on a graph.
- Sketch the Graph: Combine all information to accurately depict the function's shape in terms of concavity and behavior over different intervals.
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Description
Test your understanding of vertical and horizontal asymptotes as well as curve sketching techniques. This quiz covers key concepts and shortcuts related to asymptotes in calculus functions. Use these flashcards to solidify your knowledge for upcoming assessments.
