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AP Calculus November 9-10, 2016 Mrs. Agnew
Curve Sketching #1 AP Calculus November 9-10, 2016 Mrs. Agnew
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Essential Question Essential Vocabulary
What is the significance of the Mean Value Theorem? How do you sketch curves using derivatives? Essential Vocabulary Mean Value Theorem First Derivative Test Increasing/Decreasing Function
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Zeros & Symmetry Zeros of a function Symmetry
Where graph intersects x-axis Symmetry Y-axis (plug in –x for x) Origin (plug in –x and –y)
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Discontinuities Asymptotes Point of Discontinuity (HOLE) Examples
Vertical: where function is undefined (not hole) Horizontal: degree of numerator vs. denominator Slant: degree of numerator is one more than denominator Point of Discontinuity (HOLE) Examples
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Increasing/Decreasing
What are critical numbers? If f ‘(x) > 0 on an interval, then f(x) is increasing on that interval. If f ‘(x) < 0 on an interval, then f(x) is decreasing on that interval.
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First Derivative Test Examples Given c is a critical number of f(x)
If f ‘ changes from + to – at c, then f has a relative maximum at c. If f ‘ changes from – to + at c, then f has a relative minimum at c. If f ‘ does not change at c, then f has no extreme value. Examples
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Mean Value Theorem If f is continuous over [a,b] and differentiable over (a,b), then there exists a number c between a and b such that… Calculus slope = Algebra slope Instantaneous ROC = Average ROC Animation
![Mean Value Theorem If f is continuous over [a,b] and differentiable over (a,b), then there exists a number c between a and b such that…](http://slideplayer.com./slide/14898979/91/images/7/Mean+Value+Theorem+If+f+is+continuous+over+%5Ba%2Cb%5D+and+differentiable+over+%28a%2Cb%29%2C+then+there+exists+a+number+c+between+a+and+b+such+that%E2%80%A6.jpg "Calculus slope = Algebra slope. Instantaneous ROC = Average ROC. Animation.")
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Rolle’s Theorem Let f be continuous on [a,b] and differentiable over (a,b). If f(a) = f(b), then there exist at least one c such f ’(c) = 0. Guarantees the existence of an extreme value in the interior of a closed interval. Examples
![Rolle’s Theorem Let f be continuous on [a,b] and differentiable over (a,b). If f(a) = f(b), then there exist at least one c such f ’(c) = 0.](http://slideplayer.com./slide/14898979/91/images/8/Rolle%E2%80%99s+Theorem+Let+f+be+continuous+on+%5Ba%2Cb%5D+and+differentiable+over+%28a%2Cb%29.+If+f%28a%29+%3D+f%28b%29%2C+then+there+exist+at+least+one+c+such+f+%E2%80%99%28c%29+%3D+0..jpg "Guarantees the existence of an extreme value in the interior of a closed interval. Examples.")
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Homework: Page 176 – 177 #13, 17, 19, 21, 29, 41 – 46, 53, 64, 71 Page 186 – 189 #19, 23, 29, 35, 49, 67, 69, 85, 91
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