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Unformatted text preview: AP Calculus Answer Sheet 1/Feb./2008 Solution
1. & Answer ⒜ lim lim →
→ lim → lim → lim → ⒝ Since → lim → lim ≠ lim , → ∴ lim D N E → ⒞ lim lim sin → sin → lim → sin ⒟ lim lim → ∞ →∞ C heong S him International A cademy lim lim → → ⒠ Since lim lim ⋅ → → lim lim → → ∴ lim → ⒡ lim csc →∞ ⒢ Since lim and lim → → ∴ lim does not exist. → cos cos ⒣ lim lim → cos → cos cos lim → cos cos lim → cos cos lim → cos C heong S him International A cademy 2. for ≤ for ⅰ) lim lim → → ⅱ) lim lim → → Since is continuous at , lim exist.
→ so, lim lim i.e. → → ⇒ ⇒ lim → ⅲ) if then lim → ∴ C heong S him International A cademy 3. cos ′ lim → cos cos lim → cos ⋅ cos sin ⋅ sin lim → ⋅ cos ⋅ sin lim → cos sin ⋅ lim ⋅ lim → → cos cos ⋅ lim ⋅ ⋅ cos → cos ⋅ lim → ⋅ cos sin ⋅ lim → ⋅ cos sin sin ⋅ lim ⋅ cos → ⋅⋅ C heong S him International A cademy 4. ⒜ ⋅ ⒝ sin cos ⋅ sin cos ⋅ ⋅ cos ⋅ sin sin cos ⋅ sin ⋅ cos sin cos ⒞ sec sin ⋅⋅ sec sin ⋅ sec sin ⋅ tan sin ⋅ cos ⋅ ⋅ sec sin ⋅ tan sin ⋅ cos C heong S him International A cademy 5. ′ ⋅ ′ ⋅⋅ C heong S him International A cademy 6. ⇒ ⋅ ⋅ ⋅ ⇒ ⋅ ⇒ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ C heong S him International A cademy 7. ⋅ cos ⋅ cos ⋅ sin ⋅ ⇒ ⋅ sin ⋅ cos ⇒ cos ⋅ sin Slope of tangent at : cos ⋅ sin ⋅ Slope of normal at : normal line : ∴ C heong S him International A cademy 8. ⋅ sin ⋅ sin ⋅ cos ⋅ ⇒ sin ⋅ cos ⇒ sin ⋅ cos ⋅ cos ⋅ ⋅ cos ⋅ sin ⋅ ⋅ cos ⋅ sin ⋅ ⋅ cos ⋅ ⋅sin ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ C heong S him International A cademy 9. Let point of tangency is . Since the normal line is parallel to the line , the slope of normal line is So, the slope of tangent line is Since the slope of tangent ⇒ ⇒ ⇒ ∴ C heong S him International A cademy 10. ⒜ ⇒ ⋅ ⋅ ⇒ ⋅ ⇒ ⒝ Slope of tangent at : Tangent : ∴ or C heong S him International A cademy ⒞ Slope of normal at : Normal : ∴ or ⒟ Since the tangent line is , the slope of tangent line is By ⒜, slope of tangent ⇒ ⇒ ⇒ , Substitute ⇒ ∴ P C heong S him International A cademy 11. By the Mean Value Theorem, ′ ≈ C heong S him International A cademy ...
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 Spring '11
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