BMCC MTH 251 - Final Exam Review-OLD

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MTH 251 Differential CalculusChapter 5 – Related RatesChapter 5 – Analysis of FunctionsChapter 5 – Relative Maxima & MinimaChapter 5 – Absolute Maxima & MinimaChapter 5 – Optimization ProblemsChapter 5 – Mean Value TheoremChapters 2-4: DefinitionsChapters 2-4: Differentiation FormulasChapters 2-4: Other ProblemsSlide 11Slide 12MTH 251Differential CalculusReview for the Final Exam 64% – Chapters 2-4 36% – Chapter 5Copyright © 2005 by Ron Wallace, all rights reserved.Chapter 5 – Related Rates)(xfy dtdxxfdtdy)('If then That is, find an equation that relates the two changing values, differentiate implicitly in terms of t, substitute in the known information, and solve for the unknown rate.Chapter 5 – Analysis of FunctionsCritical points: f’(x) = 0 or DNEIncreasing: f’(x) > 0Decreasing: f’(x) < 0Constant: f’(x) = 0Concave Up: f’’(x) > 0Concave Down: f’’(x) < 0Inflection Point: f’’(x) = 0 and concavity is changingChapter 5 – Relative Maxima & MinimaStationary Critical Points: f’(x)=0f’’(x) < 0  relative maximum (concave down)f’’(x) > 0  relative minimum (concave up)f’’(x) = 0  nothing (see singular points)Singular Critical Points: f’(x) DNE & f(x) definedf’(x-h) > 0 & f’(x+h) < 0 for some h>0  relative maxi.e. changing from increasing to decreasingf’(x-h) < 0 & f’(x+h) > 0 for some h>0  relative mini.e. changing from decreasing to increasingChapter 5 – Absolute Maxima & MinimaClosed IntervalCheck f(x) @ critical points and endpoints Open IntervalCheck f(x) @ critical pointsCheck limits of f(x) as x approaches the endpointsInfinite IntervalCheck f(x) @ critical pointsCheck limits of f(x) as x approaches +/- infinityOr some combination of the above …Chapter 5 – Optimization Problems1. Draw & Label a Diagram2. Find a formula that involves the variable to be optimized (must include one other variable).3. Determine the domain for the other variable.4. Solve the resulting absolute max/min problem.Differentiate the formulaDomain endpointsCritical points5. Give your final answer in terms of the problemChapter 5 – Mean Value TheoremFind all values for c  (a,b) where …( ) ( )'( )f b f af cb a-=-That is, all values where the slope of the tangent line is the same as the slope of the secant line.NOTE: Rolle’s Theorem is the MVT with f(a) = f(b) = 0.Chapters 2-4: DefinitionsLimitContinuous at the point x = aDerivative[ ]0( ) ( )( ) limhd f x h f xf xdx h�+ -=1. ( ) is defined2. lim ( ) exists3. ( ) lim ( )x ax af af xf a f x��=If 0 0 such that| ( ) | whenever 0 | | ,then lim ( )x af x L x af x Le de d�" > $ >- < < - <=Chapters 2-4: Differentiation FormulasSee review notes from chapters 3 & 4.Power rules.Trigonometric FunctionsInverse Trigonometric FunctionsLogarithmic FunctionsExponential FunctionsChapters 2-4: Other ProblemsEvaluate LimitsRational ExpressionsL’Hopital’s Rule (make sure it applies)Interpret a graph in terms of points of continuity and differentiability.Evaluate DerivativesPolynomialsProduct RuleQuotient RuleChain RuleImplicit DifferentiationLogarithmic DifferentiationHigher Order DerivativesChapters 2-4: Other ProblemsEvaluate LimitsRational ExpressionsL’Hopital’s Rule (make sure it applies)Interpret graphs in terms of points of continuity and differentiability.Evaluate DerivativesPolynomialsProduct RuleQuotient RuleChain RuleImplicit DifferentiationLogarithmic DifferentiationHigher Order


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BMCC MTH 251 - Final Exam Review-OLD

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