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 FunctionsCritical points: f’(x) = 0 or DNEIncreasing: f’(x) > 0Decreasing: f’(x) < 0Constant: f’(x) = 0Concave Up: f’’(x) > 0Concave Down: f’’(x) < 0Inflection Point: f’’(x) = 0 and concavity is changingChapter 5 – Relative Maxima & MinimaStationary Critical Points: f’(x)=0f’’(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) definedf’(x-h) > 0 & f’(x+h) < 0 for some h>0 relative maxi.e. changing from increasing to decreasingf’(x-h) < 0 & f’(x+h) > 0 for some h>0 relative mini.e. changing from decreasing to increasingChapter 5 – Absolute Maxima & MinimaClosed IntervalCheck f(x) @ critical points and endpoints Open IntervalCheck f(x) @ critical pointsCheck limits of f(x) as x approaches the endpointsInfinite IntervalCheck f(x) @ critical pointsCheck limits of f(x) as x approaches +/- infinityOr 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 formulaDomain endpointsCritical points5. Give your final answer in terms of the problemChapter 5 – Mean Value TheoremFind 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: DefinitionsLimitContinuous at the point x = aDerivative[ ]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 FormulasSee review notes from chapters 3 & 4.Power rules.Trigonometric FunctionsInverse Trigonometric FunctionsLogarithmic FunctionsExponential FunctionsChapters 2-4: Other ProblemsEvaluate LimitsRational ExpressionsL’Hopital’s Rule (make sure it applies)Interpret a graph in terms of points of continuity and differentiability.Evaluate DerivativesPolynomialsProduct RuleQuotient RuleChain RuleImplicit DifferentiationLogarithmic DifferentiationHigher Order DerivativesChapters 2-4: Other ProblemsEvaluate LimitsRational ExpressionsL’Hopital’s Rule (make sure it applies)Interpret graphs in terms of points of continuity and differentiability.Evaluate DerivativesPolynomialsProduct RuleQuotient RuleChain RuleImplicit DifferentiationLogarithmic DifferentiationHigher Order
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