# Math Chapter 3

 Requirements for relative max/min? Must be continuous and if its on a closed interval, it cannot be an endpoint. f " > 0 relative MINf " < 0 relative MAX Critical point When f '(c) = 0 or undefined. Location of Extrema When derivative is undefined, or 0. So the same as a critical point Finding Extrema on Closed Interval Find Critical Numbers (f '(c))Evaluate f(c) for each critical numberEvaluate f(c) for endpoints High is MAX, Low is MIN NOTE: If it says a polynomial to the 3rd degree, there can only be 2 relative numbers. ex: 4th degree = 3 relative numbers etc What types of functions are always continuous and differentiable? PolynomialsRational FunctionsTrig Functions Rolle's Theorem Is f(x) continuous on [a,b]? why?Is f(x) differentiable on (a,b)? Why?Does f(a) = f(b)? If all of these apply: find the critical points at f '(x). Does Rolle's Theorem work for Absolute Values? Only if the closed interval does NOT include the vertex. Because it is not differentiable at a sharp point Mean Value Theorem (MVT) If f is continuous on [a,b] and differentiable on (a,b) then there exists c on (a,b) such that f '(c)= [f(b) - f(a)] / (b-a) Rates of Change and Slopes Average Rate of Change = Instantaneous Rate of ChangeSlope of Secant = Slope of Tangent Increasing/Decreasing f '(c) > 0 f is increasingf '(c) < 0 f is decreasingf '(c) = 0 f is constant When f ' changes from - to +, relative MINimumWhen f ' changes from + to -, relative MAXimum Velocity and Acceleration General Info Speeding Up/Speeding Down? s(t) = position > 0 right of origin / < 0 left of origins'(t) = velocity > 0 moving right / < 0 moving lefts"(t) = acceleration Find when s '(t) = 0 and s "(t) = 0. Put on number line.Plug numbers in for BOTH velocity and acceleration. When signs are same it is speeding UPWhen signs are different it is slowing DOWN Find polynomial of least degree such that (given relative min/max) f(x) = ax^4 + bx^3 + cx^2 + dx + e Plug in (x,y) in each f equation. Find derivative for equation and plug in (x,y)Solve for a,b,c,d,e Concavity Concave UP = f ' is INCREASINGConcave DOWN = f ' is DECREASING Concave UP = f " > 0 Concave DOWN = f " < 0 Point of Inflection When concavity changes from + to - or - to + at a point f " = 0 or undefinedTANGENT MUST EXIST(vertical/horizontal line is usually the case) 2nd Derivative Test When f '(c) = 0 f "(c) < 0 Relative MAX f "(c) > 0 Relative MIN Steps: find f '(x) plug in results to f "(x) Limits of Infinity (oo) Divide by highest power of the DENOMINATOR Denominator > Numerator f(x) = 0Denominator = Numerator f(x) = coefficientsDenominator < Numerator f(x) = +/- infinity (oo) lim (x => oo) (sinx)/x = 0 L'Hopital's Rule ONLY USE WHEN IT IS 0/0 OR oo/oo !!! Always substitute first to find out ^Find derivative, and substitute again. Keep going until it works NOTE: do NOT use quotient rule! Indeterminate Forms (!= means does not equal) oo - oo != 00 x oo != 01^oo != 1oo^0 != 10^0 != 1 Determinate Forms nonzero/0 = +/- oo0^-oo = oo0^oo = 0 -oo - oo = - oooo + oo = oo 1/0 = oo1/oo = 0 Graphing If f is increasing, f ' > 0 (above axis)If f is decreasing, f ' < 0 (below axis) If f '(x) = 0, there is a HORIZONTAL TANGENT If f '(x) does not exist, there is a SHARP TURN or VERTICAL TANGENT If f " > 0, concave UPIf f " < 0, concave DOWN If f ' does not exist, then it is a VERTICAL TANGENT When f " = 0, there is an inflection point (CONCAVITY CHANGES) Authorcraziieegurl13 ID45578 Card SetMath Chapter 3 DescriptionChapter 3: Applications of Differentiation Updated2010-10-28T04:51:05Z Show Answers