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# Exam 3 Study Guid MAC2311 MAC2311

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This 8 page Study Guide was uploaded by Monica Notetaker on Monday November 30, 2015. The Study Guide belongs to MAC2311 at University of Florida taught by Jane smith in Fall 2015. Since its upload, it has received 24 views. For similar materials see Calculus 1 in Mathmatics at University of Florida.

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Date Created: 11/30/15

MAC2311 EXAM 3 REVIEW LECTURES 23-31 Things to consider for the exam: Lecture 23: Extreme Values Extreme Value o What are Extreme Values? RTheoremTheorem is a way of proving the Points on a graph f(x) in which there is a point f Mean Value Theorem If All the Mean Value x= c that has a maximum or minimum closed Theorem is saying is that the slope of the o What is Absolute Max or Absolute Min? Intervatangent line is equal f to the slope of the An absolute max occurs at x=c if has both a max and f (c)≥ f (x) secfc) line f(d). min t Steps for using An absolute min occurs at x=c if Rolle’s theorem on Function is f (c)≤ f (x) continuouseck that all the conditions o Conditions for which extreme exist apply for Lecture 25: First Derivative T est Local Extreme Rolle’s Every continuous function on a closed interval Theorem Increasing ahas a min and max. o Plug in each of o If f'x >0onaninterval ,then (x)isincreasing Also called Relative Continuous functions on an open interval may Extreme the endpoints o If or may not have a max or minisdecreasing into the Max and Min on an When determining whether cannot ocf (x)his increasing or OPEN INTERVALion to see if decreasing, follow these steps fa = f (b) o Always use the domaFinding Local Extremenction f (x) o Find the f(c)=0∨ f (c)DNE derivative of Find wheo TaLecture 24: Mean Valuee the original o Find critical numbers f function. Theoremst bo Set up a number line (I know you don’t want to but o Then find you should!) using the critical numbers and any critical CritRolle’s Theorem Mean Value Local Extreme are found on an open intervalaTheoremhe original numbers. If a f is continuous on critical Absolut o Choose random numbers in betwenHas the following number is in o At aof the number line the domain fo Plug them into the first derivative to find the slope f (c) and all other o At the sm'llest (min) or largest (max) oaluef Is conditions are oa,On a closed intervaltive, then the function is true, than f a)= f (b) continuous on Rolle’s decreasing [a,b] Theorem Theo Ifs af '(x)ris positive, then the fuoction Is (a,b) such that f(c)=0 applies to that increasing differentiable function. FiSimplifiedl Extreme using the First Derivation (a,b) Questions might ask Testo There are points in The is a number c in you to use MVT to which the graph−atc ,then f(cisalocalmax verify that at a point o If the interval (a,b) in time that the pasf changes¿−¿+atc ,then fc isNo max or average velocity is Lecture 26: The Shape of a f’>0 Graph f’<0 Concavity o f x ) is differentiable on (a,b), f’’>0 then the function expresses concavity in the following ways o 1. f is concave up on (a,b) if f ' is increasing (curve lies above tangent line) f’’<0 o 2. f is concave down on (a,b) if f ' is decreasing (the curve lies below its tangent lines) Test for Concavity Conditions for L’Hopital’s Rule f '' o Assume that exists on (a,b) The limit must be indeterminate f'(x)>0 Applies to one sided limits and limits at o If for all x on (a,b), then f is concave up infinity f'(x)<0 Can be determined by indeterminate o If for all x on (a,b), ¿∞¿ then f is concave down products. aka if the limit is (0 Second Derivative Test If the limit is indeterminate differences ( f '' ∞−∞¿ Suppose that is continuous near c ' ' ' o If f(c)=0∧ f (c<0 then f (c) is a local max and concave down L’ Hopital’s Rule Uses definition of the derivative to find the limit of indeterminate forms. First Identify if the limit is 0∨ ∞ indeterminate, 0 ∞ Take the limit (plug in value given) If indeterminate, then use L’Hopital’s Rule Take the derivative of the top and bottom so that lim f (x=lim f '(x) x→a g(x) x→a g'(x) L’Hopital’s Rule is the limit Lecture 28: Graph Sketching f (x) Use to find Domain x-intercepts: y=0 y-intercepts: x=0 symmetry about the x or y axis asymptotes o Vertical asymptotes as the denominator =0 ax o Horizontal asymptotes bxm n<m then y=0 a n=m then b n>m then no horizontal asymptote ' Use f(x) to find Critical numbers in domain of original f o (horizontal tangents, vertical tangents/cusps) Increasing/ decreasing Local extreme Use f ''(x)to find Concavity Points of inflection Helpful Hints Make sure to sketch number lines before graphing Check your derivatives Check your signs Find critical points for extreme when graphing Do NOT plug in a point for every part of the graph. It will wastes time and you are only looking for the basic shapes and Antidifferentiation Lecture 29: Optimization Problems Function Formulas General To Solve an Optimization Problem c f(x) antiderivative 1. Read the problem carefully to determine f (x) ∓g(x) CF (x+C the quantity Q that you are trying to n F 'x)∓G' (x)+C optimize, and the conditions involved. x (n≠−1) 2. Draw a sketch if possible and assign symbols to known and unknown quantities 1 x n+1 n+1 3. Find the function representing the quantity x to be optimized (Q): the Primary Function ( Object Function) ex ln(x)+C 4. If necessary, find an equation relating the x variables involved (the Constraint) and sin(x) e +C write the primary function as the function cos(x) −cosx+C of a single variable. 5. Use calculus to find the desired max or sec x sinx min and check results. (You can use the first or second derivative test, or the tanx+C extreme value theorem if the domain is a secxtanx closed interval. csc x secx+c Use the First derivative test for the Absolute −cotx+C Extreme Values cscxcotx Lecture 30: Antiderivatives Suppose that f (x) = 3x+7. How do you find f x ? Things You Should Know! Lecture 31 Areas F (x)istheantiderivative o(xf) HowFt(x)isequal¿ f(x)soFt(x)= fix) that lies under the curve y= f(x)¿ a¿b The antiderivative is also called an integral Area under a curve An integral sign looks like this: ∫ f( )dx lim Rn Defie The indefinite (or general) Integral is ∫ f( )dx=F x( ) wherHelpful Hinttantives f x =F'(x) ¿lim ∆ x[ f x +…+ f x ]he integral of a function, you muAlways remember to add a constant C n→∞ ( Particular Solutions f (x) 's(t=position whenwoIf the grapheneral is below the x-axis, Find f(x)if f(x=some function∧ f a =b antiderivative! It applF(x)to all general Left endpoints 're representetime antiderivative questions includingcreasing and where a=xvat)=s (t=velocityodistance vis versa (above=positive) Integrand Midpoints are M n Rectilinearf (x)on and EconoF '(x) Get general solution If =0 then =0 (tangent Plug in a for all x’s and set it equal to slope of F is 0) the approximation of the areadistance2 f (x) constant b n→∞ If the slope of is positive then Solve for Curve. Thus Plug in solved C into general solution to Riemann Sum Riemann Sum is the sum of rectangles that approximate the area under a curve from a to b x2 f( 1+ f ¿+…+ f x( i+ f x ( n ]for some function f (x) Formula b−a Let R n be the sum of the areas of n rectangles with width ∆ x n→ ∞ n ∆ x b−a is n let i bethe¿ endpoint of i rectangles wheretheheight of therectangles is f (i ) Continue of Riemann Sum Summation Notation Riemann Sum is an approximation of the area under the curve It is defined by the summation notation Summation notation is used to write sums in compact form n n n→ ∞i=1f( i∆ x ∑ ai=am+a m+1+…+a n−1+an i=m where∆xcanbebefore∨afterthesummationnotation n is the number of rectangle/endpoint m is the number you are starting from When the question asks to find the exact area of examplef (x) from given a to b using Riemann Sum: 4 ∑ i =1 +2 +3 +4 3 o Write down given formula above i=1 o Solve for ∆ x o Solve for xi o Plug in ∆ x∧x i into given formula fx ) f( i xi f x) o Plug in for where represents all x’s in When question asks what area is represented by a given Riemann Sum o Locate ∆ x∧x i in the given Riemann Sum o Determine the a and b to identify the interval Lecture 32: The Definite Integral If f is defined on a ≤x≤b , divide [a,b] into n subintervals of equal width ∆ x= b−a o n The Riemann Sum as the limit of n approaches infinity is equal to the Definite Integral!! b n ∫a f(x)dx=n→∞ i=1f (i )∆x b Notation ∫ fx dx a Integral sign ∫❑ To Evaluate definite Integrals Helpful Hints Integrand fx ) using Sums Integration: process of finding the integral If the graph is a perfectly n n geometric figure, you can ∑ ca ic ∑ ai Definite Integral and Area actually just find the area using i=1f(x)dx=signed area=Areaabovexaxis−areabelowxaxis ¿ the measurements that the b graph gives you. For example, ∫ ¿ the graph creates a triangle on n n n a 1 a ∓b = a ∓ b a to b. Use the formula i=1 i i ∑i=1i i=1 i 2 bh=A to find the area. Remember that Areas are not negative but the total signed n ∑ c=cn area can be negative if i=1 for 1 −A2 c is a constant and n is a positive integer There are others but theses you Properties of Integrals

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