SECTION 2.1 - THE TANGENT AND VELOCITY PROBLEMS
SECTION 2.1 - THE TANGENT AND VELOCITY PROBLEMSTangent ProblemDefinition (Secant Line & Tangent Line)Example 1Find the equation of the tangent lines to the curve y 2at the points with x-coordinates1 3xx 0 and x 1 .1
Example 2Susie and Johnny are running a lemonade stand. The following table gives a running total ofmoney they made as added each hour. Estimate the slope of the tangent line to the graph att 3.# of hourstotal 0 01 4.502 10.253 18.754 26.505 33.752
Velocity ProblemDefinition (Average Velocity & Instantaneous Velocity)Example 3A baseball is rolled down a ramp. The distance the ball has traveled, measured in feet, ist2modeled by the function d(t ) 2t 1 where time, t, is measured in seconds. Find the4average velocity of the ball over the following time periods.[1, 3][1, 2][1, 1.5][1, 1.1]Find the instantaneous velocity of the ball when t 1.3
SECTION 2.2 – THE LIMIT OF A FUNCTIONDefinition (Limit)Example 1Find each limit or explain why it does not exist.lim f (x ) lim f (x ) x 4x 2lim f (x ) lim f (x ) x 0x 4Example 2Estimate lim x 2 2x 1 .x 6()4
Definition (One-Sided Limit)Example 3 2x 2 ifLet f (x ) x 1 if0 x 3. Find lim f (x ) and lim f (x ) .x 3x 3x 3Definition (Infinite Limit)5
Example 411Find lim and lim .x 0x 0xxExample 5Find the following limits.lim csc 2 x x 0limx 21(x 2)2 Definition (Vertical Asymptote)(See examples 4 & 5.)6
Example 6Find the vertical asymptotes of the function f (x ) x 2 3x 7.x 37
SECTION 2.3 – CALCULATING LIMITS USING LIMIT LAWSLimit LawsSuppose c is a constant and the limits lim f (x ) and lim g(x ) exist.x ax a1.) lim (f (x ) g(x )) lim f (x ) lim g(x )x ax ax a2.) lim (f (x ) g(x )) lim f (x ) lim g(x )x ax ax a3.) lim cf (x ) c lim f (x )x ax a4.) lim (f (x )g(x )) lim f (x ) * lim g(x )x a5.) limx ax ax af (x )f (x ) limif lim g(x ) 0 x ag(x ) lim g(x ) x ax a6.) lim (f (x )) nx a( limx a)f (x )nwhere n is a positive integer7.) lim c cx a8.) lim x ax a9.) lim x n a n where n is a positive integerx a10.) limnx n a where n is a positive integer. If n is even, we assume that a 0 .11.) limnf (x ) n lim f (x ) where n is a positive integer. If n is even, we assume that f (x ) 0 .x ax ax a8
Example 1Calculate the following limits.lim 3 x 2 5 x 9x 2limx 3()x2 x 32x 1x 2 4 x 12limx 2x 2 2xlimx 8x 2 6x 4x 89
Direct Substitution PropertyExample 2Find the following limits.lim 2x 2 3 x 4 x 2()limx 9 x 2x 3lim2( 3 h) 18 hx 422h 0Theorem 110
Example 3Determine if the following limits exist. 4 x 1 if x 4lim f (x ) where f (x ) 2x 4 x 1 if x 4limx 44 x4 xTheorem 2Squeeze Theorem11
Example 4Use the squeeze theorem to find the following limits. 4 x 2 2x lim sin 2 x x 03 1 lim x 2 2x cos x 2 x ()12
SECTION 2.4 – THE PRECISE DEFINITION OF A LIMITDefinition (Limit – Precise Definition)Definition (One Sided Limit – Precise Definition)Definition (Infinite Limits – Precise Definition)13
Example 1Use the precise definition to prove the following statements.lim (7 x 27 ) 8x 5()lim x 2 3 x 2x 2()lim x 3 2x 2 3 x 4 2x 114
Example 2Prove the following statement by using the precise definition of an infinite limit.2lim4 x 4x 415
SECTION 2.5 – CONTINUITYDefinition (Continuous at a number)Example 1 x Find all of the discontinuities of f (x ) 3 x (x 3 )2 if TION 4.1 – MAXIMUM & MINIMUM VALUESDefinition (Absolute Max/Min & Local Max/Min)Example 1Find the absolute max, absolute min, local max(s), local min(s), max value, and min value of thefollowing function.58
The Extreme Value TheoremFermat’s TheoremExample 2Find the absolute max/min of the following functions.2f (x ) (x 2) 1g(x ) 1 x 259
Definition (Critical Number)The Closed Interval Method60
Example 3Find the extreme values of the functions on the given intervals.f (x ) x x on 0, 4[ ](g(x ) x 2 2xh(x ) )3[]on 2, 1[ ]ln xon 1, 3x261
SECTION 4.2 – THE MEAN VALUE THEOREMRolle’s TheoremThe Mean Value TheoremTheorem 5Corollary 762
Example 1Verify that the functions satisfy the hypotheses of Rolle’s Theorem on the given interval. Thenfind all numbers c that satisfy the conclusion of Rolle’s Theorem.f (x ) x 3 3 x 2 7 on 0, 3[ ] π 3π g(x ) ln(sin x ) on , 4 4 Verify that the functions satisfy the hypotheses of the Mean Value Theorem on the giveninterval. Then find all numbers c that satisfy the conclusion of the Mean Value Theorem.h(x ) 2x 2 3 x 1 on 0, 2[ ]k (x ) xx2 1[]on 1, 163
Example 2Show that the equation x 101 x 51 x 1 0 has exactly one real root.Show that the equation x 3 x 2 x 1 0 has at most one real root.64
SECTION 4.3 – HOW DERIVATIVES AFFECT THE SHAPE OF THE GRAPHIncreasing/Decreasing TestFirst Derivative Test65
Example 1Find where each of the following functions is increasing/decreasing and the localmaxima/minima.f (x ) x 3 6 x 2 15 x 4g(x ) x 1 xh(x ) e 2 x x266
Definition (Concave Up & Concave Down)Concavity TestDefinition (Inflection Point)Second Derivative Test67
Example 2Find the intervals of concavity and inflection points of the following functions.f (x ) x 3 6 x 2 15 x 4g(x ) x 1 xh(x ) e 2 x x268
SECTION 4.4 – INDETERMINATE FORMS AND L’HOSPITAL’S RULEDefinition (Indeterminate Forms)L’Hospital’s Rule69
Example 1Find the limits.1 cos xlim x 0x2 xe 4x 1 4xlim x 0x2limx e x 1 4x x2lim x 3 e x x 70
Definition (Indeterminate Products & Differences)StrategyDefinition (Indeterminate Powers)Strategy71
Example 2Find the limits.x1lim x 1x 1 ln xlim x 2 ln x x 0 lim xx 111 x lim (tan x )x π2cos x 72
SECTION 4.5 – SUMMARY OF CURVE SKETCHINGChecklist Of Important Info Needed To Sketch A Curve73
Example 1Sketch the following curve.y x 4 3x 3 3x 2 x74
Example 2Sketch the following curve.x2y x 875
Example 3Sketch the following curve.ππy 4 x tan x, x 2276
SECTION 4.7 – OPTIMIZATION PROBLEMSStrategyFirst Derivative Test For Absolute Extreme ValuesDo Problems 4, 12, 18, & 26 in class.77
SECTION 4.8 – NEWTON’S METHODWhat is Newton’s Method used for?How does Newton’s Method work?78
Example 1Use Newton’s Method to find the root of the equation x 5 x 4 3 x 2 3 x 2 0 in the interval1, 2 correct to six decimal places.[ ]Example 2Use Newton’s Method to find the absolute maximum value of the function f (t ) cos t t t 2correct to eight decimal places.79
SECTION 4.9 – ANTIDERIVATIVESDefinition (Antiderivative)Theorem 1.Example 1Find the general antiderivative for the following functions.f ′(x ) 3 x 2 4 x 3g′(x ) 3e x csc x cot xh′(x ) 5 x 2 x 380
Example 2For each of the following, find the particular antiderivitive.f ′(t ) 2t 3 sin t with f (0 ) 5g′′(x ) 2x 3 3 x 2 4 x 5 with g(0 ) 2 and g(1) 081
TABLE OF ANTIDERIVATIVES82
SECTION 5.1 – AREAS AND DISTANCESThe Area ProblemExample 1Estimate the area under the curve y x 3 1 from 0 to 2.83
Definition (Area)NotationExample 2Find the area under the curve y x 3 1 from 0 to 2.Find the area under the curve y x 2 x from 1 to 3.84
The Distance ProblemExample 3A radar gun was used to record the speed of a runner. Use the data to estimate the distance therunner covered during those 5 seconds.time (s)velocity (ft/s)00.54.6717.341.58.8629.732.533.544.5510.22 10.51 10.67 10.76 10.81 10.8185
SECTION 5.2 – THE DEFINITE INTEGRALDefinition (Definite Integral)Example 1nExpress limn sin x xi[]as a definite integral on the interval 0, π .i 1Theorem 386
Theorem 4Example 2 (3Evaluate the integral by interpreting it in terms of area;)9 x 2 dx . 3Equations Involving Sums of Powers of Positive Integers (See Appendix E)87
Example 3Evaluate the Riemann sum for f (x ) x 3 1 by taking sample points to be right endpoints and2a 0 , b 2 , and n 8 , then evaluate (x3) 1 dx .0.88
Example 4Evaluate the Riemann sum for f (x ) x 2 x by taking sample points to be right endpoints and3a 1 , b 3 , and n 6 , then evaluate (x2) x dx .189
The Midpoint RuleExample 51Use the Midpoint Rule with n 5 to approximate 1 x 3 dx .0Properties of the Definite Integral90
Example 6 (4 2 3Evaluate the following integral)9 x 2 dx .391
SECTION 5.3 – THE FUNDAMENTAL THEOREM OF CALCULUSExample 1xif 0 x 2 3 (x 3 )2 if 2 x 4 Let g(X ) f (t ) dt where f (x ) . 2x 10if4 x 6 0 x 8if 6 x 8Find the values of g(0 ) , g(1) , g(2) , g(3 ) , g(4 ) , g(5 ) , g(6 ) , g(7 ) , and g(8 ) then sketch a roughgraph of g(x ) .x92
The Fundamental Theorem of Calculus (Part 1)Example 2Find the derivative of each of the following functions.g(x ) x 1 t 4 dt1x( )h(x ) tan t 2 dtπk (x ) x2 0t1 t3dtThe Fundamental Theorem of Calculus (Part 2)93
Example 3Evaluate the following integrals.4 (1 6x 48 x ) dx229 e03πx 2 dxx 2 (2t 3 sin t ) dtπ2The Fundamental Theorem of Calculus94
SECTION 5.4 – INDEFINITE INTEGRALS AND THE NET CHANGE THEOREMDefinition (Indefinite Integral)Table of Indefinite Integrals95
Example 1Evaluate the following integrals. 1 x45 dx (2 cos t sec t ) dt2 2 x dxThe Net Change TheoremExample 2A particle moves along a line with a velocity function of v (t ) t 2 t , where v is measured inmeters per second. Find the displacement and the distance traveled by the particle during thetime interval 0, 5 .[ ]96
SECTION 5.5 – THE SUBSTITUTION RULEThe Substitution Rule (U-Substitution)Example 1Evaluate the following integrals.x 2 x 2 4x dx sin x cos(cos x )dx tan x ln(cos x )dx97
Substitution Rule for Definite IntegralsExample 2Evaluate the following integrals.1 y(y2) 14dy01 v2( )cos v 3 dv0π4 (1 tan t )3sec 2 t dt098
Integrals of Symmetric FunctionsExample 3Evaluate the following integrals.2 (14 x ) dx2 227 27sin xdx1 x 299
1 SECTION 2.1 - THE TANGENT AND VELOCITY PROBLEMS Tangent Problem Definition (Secant Line & Tangent Line) Example 1 Find the equation of the tangent lines to the curve 1 3x 2 y at the
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