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Name: _________________ Period: ______ Date: ________________ AP Calc AB

Mr. Mellina

Chapter 3: Derivatives

Sections:v 2.4 Rates of Change & Tangent Lines

v 3.1 Derivative of a Function v 3.2 Differentiability

v 3.3 Rules for Differentiation v 3.4 Velocity and Other Rates of Change

v 3.5 Derivatives of Trigonometric Functions

HWSets

SetA(Section2.4)Page92,#’s1-31odd.

SetB(Section3.1)Pages105-108,#’s1-11odd,13-17all,19-23odd,28.

SetC(Section3.2)Page114&115,#’s1-16all.31-35odd.

SetD(Section3.3)Pages124,#’s1-35odd.

SetE(Section3.5)Pages146&147,#’s1-35odd.

SetF(Section3.4)Pages135-138,#’s1-19,odd,25,&27.

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2.4 Rates of Change and Tangent Lines Topics

v Continuity at a Point v Continuous Functions v Algebraic Combinations

v Composites v Intermediate Value Theorem for Continuous Functions

Warm Up! Evaluateeachlimitwithoutacalculator.

a. lim$→&

'($ )*'$

b. lim$→&

+),-*

+)

$ c. lim

$→&

.($*/$

Example1:FindingAverageRateofChangeFindtheaveragerateofchangeofthefunctionoverthegiveninterval.a. 𝑓 𝑥 = 𝑥/ − 𝑥,[1,3] b. 𝑓 𝑥 = 4𝑥 + 1,[10,12]

Average Rates of Change We encounter average rates of change in such forms as average speed (in mph), growth rates of populations (in % per year), and average monthly rainfall (inches per month). The Average Rate of Change of a quantity over a period of time is the amount of change ____________ by the ________ it takes. In general, the average rate of change of a function over an interval is the amount of ___________ divided by the _________ of the interval.

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Example2:ConsideringtheSlopeataPointLet’sconsider𝑓 𝑥 = 𝑥7,[0,4].Approximatetheslopeat(1,1)a. SlopeofSecantlinethrough(1,1)&(4,16)b. SlopeofSecantlinethrough(1,1)&(3,9)c. SlopeofSecantlinethrough(1,1)&(2,4)d. SlopeofSecantlinethrough(1,1)&(1.1,1.21)Howfarcanwego?

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Example3:ConsideringtheSlopeataPointFindtheslopeoftheparabola𝑓 𝑥 = 𝑥7atthepointP(2,4)Example4:ExploringSlopeandTangentLet𝑓 𝑥 = '

8.

a. Findtheslopeofthecurveatx=ab. Wheredoestheslopeequal-1/4?

Slope of a Curve y = f(x) at a Point P(a, f(a)) M = Provided the limit exists. What you are taking the limit of above is called the ______________ ____________. If you are asked to find the slope using the definition or using the difference quotient, this is the technique you will use.

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Example5:FindingaNormalLineWriteanequationforthenormaltothecurveatthevalueprovided.a. 𝑓 𝑥 = 4 − 𝑥7atx=1 b. 𝑓 𝑥 = '

8*'atx=2

Normal to a Curve The normal line to a curve at a point is the line ____________________ to the tangent at that point.

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Example6:FindingInstantaneousRateofChangeFindtheinstantaneousrateofchangeforthefunctionsatthegiventime.a. 𝑓 𝑡 = 2𝑡7 − 1att=2 b. 𝑓 𝑡 = 3𝑡 − 7att=1

Speed Revisited: Instantaneous Rate of Change The function 𝑦 = 16𝑡7 that gave us the distance fallen by a rock in Section 2.1, was the rock’s ____________ function. A body’s average speed along a coordinate axis (here, the y-axis) for a given period of time is the average rate of change of its position 𝑦 = 𝑓(𝑡). Its instantaneous _______ at any time t is the instantaneous rate of change of position with respect to time at time t, or lim

$→&

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3.1 Derivative of a Function Topics

v Definition of Derivative v Notation

v Relationships between the graphs of f and f’ v Graphing the Derivative from Data

v One-sided Derivatives

Warm Up! Evaluatetheindicatedlimitalgebraically

a. lim$→&

(7($))*B$

b. lim8→7,

8(/7

c. lim

8→B

78*C8*7

Definition of Derivative In section 2.4, we defined the slope of a curve y = f(x) at the point where x = a to be 𝑚 = lim

$→&

E()*E()

When it exists, this limit is called the ____________ of ____ at ___. The derivative of the function f with respect to the variable x is the function ____ whose values at x is 𝑓′(𝑥) = lim

$→&

E()*E()

, provided the limit _________. The domain of f’ may be smaller than the domain of f. If f’(x) exists, we say that f has a derivative (is differentiable) at x. A function that is differentiable at every point of its domain is a differentiable function.

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NotationWhatitsoundslike Whatitlookslike Whatitmeans“fprimeofx”

Thederivativeof___withrespectto___

“yprime”

Thederivateof___

“deewhydeeecks”

Thederivativeof___withrespectto___

“deeeffdeeecks”

Thederivativeof___withrespectto___

“deedeeecksuveffuvecks”

Thederivativeof___

Example1:ApplyingtheDefinitionUsethedefinitionofthederivativetofindf’(x)a. 𝑓 𝑥 = 2𝑥 + 3b. 𝑓 𝑥 = 𝑥/ − 3

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Example2:ApplyingtheAlternateDefinitionDifferentiatef(x)a. 𝑓 𝑥 = 𝑥atthepointx=a b. 𝑓 𝑥 = '

8,ata=2

Example3:RelationshipsBetweentheGraphsoffandf’Considerthefollowingfunctionf(x)a. “Thederivativeisthe__________oftheoriginalfunction.”

Alternate Definition of Derivative at a Point The derivative of the function f at the point x = a is the limit f’(a) = provided the limit exits.

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b. Graphthederivativeofthefunctionfwhosegraphisprovidedbelow.

Example4:Graphingffromf’Sketchthegraphofafunctionfthathasthefollowingpropertiesa. i.f(0)=0 ii.Thegraphoff’,thederivativeoff,isshownbelow. iii.fiscontinuousforallx

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b. Sketchthegraphofacontinuousfunctionfwithf(0)=1and𝑓′ 𝑥 = 2,𝑥 < 2−1,𝑥 > 2

Example5:Comparingf’withfSupposethegraphbelowisthegraphofthederivativeofh.a. Whatisthevalueofℎ′(0)?Whatdoesthistellusaboutℎ 𝑥 ?b. Usingthegraphofℎ′ 𝑥 ,howcanwedeterminewhen thegraphofℎ 𝑥 isgoingup?Howaboutgoingdown?c. Thegraphofℎ′ 𝑥 crossesthex-axisat𝑥 = 2and𝑥 = −2.Describethebehaviorofthe

graphofℎ 𝑥 atthesepoints.

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Example6:One-SidedDerivativesCanDifferataPointShowthatthefollowingfunctionhasleft-handandright-handderivativesatthegivenxvalue,butnoderivativethere.

a. 𝑦 = 𝑥7,𝑥 ≤ 02𝑥,𝑥 > 0 atx=0

b. 𝑓(𝑥) = 𝑥7 + 𝑥,𝑥 ≤ 13𝑥 − 2,𝑥 > 1 atx=1

One-Sided Derivatives A function y = f(x) is differentiable on a ___________ interval [ , ] if it has a derivative at every interior point of the interval, and if the following limits _________ at the ______________: lim

$→&,E(M($)*E(M)

$ [the ________-hand derivative at a]

lim

$→&NE(M($)*E(M)

$ [the ________-hand derivative at a]

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Example7:EstimatingDerivativeswithaTable

Thefollowingtableliststhepositions(t)ofaparticleattimetontheinterval0 < 𝑡 < 4.

a. Findtheaveragevelocitybetweentimest=0.5andt=1.5.

b. Estimate𝑠′(1.5).Includeunitsofmeasure.

c. Onwhatinterval(s)does𝑠′(𝑡)appeartobepositive?

d. Onwhathalfsecondintervalistherateofchangeof𝑠(𝑡)thegreatest?

Afunction𝑇(𝑥)iscontinuousanddifferentiablewithvaluesgiveninthetablebelow.Usethevaluesinthetabletoestimatethefollowing.

e. 𝑇′(1.4) f. 𝑇′(2.4)

g. Theaveragerateofchangeof𝑇 𝑥 betweenx=1.4andx=2.2

h. Theinstantaneousrateofchangeof𝑇(𝑥)atx=1.

i. Theequationofthetangentlineto𝑇(𝑥)atx=1.

t(sec) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

s(ft) 12.5 26 36.5 44 48.5 50 48.5 44 36.5

x 1.0 1.4 1.8 2.2 2.6

T(x) 1.06 2.2 3.2 2.8 3.1

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3.2 Differentiability Topics

v How f’(a) Might Fail to Exist v Differentiability Implies Local Linearity v Numerical Derivatives on a Calculator v Differentiability Implies Continuity

v Intermediate Value Theorem for Derivatives

Warm Up! Tellwhetherthelimitcouldbeusedtodefinef’(a)(assumingthatfisdifferentiableata)a. lim

$→&

E M($ *E(M)$

b. lim$→&

E M($ *E($)$

c. lim8→M

E 8 *E(M)8*M

d. lim

8→M

E M *E(8)M*8

e. lim$→&

E M($ (E(M*$)$

Example1:Whenf’(a)failstoexistgraphically Afunctionwhosegraphisotherwisesmoothwillfailtohaveaderivativeatapointwherethegraphhas:a. A_________,wheretheone-sided b. A________,wheretheslopesofthe

derivativesdiffer secantlinesapproach∞fromone sideand−∞fromtheotherex: ex:

𝑓 𝑥 = 𝑥 𝑓 𝑥 = 𝑥)T

How f’(a) Might Fail to Exist A function will not have a derivative at a point P(a, f(a)) where the slopes of the secant lines, E(8)*E(M)

8*M fail to approach a ________ as x approaches a.

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c. A___________________________, d. A____________________,

wheretheslopesofthesecantlines whichwillcauseoneorbothofthe

approacheither∞or−∞fromboth one-sidedderivativestobe

sides. Non-existent.

ex: ex:

𝑓 𝑥 = 𝑥T 𝑓 𝑥 = −1, 𝑥 < 01,𝑥 ≥ 0

Example2:FindingWhereaFunctionisNotDifferentiable Findallpointsinthedomainoff(x)wherefisnotdifferentiable.

a. 𝑓 𝑥 = 𝑥 − 2 + 3 b. 𝑓 𝑥 =𝑥,𝑥 < 1'8,𝑥 ≥ 1

Most of the functions we encounter in calculus are differentiable wherever they are defined. They will _____ have corners, cusps, vertical tangent lines, or points of discontinuity within their domains. Their graphs will be ___________ and _________, with a well-defined slope at each point. Polynomials are differentiable, as are rational functions, trig functions, exponential functions, and logarithmic functions. Composites of differentiable functions are differentiable, and so are sums, products, integer powers, and quotients of differentiable functions, where defined.

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Example3:UsingtheCalculatortoEvaluateDerivatives UseyourCalculatortofindthederivativeatthegivenpoints.a. 𝑦 = 𝑥/,findVW

V8atx=2. b. 𝑓 𝑥 = 5𝑥7 + 4𝑥,find𝑓′(𝑥)atx=1.

Example4:UsingtheCalculator Graphthefollowingfunctions,usethevalueandVW

V8featuretofillinthetable.Thenguess

whatyouthinkthederivativeis.a. 𝑓 𝑥 = 𝑒8 b. 𝑓 𝑥 = ln𝑥

V

V8𝑒8 = V

V8ln𝑥 =

x f(x) f’(x)-1 0 1 2 3

x f(x) f’(x)1 2 3 4 5

WARNING The calculator may return an incorrect value if you evaluate a derivative at a point where the function is not differentiable. Examples: d(1/x,x)| x=0 returns -∞ d(abs(x),x)| x = 0 returns ±1

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Example5:DifferentiabilityimpliesContinuityIffisafunctionsuchthat lim

8→*/

E 8 *E(*/)8(/

= 2,whichofthefollowingmustbetrue?a. Thelimitof𝑓(𝑥)asxapproaches-3doesnotexist.b. fisnotdefinedatx=-3.c. Thederivativeoffatx=-3is2.d. fiscontinuousatx=2.e. f(-3)=2.Example6:UsingtheIVTSketchagraphofafunctionf(x)thatmeetsthefollowingcriteriaforf’(x).a. f’isnegativeandincreasingforallx b. f’ispositiveandincreasingforallx c. f’isnegativeanddecreasingforallx d. f’ispositiveanddecreasingforallx

Differentiability Implies Continuity We began this section with a look at the typical ways that a function could fail to have a derivative at a point. As one example, we indicated graphically that a discontinuity in the graph of f would cause one or both of the one-sided derivatives to be nonexistent. Theorem 1: Differentiability Implies Continuity If f has a derivative at x = a, then f is _______________ at x = a. (the converse is not true)

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3.3 Rules for Differentiation Topics

v Positive Integer Powers, Multiples, Sums, and Differences v Products and Quotients

v Negative Integer Powers of x v Second and Higher Order Derivatives

Warm Up! Writetheexpressionasasumofpowersofx.

a. 𝑥7 − 2 𝑥*' + 1 b. 88)('

*' c. 3𝑥7 − 7

8+ [

8)

d. /8\*78T(B

78) e. 𝑥*' + 2 𝑥*7 + 1 f. 8N+(8N)

8NT

g. Forthegivenfunctionbelow,graphVW

V8 h. Usethealternateformofaderivative

tofind𝑦′for𝑦 = 5.

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Example1:DifferentiatingaPolynomialFindV]

V^

a. 𝑝 = 𝑡/ + 6𝑡7 − [

/𝑡 + 16 b. 𝑝 = ^T

/+ ^)

7+ 4𝑡 − '

^+ 1

Example2:UsingDerivativesFindthefollowing.a. Findtheslopeatx=2of b. Atwhatx-value(s)arethereany 𝑦 = 𝑥B − 4𝑥7 + 1 horizontaltangentsof 𝑦 = 𝑥B − 2𝑥7 + 2

Rules for Differentiation 1. Derivative of a Constant Function: If f is the function with the constant value c,

then, VEV8= V

V8(𝑐) =

2. Power Rule for Integer Powers of x: If n is a positive or negative integer, then,

VV8(𝑥a) =

3. The Constant Multiple Rule: If u is a differentiable function of x and c is a constant,

then, VV8(𝑐𝑢) =

4. The Sum and Difference Rule: If u and v are differentiable functions of x, then their

sum and difference are differentiable at every point where u and v are differentiable. At such points,

𝑑𝑑𝑥

(𝑢 ± 𝑣) =

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Example3:UsingtheProductandQuotientRuleFindf’(x)andsimplify.a. 𝑓 𝑥 = 𝑥7 + 1 𝑥/ + 3 b. 𝑓 𝑥 = 3 + 2 𝑥 5𝑥/ − 7

Rules for Differentiation Continued: Products and Quotients 5. The Product Rule: The product of two differentiable functions u and v is differentiable, and

VV8(𝑢𝑣) =

6. The Quotient Rule: At a point where 𝑣 ≠ 0, the quotient 𝑦 = f

g of two

differentiable functions is differentiable, and V

V8hfgi =

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c. 𝑓 𝑥 = 8)*'8)('

d. 𝑓 𝑥 = '(ln8

8)*ln8

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Example4:WorkingwithNumericalValuesLet𝑦 = 𝑢𝑣betheproductofthefunctionsuandv.Find𝑦′(2)ifa. 𝑢 2 = 3, 𝑢j 2 = −4, 𝑣 2 = 1,and𝑣j 2 = 2

Example5:UsingthePowerRule

Let𝑦 = 8)(/78

.Findthefollowinga. 𝑦jusingthequotientrule b. 𝑦′bymakingthefractionasum oftwopowersofx.c. Findanequationforthelinetangenttoyatthepoint(1,2)

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Example6:FindingHigherOrderDerivativesFindthefirstfourderivativesofthefunctionprovided.a. 𝑦 = 𝑥/ − 5𝑥7 + 2 b. 𝑦 = 8('

8

Second and Higher Order Derivatives The derivative 𝑦′ = VW

V8 is called the ______ derivative of y with respect to x. The first

derivative may itself be a differentiable function of x. If so, its derivative 𝑦jj = VWj

V8= V

V8hVWV8i = V)W

V8)

is called the __________ derivative of y with respect to x. If y’’ is differentiable its derivative is called the _________ derivative of y with respect to x. The names continue as you might expect they would, except that the multiple-prime notation begins to lose its usefulness after about 3 primes. We use

𝑦(a) = VV8𝑦(a*')

to denote the ____ derivative.

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3.5 Derivatives of Trigonometric Functions Topics

v Derivative of the Sine Function v Derivative of the Cosine Function

v Simple Harmonic Motion v Jerk

v Derivatives of the Other Basic Trig Functions

Warm Up! Evaluatethelimitwithoutacalculator.a. lim

8→&

[ mno /8B8

b. lim8→&

pqm 8*'8

Example1:DerivativeofSineConsiderthefollowinggraphofSine.Fillinthechartwiththeslopesatthegivenvaluesof𝜃.

Derivative of Sine 𝑑𝑑𝑥 𝑠𝑖𝑛𝑥 =

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Example2:DerivativeofCosineConsiderthefollowinggraphofCosine.Fillinthechartwiththeslopesatthegivenvaluesof𝜃. Example3:RevisitingtheDifferentiationRulesFindthederivativea. 𝑦 = 𝑥7 sin 𝑥 b. 𝑢 = pqm 8

'*mno 8

Derivative of Cosine 𝑑𝑑𝑥 𝑐𝑜𝑠𝑥 =

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Example4:DerivativeofTangent

Findthederivativeoftan 𝑥usingthequotientrule.

a. VV8

tan 𝑥

Example5:ATrigonometricDerivativeFindthederivativeofeachfunction.a. 𝑓 𝑡 = 𝑡 + 4 sec 𝑡 b. ℎ 𝜃 = 5 sec 𝜃 + tan 𝜃c. ℎ 𝑠 = '

z− 10 csc 𝑠 d. 𝑦 = 𝑥 cot 𝑥

Derivatives of the Other Basic Trig Functions V

V8tan 𝑥 = V

V8sec 𝑥 =

V

V8cot 𝑥 = V

V8csc 𝑥 =

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3.4 Velocity and Other Rates of Change Topics

v Instantaneous Rates of Change v Motion Along a Line v Sensitivity to Change

v Derivatives in Economics

Warm Up! Answerthequestionsaboutthegraphofthequadraticfunction𝑓 𝑥 = −16𝑥7 + 160𝑥 − 256byanalyzingtheequationalgebraically.(Nocalculator)a. Doesthegraphopenupwardordownward?b. Whatisthey-intercept?c. Whatarethex-intercepts?d. Whatistherangeofthefunction?e. Forwhatx-valuedoesVW

V8= 100?

f. FindV)W

V8)atx=7

Example1:Averagevs.InstantaneousVelocityConsideragraphofdisplacement(distancetraveled)vs.time AverageVelocitycanbefoundbytaking: changein

changein=

𝑉Mg� =

= E *E

Instantaneous Rate of Change The speedometer in your car does not measure average velocity, but instantaneous velocity. The velocity at one ____________ in _______. 𝑉(𝑡) = Vz

V^= lim

∆�→&

E()*E()

It is conventional to use the word instantaneous even when x does not represent time. The word, however is frequently omitted in practice. When we say “rate of change,” we mean instantaneous rate of change.

Velocity is the ________ derivative of position.

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Example2:PVT(CalculatorAllowed)Anobjectwasdroppedoffacliff.Theobject’spositionisgivenby𝑠 𝑡 = −16𝑡7 + 16𝑡 + 320,wheresismeasuredinfeetandtismeasuredinseconds.a. Whatistheobjectsdisplacementfromt=1tot=2seconds?b. Whenwilltheobjecthittheground?c. Whatistheobject’svelocityatimpact? d. Whatistheobject’sspeedatimpact?e. Findtheobject’saccelerationasafunctionoftime.

Relationships between Position, Velocity, and Acceleration Position: s(t) describes the position s of an object after t seconds.

• Displacement of an object is the ____________ ___ ___________ over a given interval of time. • On a position graph, we can easily determine the direction an object moves:

o When s(t) is increasing, an object moves ________ or _________. o When s(t) is decreasing, an object moves ________ or _________.

Velocity (Instantaneous): v(t) describes the rate of change in the position. v(t) = ____ Unless term “average velocity” is used, we will assume velocity refers to instantaneous velocity. It is the slope of a tangent line to the position function… aka the ____________.

• On the velocity graph, we can also determine the direction an object moves:

o When v(t) > 0, an object moves in a positive direction (______ or ______). o When v(t) < 0, an object moves in a negative direction (______ or ______). o When v(t) = 0, an object is _____________.

• Velocity is a ____________ quantity and must have magnitude and direction. • Speed is ______________ and is always positive.

Acceleration (Instantaneous): a(t) describes the rate of change in the velocity. a(t) = ______ = ______

• We can use the velocity and acceleration to determine the following: o An object is speeding up when v(t) and a(t) have the _________ sign. o An object is slowing down when v(t) and a(t) have _____________ signs.

Don’t forget units!

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Example3:PVTSupposethegraphprovidedshowsthevelocityofaparticlemovingalongthex-axis.Justifyeachresponse.a. Whichwaydoestheparticlemovefirst?b. Whendoestheparticlestop?c. Whendoestheparticlechangedirection?d. Whenistheparticlemovingleft?…movingright?e. Graphtheparticle’saccelerationfor

0 < 𝑡 < 10.f. Whenistheparticlespeedingup?g. Whenistheparticleslowingdown?h. Graphtheparticle’sspeedfor 0 < 𝑡 < 10i. Whenistheparticlemovingthefastest?j. Whenistheparticlemovingata constantspeed?

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Example4:UsingDerivativesAdynamicblastpropelsaheavyrockstraightupwithalaunchvelocityof160feetpersecond(about190mph).Itreachesaheightof𝑠(𝑡) = 160𝑡 − 16𝑡7ftaftertseconds.a. Howhighdoestherockgo?b. Whatisthevelocityandspeedoftherockwhenitis256ftabovethegroundontheway

up?Onthewaydown?c. Whatistheaccelerationoftherockatanytimetduringitsflight(aftertheblast)?d. Whendoestherockhittheground?

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Example5:StudyingParticleMotionAparticlemovesalongalinesothatitspositionatanytime𝑡 ≥ 0isgivenbythefunction𝑠 𝑡 = 𝑡7 − 4𝑡 + 3,wheresismeasuredinmetersandtismeasuredinseconds.a. Findthedisplacementoftheparticleduringthefirst2seconds.b. Findtheaveragevelocityoftheparticleduringthefirst4seconds.c. Findtheinstantaneousvelocityoftheparticlewhent=4.d. Findtheaccelerationoftheparticlewhent=4.e. Describethemotionoftheparticle.Atwhatvaluesoftdoestheparticlechange

directions?f. Graphs(t)anditsderivative.

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Itisimportanttounderstandtherelationshipbetweenapositiongraph,velocityandacceleration: