AP Calculus AB Notes 2018‐2019 Arbor View HS
AP Calculus ABNotes 2018‐2019Arbor View HSName:Period:1
Table of Contents1.2 Finding Limits Graphically and Numerically (48).41.3 Evaluating Limits Analytically (57) .71.4 Continuity and One-Sided Limits (68).101.5 Infinite Limits (80) .143.5 Limits at Infinity (192) .182.1 The Derivative and the Tangent Line Problem (94) .202.2 Basic Differentiation Rules (105) .232.2 Day 2 Rates of Change (109) .262.3 Product and Quotient Rules (117) .292.3 Day 2 Trigonometric & Higher-Order Derivatives (121) .322.4 Day 1 Chain Rule (127) .362.4 Day 2 Chain Rule (132) .392.5 Day 1 Implicit Differentiation (132) .422.5 Day 2 Implicit Differentiation (132) .452.6 Day 1 Related Rates (144) .472.6 Day 2 Related Rates (144) .503.1 Extrema on an Interval (160) .533.2 Rolle’s & Mean Value Theorems (168) .553.3 Increasing/Decreasing f(x)s and the 1st Derivative Test (174) .583.4 Concavity and the 2nd Derivative Test (184) .613.6 Summary of Curve Sketching (202) .643.7 Optimization (211) .682
3.8 Newton’s Method (222) .723.9 Differentials (228) .744.1 Antiderivatives and Indefinite Integration (242).794.2 Area (242) .834.3 Riemann Sums and Definite Integrals (265) .874.4 The Fundamental Theorem of Calculus (275) .90Slope Fields (Appendix pg. A6).944.5 Integration by Substitution (288) .974.6 Numerical Integration (300).1005.1 The Natural Log Function: Differentiation (314) .1035.2 The Natural Log Function: Integration (324).1065.3 Inverse Functions (332) .1095.4 Exponential f(x)s: Differentiation & Integration (341) .1125.5 Bases Other than e and Applications (351) .1155.6 Differential Equations: Grow and Decay (361) .1185.7 Differential Eqs: Separation of Variables (369).1205.8 Inverse Trig Functions - Differentiation (380).1235.9 Inverse Trig Functions - Integration (388) .1267.7 Indeterminate Forms and L'Hopital's Rule (530) .1286.1 Area of Region Between Two Curves (412) .1306.2 Volume the Disk Method (421) .1336.3 Volume the Shell Method (432).1373
Notes #1-1Date:1.2 Finding Limits Graphically and Numerically (48)Letter of recommendation:participate in class, standout – in a good way!1) A penny: .01 2) Go ½ the distanceeach time over 10 ft.lim f(x) Lx 2 3x 2 ?limx 2x 2xy1.75Ex.1 Find limx “hiccup” function* The limit (L) of f(x) as x approaches c.x c1.91.9992?2.0012.12.251 cos xnumerically and graphically.x0 3 x -2Ex.2 lim f ( x) ?1x-2 x 2 * Existence at thepoint is irrelevant.Limits that fail to exist:1. f(x) approaches different values from the left and rightsides of c.lim f ( x) lim f ( x)x cfrom the leftDiving board f(x)Exists everywhere else.lim f ( x) x cfrom the right 2 x 1Ex.3 lim f ( x) if f ( x) x 1 0 x 1x 0lim f ( x) x 34
2. Unbounded BehaviorEx.4 limx 01 xxy3. Oscillating BehaviorEx.5 limx 0 1 tan x The Formal Definition of a Limit (52)We start with the informal: L is the limit of f(x) as xapproaches c if f(x) gets close to L as x gets close to c.lim f(x) Lx cThe description is informal because the underlined phrasesdo not have precise meaning.Lowercase Greek:δ – delta (x)ε – epsilon (y)ε-δ Definition of a Limit: for each ε 0 there exists a δ 0such that if 0 x c , then f ( x) L .excludes cAlphabetical order.L εLL–εc–δcc δ0 x c 5
y xSo for f(x) x, lim x if ε .5,x 2then δ .5 and so on .44Ex.6 Find the limit L. Then find δ 0 such thatf ( x) L 0.01 whenever 0 x c .a) lim xx 3b) lim 2 x 6x 4Ex.7 Find the limit L. Then use the ε-δ Definition to provethat the limit is L.a) lim 5xx 1Summary:b) lim 2 x 1x 4Ex.8 The sum of the prime divisors of 2010 is:a) 211b) 208c) 76d) 77e) 786
Notes #1-2Date:1.3 Evaluating Limits Analytically (57)What are we class?Techniques of Finding Limits1. Direct SubstitutionSee continuity in 1.4.lim f(x) f(c) if f(x) is continuous at c.x cEx.1 Find the limit:b) lim 3 x 22a) lim 2 x 7x 2c) lim sin xx 3Composite functionssee (59) Ex.4x 5d) lim 42x 5 lim g(x) L & lim f(x) f(L), then lim f(g(x)) f lim g ( x) f(L)x cx Lx cx cProperties of Limits (57)Ex.2-5 lim f(x) 5, lim f(x) 4 and lim g(x) 8x 3x 8x 3a. lim [f(x) g(x)] b. lim [f(x) – g(x)] c. lim [f(x)]2 d. lime. lim 6f(x) f. lim g x g. lim f(g(x)) h. lim sec xx 3x 3x 3x 3x 3f ( x) g(x)x 3x 3x 7
2. Dividing Out TechniqueKnow the sum anddifference of cubes!x3 125 0 0 indeterminate form3Ex.6 Find the lim8 x 27x c2 x 3a) c 0for:b) c 2c) c 1.5Ex.7 Find the limit:664x – 1x 2 4 x 5a) limx 5x 53( x x) 3x x x 0b) lim3. Rationalizing Technique (the numerator)Ex.8 Find the limit:Don’t multiply outthe denominator.Memorize these!sin x 1limx 0x1 cos x 0limx 0xa) limx 3x 1 2x 3b) limz 07 z 7zEx.9 Find the limit using the Special Trig Limits (64)1 cos 2 xa) lim3xx 0b) limx 0xsin(5x)8
Complex fraction1 1 2.xEx.10 Evaluate the limit, if it exists: limx 2x 2a)tanθ 14b) 14c) 1d) -1e) dnetan 1 xEx.11 Evaluate the limit, if it exists: lim.x 1 1sin x 1a) 0b)14c)12d) e)2 2 4The Squeeze Theorem aka Sandwich Theorem (63)If h(x) f(x) g(x) & lim h(x) L & lim g (x) Lx cx cthen lim f (x) L .x c1and the SqueezexTheorem to find lim f (x) if -x2 f(x) x2.Ex.12 Use the graph of f(x) x2sinx 02008 #5Summary:5 x 4 8 x 2Ex.13 limx 03x 4 16 x 2(A) 12(B) 0(C) 1(D)53(E) dne9
Notes #1-3Date:1.4 Continuity and One-Sided Limits (68)Can be traced withoutlifting your pencil.A function is continuous (uninterrupted) at c if:1. f(c) is defined.2. lim f(x) exists.x c3. lim f(x) f(c).x cA discontinuity is removable if f can be made continuous byredefining f(c), i.e. the value at that single point.cccDefined at c?Limit at c?Removable?A function is continuous:A) on an open interval if it continuous at each point in the interval.B) everywhere if it is continuous on (- , ). from the right– from the leftC) on a closed interval [a, b] if it is continuous on (a, b) andlim f ( x) f (a) andlim f ( x) f (b) .x a x b aMatch (A-C)babab10
DiscontinuitiesRemovableJumpInfiniteThe Greatest Integerf(x) is a Step functionor staircase.Ex.1 Discuss the continuity of the function:1x 2 5 x 6a) f ( x) 2b) f ( x) x 35 xc) f ( x) x d) f(x) Graphing calc undercatalog int(2 x 15x 2x 21x 2 x 22Ex.2 Find the limits from the graph of f(x):d) lim f ( x)a) lim f ( x)x 1 b) lim f ( x)x 1 c) lim f ( x)x 1x 1 4321e) lim f ( x)x 1 12f) lim f ( x)432112x 111
Ex.3 Find the limit of:a) lim x 5 x 5See Properties ofLimits (57).(73)Existence theoremstell you somethingexists, but do not giveyou a method forfinding them.A person was 2 feettall f(a) when she was1 year old (a) and shewas 5 feet tall f(b)when she was 14years old (b). Atsome point (c) shemust have been 4 feettall (k) becausehuman growth iscontinuous over aninterval of time.b) lim 2 x 6 x 3c) lim x 1xxIf f & g are continuous at c, then 1. af2. f /– g3. f · gare continuous at c.f4. , g(c) 0g5. f(g(x)), restrictionsSome functions are continuous at every point in their domain:1. Polynomial2. RationalGiven #2 above and #1 & #43. Radicaly 2x5 5 – cosx is continuous.4. TrigonometricThe Intermediate Value Theorem (often 1 point on AP exam):Suppose f(x) is continuous on [a, b]. If k is any numberbetween f(a) and f(b), then there is at least one number c in[a, b] such that f(c) k.abIf f(a) 2 and f(b) 5, thenthere is a f(d) 3 (three inthis case) and a f(c) 4somewhere on [a, d].Ex.4 Explain why the function has a zero in the giveninterval: f(x) x3 x2 – 1; [0, 1].Ex.5 Use the Intermediate Value Theorem and the bisectionmethod to estimate the zero of f(x) ex – 3x; [0, 1].12
Ex.6 Find value(s) for a, so that the functionf x2 a2 x x 4 2 x2 x 2is continuous.x 2Ex.7 Find values for a and b, so that the function x 45, f ( x) 2ax b, bx 5 2008 #77 4 x 3 is continuous.x 3Ex.8 The figure shows the graph of a function f withdomain 0 x 4. Which of the limits exist?I.II.lim f ( x)x 2lim f ( x)x 2III. lim f ( x)x 2Summary:(A) I(B) II(C) I & II (D)I & III (E) I, II, & III13
Notes #1-4Date:1.5 Infinite Limits (80)Infinite Limit: a limit in which f(x) increases or decreaseswithout bound as x approaches c.It shows that f(x) isunbounded, so thelimit dne. The signis misleading. by 0The notation limx 01 does not mean that the limit exists!xEx.1 Find the real number c that is not in the domain.Determine whether f(x) approaches - or as xapproaches c from the left and from the right.31b) f ( x) a) f ( x) 2 xx 4 c c lim f ( x) x c lim f ( x) x c c) f ( x) 2 x 3 2c lim f ( x) x c lim f ( x) x c lim f ( x) x c lim f ( x) x c d) f ( x) 3 x 2 2c lim f ( x) x c lim f ( x) x c Vertical Asymptote: x c, if f(x) approaches - or as xapproaches c from the left or the right.14
Precalc notes Ch.2.6DegreeABN Dy N Dy 0N Dno H.A.Ex.2 Find the vertical asymptotes and removablediscontinuities of the functions:3x 2 1x 2 2 x 8a) f ( x) 2b) f ( x) x 9x2 4Past AP Type Problems Covered by this Chapter:The AP Calc test has4 parts:Multiple choice:30 Qs in 60 minutesno calculator.15 Qs in 45 minutes –some require acalculator.Free response:2 Qs in 30 minutessome parts of Qs mayrequire a calculator.x 2 x 6Ex.3 Evaluate the limit: lim.x 22 xa) 5Ex.4a)14b) 3c) -3d) -5Evaluate the limit, if it exists: limx 9b) 14c) 1e) dnex 5 2.x 9d) 0e) dne4 Qs in 60 minutesno calculator.Ex.5a) 0How many vertical asymptotes exist for1in (0, 2π)?f ( x) 22sin x sin x 1b) 1c) 2d) 3e) 415
IVTEx.6If p(x) is a continuous function on [1, 3] withp(1) K p(3) and c is in the closed interval [1, 3],then what must be true?Ex.7 Identify the vertical asymptote(s) for f ( x) GCa) x -2, x 1b) x -2d) y -2, y 1e) y -2Ex.8a) 0x 2 3x 4.2x x 2c) x 1 x 1 x Find the limit: lim e x .x 0 b) 1c) 2d) dnee) noneEx.9Is the function continuous at x 1? Why or why not?2for x 1 xf ( x) 2 x for x 1Ex.10 sin(2 x), x f ( x) 2x k , x a) -2πb) -πc) 0what value of k will makethis function continuous?d) πe) 2π16
Free ResponseEx.10 Calculators may not be used. Use the graphs of f(x)and g(x) given below.f(x)g(x)a) Is f [g(x)] continuous at x 0? Explain.b) Is g[f(x)] continuous at x 0? Explain.c) What is the limf [ g ( x)] ? Explain.x 1 f ( x) g ( x), 2 x 0, what is k sod) If h( x) ()(),0kgxfxx that h(x) is continuous at x 0?Summary:17
Notes #1-5Date:End Behavior(left & right)3.5 Limits at Infinity (192)Horizontal Asymptote: y L if xlimf ( x) L or limf ( x) Lx Note: from this definition, the graph of a function of x canhave at most two horizontal asymptotes – one to theright and one to the left. indeterminate form ( numerator & denominator by the highest power of x in the denominator).n: degree ofnumeratord: degree ofdenominatorGuidelines for Finding Limits at Infinity of RationalFunctions If n d:the limit is 0. If n d:limit If n d:the limit does not exist, we may writean(ratio of the leading coefficients).bdlim f ( x) to show that f(x) increases orx decreases without bound.Ex.1 Find the limits: lim f ( x) and lim f ( x)x x 1a) f ( x) 3x 3x 2x 2x2 6b) g ( x) 2x 418
Ex.2 Find the limits: lim g ( x) and lim g ( x)x a) h( x) 6 x 13x 2 5x x2 xb) g ( x) x 1Ex.3 Sketch the graph of the equation. Look for intercepts,symmetry, and asymptotes.3x 2g ( x) 2x 16(2 x 1)(3 x)Ex.4 limisx ( x 1)( x 3)2008 #77(A) -3Ex.5(B) -2(C) 2(D) 3(E) dne6 6 6 .Summary:19
Notes #1-6Date:2.1 The Derivative and the Tangent Line Problem (94)Tangent lines to acircle are a specialcase, because theradius is perpendicularto the tangent line.Leave your answer inpoint-slope form!Converting to slopeintercept form justcreates more chancesto make errors.Secant comes fromthe Latin secare,meaning to cut, andis not a reference tothe trig function.Ex.1 Find the equation of a circle thathas (0, 0) as its center and passesthrough (1, 2). Graph.Ex.2 Find an equation of the tangent line to the circle thatpasses through (1, 2).One way to find the tangent to a general curve is to use asecant line to approximate the slope.Ex.3 Approximate the slope ofy x2 3 at (1, 4).(c x, f(c x))(c, f(c))Calculus in Motion!Difference QuotientThe quotient of twodifferences.The smaller x thebetter.f(c) xcmsec : y x f ( c x ) f ( c)( c x ) c f ( c x ) f ( c) xmtan : lim x 0f (c x ) f (c) xEx.4 Find the slope of the tangent line to the graph off(x) 3x 1 at (3, 10).Stuff cancels!20
Ex.5 Find the formula for the slope of f(x) 2x2 and useit to find the equation of the tangent line at (-2, 8).The derivative of a function f with respect to x:limit definitionof the derivative!f '( x) lim x 0f ( x x) f ( x)“f prime of x”f ' (x) is also a function of x.Other notations (97): Dx[y]y derivative – slope oftangent lineAP Exam questionNOT regression!Box before simplify!provided the limit exists. x“the derivative of y with respect to x”"y prime"dydxdfdxdf ( x)dx"dy dx""the derivative of f with respect to x""d dx of f at x " or "the derivative of f at x "Ex.6 Let f be a function that is differentiable for all realnumbers. The table below gives values of f for selectedpoints in the closed interval 2 x 13. Estimate f '(4) .xy21345-283136Ex.7 Find the derivative of f (x) 1by the limit process.x21
conjugateThe converse is notalways true.(103) #61-70Ex.8 Find the derivative of f (x) x 3 by the limitprocess. Find the equation of the tangent line if x 1.If f is differentiable at x c, then f is continuous at x c.Alternative form of derivative: f '(c) limx cf ( x) f (c)x cEx.9 Use the alternative form to find the derivative at x c.f(x) x2, c 3Ex.10 Find the lim f ( x) .x 3Is f (x) x 3 continuous?Find the left & right derivatives of f (x) at x -3.When is a function not differentiable at a point?1) If it is not continuous at the point.2) If the graph has a sharp turn at the point.3) If there is a vertical tangent at the point.Summary:Evolution:xsin(3x )4x xsin xsin(5 x )22
Notes #1-7Date:2.2 Basic Differentiation Rules (105)Ex.1 Find the derivative of these constant functions:a) f(x) -3b) s(t) 0The Constant Rule:Ex.2 Find the derivative of these functions:a) y xb) f(x) x211 11 2 11 3 3 1c) f(x) x3d) y x-1The Power Rule:23
Ex.3 Find the derivative of the function (rewrite):15b) y 3c) g(x) x3a) f(x) x7xEx.4 Find the derivative of these functions using limits:y 3x2The Constant Multiple Rule:derivativeslope of tangent linerate of changeEx.5 Prove the Constant Multiple Rule:d[c f ( x)] c f '( x)dxEx.6 Find the slope of the graph of f(x) 2x2 when x a) -2b) 0Ex.7 Find the equation of the tangent line to the graph of1f(x) x4 when x -2.224
Ex.8 Using parentheses when differentiating:Originala) y RewriteDifferentiateSimplify37x2b) f ( x) c) y 54 x 332 4 x3The Sum & Difference Rules:Derivative of Sin & Cos:Memorize!!!d[ f ( x) g ( x)] dxd[sin x] dxd[ f ( x) g ( x)] dxd[cos x] dxEx.9 Find the derivative of the function:7 x4 2x25-2a) y x(3x – 2x )b) f ( x) 3x 2c) f ( x) If in doubt, take thederivative and to 0.3cos x 2 x4d) y -2x2 – 3sinxEx.10 Find the derivative of the quadratic functionf(x) ax2 bx c. Find the value of x that makesf '(x) 0. The corresponding point on the graphy f(x) is special. Why?Summary:25
Notes #1-8Date:2.2 Day 2 Rates of Change (109)Utah StateMath 1210Calculus 1Exam 2cos(x y) Ex.1 Prove the Difference Rule:Ex.2 Prove the Derivative of cosx:sin(x y) cos(2x) An unusual cloudmight form as a planeaccelerates to justbreak the soundbarrier ( 769 mph atsea-level and 70 F innormal atmosphericconditions). A theoryis that a drop in airpressure at the planeoccurs so that moistair condenses there toform water droplets.Average velocity: schange in positionor va change in time tEx.3 What is the average velocity of a jet between 5pm and5:12 pm if it travels 154 miles?26
Free-fall Constants on the EarthAverage velocity: slopeInstantaneous vel:derivativeAcceleration due to gravity: g 32ftmorg 9.8sec2sec2Position Function: s(t)Velocity Function: v(t) s′(t) Speed is the velocity .The position of a free-falling object (neglecting air1s(t ) gt 2 v0t s0resistance) can be represented by:2Ex.4 A gold coin is dropped from the top of the 1149 footStratosphere. Indicate units of measure.a) Determine the position and velocity functions.b) Determine the average velocity on [3, 7].c) Find the instantaneous velocities when t 3 & 7.d) How long does it take to hit the ground?e) Find the velocity at impact.27
Derivative1. formula-slope oftangent line2. rate of change3. velocityEx.5 Find the derivative of the area A of a circle withrespect to its radius r.A(r) A'(r) Ex.6 Find the derivative of the volume of a sphere withrespect to its radius r.AP Exam!Ex.7 Evaluate using derivatives:1 ( x h)7 x 7 a) limb) lim sin h sin h 6 6 h 0h 0h (114) #63Ex.8 Find k such that the line y 5x – 4 is tangent to thegraph of the function: f(x) x2 – kx.2008 #6Ex.9 Which of the statements about f are true?I. f has a limit at x 2II. f is continuous at x 2III. f is differentiable at x 2(A) ISummary:(B) II(C) I & II x2 4 f ( x) x 2 1if x 2if x 2(D) I & III (E) I, II, & III28
Notes #1-9Date:2.3 Product and Quotient Rules (117)The algebra withinthe calculus can bemore challenging thanthe calculus itself.The Product Rule:ddudvuv v u dxdxdxIf y uv, then y' uv' u'v.Ex.1 Use the product rule to find f '(x) if f (x) x x.How could we answer this question a different way?Ex.2 Find the derivative of:a) k(x) sinx cosxb) f(x) x2cosxThe Product Rule can be used with more than two functions:d f ( x) g ( x)h( x) f '( x) g ( x)h( x) f ( x) g '( x)h( x) f ( x) g ( x)h '( x)dxEx.3 k(x) xsinx cosx29
low d high minushigh d low over lowsquareddu dv ud u dxdx , v 0The Quotient Rule: dx v v2vIf y vu ' uv 'u-1, then y' orlety uv.vv2Ex.4 Find k'(x):9 x7a) k ( x) x 1You may be ableto rewrite to avoidthe Quotient Rule.b) k ( x) tan xEx.5 Find f '(x) without using the Quotient Rule:3x 5 22 x 4 7 x3a) f ( x) b) f ( x) 22x5x 2Ex.6 Find the equation of the tangent line to the graph of fat the indicated point:2 x 4 a) y 2at 2, 7 x 3 b) y (x2 – 4x 2)(4x – 1) when x 1.30
When in doubt, findthe derivative and setit 0!Ex.7 Determine the point(s) at which the graphs of thefollowing functions have a horizontal tangent.x 2 3x 1a) y 2b) y 2x 3x 1On AP Exam and alot of our tests!Ex.8 Use the information to find f '(3):g(3) 4g '(3) -2h(3) 3a) f (x) 4g(x) –h '(3) π1h(x) 12b) f (x) g(x)h(x)c) f ( x) g ( x)2h( x)d) f ( x) g ( x) h( x)g ( x)Summary:31
Notes #1-10Date:Can we avoid theQuotient Rule here?2.3 Day 2 Trigonometric & Higher-Order Derivatives (121)2 x 5.Ex.1 Find k'(x) using the Quotient Rule k ( x) 3xEx.2 Use an identity & the Prod. or Quot. Rule to findd[sec x] .dxNotice that bothcofunctions havenegative derivatives.Tangent Function:dtan x sec2 xdxSecant Function:dsec x dxCotangent Function:dcot x -csc2 xdxCosecant Function:dcsc x -csc x cot xdxEx.3 Find y':a) y sec x1 tan xb) y 1 sin xcos x32
Ex.4 Find the equation of the tangent line at the point:a) y tan x, (π/4, 1) b) y x·cos x, (π, - π)c) y secx – 2cosx, (π/3, 1)dny(123) Higher-Order Derivatives f ( n ) ( x)ndxy 2x4 – 5x2 – 17 f (x)y' nth derivative: thederivative taken ntimesdydx f '(x) y" f "(x)d2y dx 2y'" d3y f '" (x) dx3(4)y fd4y(x) dx 4(4)Ex.5 Find y" for: y x·cos x .33
Ex.6 Find y" for: y 4xx 1Ex.7 Find f (27)(x) for f (x) cos x.Throw pen up in theair and discuss v(t)a(t).Speed v(t )Position Function:s(t)Velocity Function:s′(t) v(t)Acceleration Function: s"(t) v′(t) a(t)Speed increases when v(t) & a(t) have the same sign.Speed decreases when v(t) & a(t) have the opposite sign.Ex.8 Find the velocity and acceleration when t 4 sec.s(t) t3 – 6t2 9t and s is in meters. Indicate units of measure.Units!Is the speed increasing or decreasing at t 4 sec?34
Ex.92008 #82Ex.10 A particle moves along a straight line with velocity2given by v(t) 7 – 1.01 t at time t 0. What is theacceleration of the particle at time t 3?(A) -0.914 (B) 0.055(C) 5.486(D) 6.486(E) 18.087Summary:35
Notes #1-11Date:2.4 Day 1 Chain Rule (127)Ex.1Find the derivative of C(x) (x2 5)3.Ex.2 Complete the table by decomposing the functions:y f(g(x))y u g(x)y f(u)x 2 1y sin 6xy (3x 2)5y tan2 xTHE CHAIN RULE (the derivative of a composite function):If f is differentiable at the point u g(x), and g isdifferentiable at x, then the composite function f g ( x) f ( g ( x)) is differentiable at x, and f g x f g x g ( x).Or, if y f(u) and u g(x), thendy dy dudy dx du dx where du is evaluated at u g(x)Ex.3 Find C'(x) of C(x) (x2 5)3 using the Chain Rule.36
“Outside-Inside” Differentiation:Ex.4 Find y':a) y (3x2 1)2b) y sin(x2 x)Ex.5 Differentiate with respect to x:5 2 x 1 1b)a) y y 32x 1(2 x 3) Power Chain Rule:d nduu nu n 1dxdxEx.6 Differentiate with respect to x:a) y 5 x 2 4 x 12008 #25b) y Ex.7 f is differentiable at x 2,what is the value of c d?(A) -4(B) -2(C) 0 246 x 1 cx dfor x 2 x 2 cxfor x 2f ( x) (D) 2(E) 437
2003 AP Multiple Choice Questions 21. If y x3 1 , then dy dx(A) (3x2)2(B) 2(x3 1)(D) 3x2(x3 1)(E) 6x2(x3 1)(C) 2(3x2 1)14. If y x2 sin 2x, then dy dx(A) 2x cos 2x(B) 4x cos 2x(C) 2x(sin 2x cos 2x)(D) 2x(sin 2x – xcos 2x)(E) 2x(sin 2x xcos 2x)From now on ask yourself, “Do I need to use the chain rule?”Do (a) & (b):Summary:38
Notes #1-12Date:Reminder:2.4 Day 2 Chain Rule (132)Tangent Function:dtan x sec2 xdxSecant Function:dsec x sec x tan xdxCotangent Function:dcot x -csc2 xdxCosecant Function:dcsc x -csc x cot xdxEx.1 Differentiate with respect to xa) sin(3x)b) tan xc) csc(x2 x)Shreksin(x x)sin(2x) 2sinxcosxEx.2 Repeated use of the Chain Rule:a) sin[1 tan(2x)]b) cos2(3x)Ex.3 Evaluate the derivative at the given point:1 1 2 a) y (2x 1)-3, 1, b) y sin x , , 1 x 27 2 39
Ex.4 Find an equation of the tangent line to the graph of f atthe given point or value:a) y x 2 4 x 1 , (4, -1)b) y 2tan x cos2 (2x), when x π.Ex.5 g(2) -3, g'(2) -6, h(2) 3, h'(2) -2, g'(3) 4Find the derivatives at x 2.a) [g(x)]3b) f (x) g(h(x))3[g(2)]2 g'(2)g(h(x)) g'(h(2))h'(2)3(9)(-6) g'(3) h'(2) 4(-2) -162c) g(x)h(x)-8g ( x)d)h( x)40
Ex.6 Find the derivatives:xa) y x 2 1b) y x4 x 1Ex.7 Find the derivative of y sin3(2x).2008 #8Ex.8 If f ( x) cos(3x) then f '(A)2008 #33 32(B)32(C) 32 9(D) 32(E) 3 32Ex.9 If f ( x) ( x 1)( x2 2)3 then f '( x) (A) 6 x( x2 2)2(B) 6 x( x 1)( x2 2)2(C) ( x2 2)2 ( x2 3x 1)(D) ( x2 2)2 (7 x2 6 x 2)(E) 3( x 1)( x2 2)2Summary:41
Notes #1-13Date:2.5 Day 1 Implicit Differentiation (132)W.1 Determine the derivative of y (3 – x)4.Ex. Conic SectionsNote: An explicitlydefined function is onethat is written infunction form, y f(x).Implicitly-Defined Function: a function with multiplevariables that is not solved for one of the variables2For example: 4 x 1 y 2 25Implicit Differentiation: differentiating a function that isnot written as an explicit formula.Use the following steps:1. Differentiate both sides of the equation with respect to x.2. Collect all terms with2y xdydx 2xdxdx3. Factor outNote:dydx.dydxon one side of the equation.4. Solve fordydx.When differentiating with respect to x, the derivativeof x isEx.1 Finddydxdxdx 1, and the derivative of y isdydx.by implicit differentiation x2 – xy y2 7.42
Note: There are many waysof writing the correct answer.Watch for these on multiplechoice selections.Ex.2 Finddydxby implicit differentiation & then evaluate thederivative at the indicated point. xy2 2y4 x2y, (2, 1).Ex.3 Calculate the derivative of y with respect to x of:sin(x y) x cos ydy1 cos( x y) dx cos( x y) sin y2008 #16Ex.4 If sin(xy) x, then(A)1cos( xy)(C) 1 cos( xy)cos( xy)dydx (B)1x cos( xy)(D) 1 y cos( xy)x cos( xy)(E) y(1 cos( xy))x43
Ex.5 x2 4y2 36a) Find two explicit functions.b) Sketch the graphs of theequations and label the partsgiven by the correspondingexplicit functions.c) Differentiate the explicit functions.d) Differentiate implicitly and show that the result isequivalent to part c.e) Find the points at which the graph of the equationhas a vertical or horizontal tangent line.Summary:44
Notes #1-14Date:2.5 Day 2 Implicit Differentiation (132)Opposite reciprocal slopeThe normal line at a point is to the tangent line at the point.Ex.1 Find the tangent & normal line of 2xy πsiny 2π at 1, . 2 x2dyEx.2 Findif: y . Hint: rewrite.dx2 x 3 yFinding a Second Derivative2Ex.3 Find d 2y for 4 y3 9 5x2 .dxDifferentiating again:Now substitute dy 5x2 and simplify:dx6yEliminate thecomplex fraction:45
26Ex.4 Find d 2y if: y .dxx yUsing the Chain Rule with a Table of ValuesEx.5 Evaluate the derivatives using the table below.x f(x) g(x) f x g x 2 231/3 33 3 42π5a) f(g(x)) at x 2c)1 g x 2at x 3b) g(f(x)) at x 2d)f x at x 2Summary:46
Notes #1-15Date:2.6 Day 1 Related Rates (144)Related Rates Equation: an equation that relates thec
Aug 10, 2018 · 1.3 Evaluating Limits Analytically (57) Techniques of Finding Limits 1. Direct Substitution lim x c f(x) f(c) if f(x) is continuous at c. Ex.1 Find the limit: a) 2 lim 2 7 x x b) 5 lim 223 x x c) 3 limsin x 2 x d) 5 lim4 x lim x c g(x) L & lim x L f(x) f(L), then lim x c f(g(x)) f lim ( ) xc gx f(L) Properties of Limits (57) Ex.2-5 3
Perpendicular lines are lines that intersect at 90 degree angles. For example, To show that lines are perpendicular, a small square should be placed where the two lines intersect to indicate a 90 angle is formed. Segment AB̅̅̅̅ to the right is perpendicular to segment MN̅̅̅̅̅. We write AB̅̅̅̅ MN̅̅̅̅̅ Example 1: List the .
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AP Calculus AB Summer Homework Name _ Dear Future Calculus Student, I hope you are excited for your upcoming year of AP Calculus! . concepts needed to be successful in calculus. Please submit this packet during your second calculus class period this fall. It will be graded. . final answers MUST BE BOXED or CIRCLED. 2 Part 1 - Functions .
