I. Self-intro II. Course info III. Wat is calculus? Te biggest acievement of mankind in te past 000 years. eg. Trending now on internet wit fake news! Sec2. Tangent and velocity Tangent eg. Circle: moving way, approimate by secant line way. eg. cubic curve y = 3 by secant approimation. Application. Velocity (and also trending on internet) eg. d = t 2. Geometry and pysics correspondence P slope of P P average speed P P P tangent slope at P instantaneous speed Sec2.2 Te limit of a function Def. If te function value f () approaces L as approaces c, ten we say te limit of function f () as goes to c is L. Notation: lim f () = L. c eg. regular, point jump, two sided jump Conclusion. Te eistence of te limit of a function as goes to c as noting to do wit te function value at c. eg. lim 0, eg. t2 9 t 3, eg. wild lim 0 sin π. One sided limit rescue mild jump Te left-and limit lim a f () = L, if function value f () approaces number L as input approaces a from left and side. Similarly for rigt-and limit eg7. Infinite limit eg.,, 2 2 lim a f () = +, if fcn value f () becomes as large as possible, as approaces a. (lim a f () = as large negative as possible) lim a ± f () = ± eg. lim 0 + ln eg. lim π tan 2 Application: Vertical asymptote, te grap looks like a vertical line = a if lim a/a /a + f () = ± HW2.2 #8 lim +0 9 +9 Q. orizontal asymptote? Sec0. Curves defined by parametric equations Graps good, not general enoug, eg. circle needs two graps. A unified way, parametric equations { = + 5 cos t eg. wat is tis curve? y = 2 + 5 sin t eg.hw0. #4 parametric and usual (non-parametric) equations for te line troug A (, ) and B (3, 5).
Sec2.3 Calculating limits using limit laws + laws sum difference Te limit of te is te product quotient te limit of te bottom is not 0. sum difference product quotient* of te limits. * provided lim [f () + g ()] = lim f () + lim g () a a a case lim g () 0 a ( eg2. a) lim 2 2 3 ) 3 + 2 2 + 4 b) lim (power law) 5 2 5 3 2 4 eg3 lim 2 2 eg5 lim (2 + ) 2 4 Recall slope/instantaneous eg from Sec2.2 0 t2 + 9 3 eg6. lim, warm-up eg. lim t 0 t2 + 9 (via grap) t 0 t( HW2.3#6 lim t 0 t + t ) (save all -work on a single notepad) t squeeze t m preparing for sine derivative f () a L g () () a L lim a g () = L. RMK. You re te middle of a triplet, your younger broter is 9 and your older sister is also 9, ow old are you? eg. lim 0 2 sin Sec2.5 Continuity applications of limit Continuity Geo: grap as no break/jump Abs/analytically: lim a f () = f (a) Def. continuous at a, continuous over interval (b, c), [b, c], [b, c). T m + If f and g are bot continuous at a, ten f + g is also continuous at a. Case, g (a) 0. Applications: Polynomials, rational function over teir domains, in fact, all elementary functions over teir domains. RMK. y = 2 & y = ; y = e & y = ln graps are symmetric w.r.t. bisecting line y =. Composition limit law lim a f (g ()) = f (lim a g ()), provided f is continuous at b = lim a g (). eg. f () = continuous, so lim a g () = lim a g (). 2
c.eg. H () Heaviside function not continuous at 0, g () = 2, lim 0 H (g ()) = 0 = H (g (0)). T m. g cont. at a, f cont at g (a) ten f (g) cont at a. Intermediate Value T m. Suppose f is cont on closed interval [a, b] and N is between f (a) and f (b), ten tere eists a c on [a, b] suc tat te intermediate value N is realizes at c, tat is f (c) = N. eg0 2 = cos on [0, 2]. RMK. man cow potosyntesis = man cow(grass) =man(milk) = ceese HW2.5 {#8 Find te constant c tat makes g continuous on (, ) were g () = 2 c 2 < 4 c + 20 4 Sec2.6 Limits at infinity; orizontal asymptotes eg. y = /, y = + Def. lim + f () = L means te function value approaces L as becomes suffi ciently larger and larger, or approaces + from left and side. Appl. Geo H-line y = L is called a orizontal asymptote of te grap/curve y = f () at. eg. y = arctan eg. y = e sin eg. lim 3 2 2 eg3. lim 5 2 + 4 + eg4. f () = 2 2 + find its orizontal asymptotes 3 5 ( ) HW2.6 #4 lim 252 + 5 RMK. limit at, lim f () = means f () becomes arbitrarily large (negative) as becomes suffi ently large (negative). Sec2.7 Derivatives and rate of canges f() f(a) Tangent line: limit definition of te slope at (a, f ()), lim a a eg. f () = 2, at (, ). f(t) f() Instantaneous velocity/speed: limit definition v (a) = lim t a t a eg3 d (t) = 4.9t 2, at t = = lim 0 f(a+) f(a) Slope, instantaneous velocity, instantaneous rate of oter canges lim 0 y,... Universal name f() f(a) Derivative of f () at = a lim a a eg4. f () = 2 8 + 9, find f (a). HW2.8 #0 y = tangent line at (6, 4). = lim 0 f(a+) f(a) Sec2.8 Te derivative as a function f f(a+) f(a) (a) = lim 0 a >> 9 >> 0= f () a function eg. Sketc grap of f () w/ f () = cos (symmetry) eg3. f () =, find f () and its domain = f (a) eg4. f () =, find f () 2+ Q. symmetry properties, odd/even (HW2.8 #7, #8 & Workseet3 ), inverse fcn Oter notations y = f () y = f () = = df = d f () = Df () = D d d d f () ; = lim d 0 y 3
Def of differentiability: differentiable at a if f (a) eists, over (a, b) if differentiable at every pt in te interval. eg. f () =, were f () is differentiable? eg y =Heaviside, y =. Q. 2 sin, sin differentiable? (later) T m. f differentiable f continuous Geometric argument, f differentiable at a, grap as a tangent line near at a, grap looks like tangent line wic as no break grap itself as no break. Analytic: f () f (a) = f() f(a) ( a) f (a) ( a) as small as possible a wen a. Sec3. Derivatives of polynomials and eponential functions Q. p () = 00 50 99 + 30 0 + 4 3 + 2 + +, p () = d p () =? d Powers& general power rule sum/difference rule + [cf ()] = cf () eg&q Q. E () = e, E () =? Wy e? Oters, f () = 2, or 3, f () =? e + e 0 e E (0), "Intermediate value teorem" tere eists a good/nice/natural = e e base so tat slop is. Q. (e ) =? (e e ) = (e 2 ) =? Sneak in iger order derivatives Notation: y = f () y, y, y, y, y (5), y (6), ; f, f, f, f, f (5), f (6), ;, d2 y = d d d 2 d ( d d y), eg. distance d (t), velocity v (t) = d (t), acceleration a (t) = v (t) = d (t), Jerk j (t) = a (t) = d (t). eg. d 208 d 208 ( 207 ) =?; (e ) (5) =? ( ) (3) Q. 2 =? Sec3.2 Product and quotient rule Q. (e ), (e e ) = (e 2 ), ( ) ( t+, Known f, g, f, g,want (fg), t ) 2 + 2 3 +6 ( f g ) in terms of f, g, f, g. f f(+) f() () = lim 0 ; g g(+) g() () = lim 0 f ( + ) g ( + ) f () g ( + ) + f () g ( + ) f () g () [f () g ()] }{{} = lim 0 f ( + ) f () g ( + ) g () = lim g ( + ) + lim f () 0 0 = f () g () + f () g () As g is differentiable g is cont. lim t 0 g ( + ) = g (). 4
[ ] [ ] [ f() g() = f () g() = f () + f () g() [ ] = lim g () 0 = lim 0 g(+) g() g() ] = f ()g() f()g () g 2 () g () g ( + ) = lim 0 g ( + ) g () [ ] g ( + ) g () eg. (e ),eg. ( 4 = ( 3 ), ( 00 ) eg. (c 00 ) eg. ( ) ( = d ) 2 d, ( n ) = d n 2 d HW3.2 #7 f (v) =, find f (v). Simplify first. v v+ a v eg5. Find an equ of te tangent line to te curve y = Review trig formulas g ( + ) g () = g () e at =. + 2 g 2 () Sec3.3 Derivatives of trignometric functions d Py/Geo: (cos t, sin t) = ( sin t, cos t), or (sin ) = cos and (cos ) = sin Justify: slopes at 0 first (sin ) =0 and (cos ) =0, ten at via sum trig formulas (similar to (e ) : derivative at 0, ten at ). Step2. (sin ) : sin(+) sin sin cos +cos sin sin = = cos sin cos + sin (cos ) : cos(+) cos cos cos sin sin cos = = sin sin cos + cos Step. (sin ) =0 = sin sin sin 0 sin θ 0 = lim 0 or lim θ 0 θ Go w/ area comparison, θ sector and outer θ-triangle θ > 0 sin θ < θ < tan θ Area OP B < Area secoab 2 cos θ sin θ < θ sin θ θ < 0 θ sin θ = < cos θ θ = sin( θ ) θ as θ 0 so lim θ 0 θ sin θ θ 0 θ = sin θ Tus lim θ 0 = sin θ sin θ Now lim θ 0 = lim θ θ 0 θ = = sin θ = sin lim 0 Conclusion: slope of tangent line to y = sin at 0 is. θ in radians < Area OAC 2 θ 2 tan θ (cos ) cos cos 0 cos =0 = lim 0 = lim 0 = 0 Geo. ( cos Analytic. cos + ) cos + = (cos ) 2 sin sin = = sin sin 0 0 (cos +) (cos +) (cos +) 2 eg. (tan ) = ( ) sin cos, (cot ) = ( ) cos sin, (sec ) = ( ) cos, (csc ) = ( ) sin Mr. sin founding fater, Mrs. cos founding moter eg4. Find te 27t derivative of cos sin 7 eg5. Find lim 0 4 Review for Mierm I 5
Functions: Elementary ones Operations: algebraic + composition Properties: Limits of functions Laws odd/even/periodic inverse graps relation { algebraic + composition Applications: Derivatives { of functions Rule: continuity tangents/velocity/oter rate of cange algebraic + composition Cain rule (later) More tecniques later; Applications later. Review probs Sec2.3 eg8 lim 0 eg lim 0 2 sin Sec2.6 eg5 lim 2 + Sec2.7 eg2 tangent line at (3, ) to y = 3/ Sec3. eg6 Find points on te curve y = 4 6 2 +4 were tangents are orizontal. eg7 Distance in terms of time function s (t) = 2t 3 5t 2 + 3t + 4. Find acceleration a (t). HW3.-2 #7 Differentiate v = ( + 3 ) 2. HW3.-2 #9 Find an equation of te tangent line to te given curve y = 2 2 + + at (, 0). Sec3.3 eg4 Find (cos ) (27). eg5 Find lim 0 sin 7 4. Sec3.4 Te cain rule Q. 2 +, 25 2 sin 2, sin 2 2, 3, π, e 2, e π ( 2 + ) 000 Now F () = f g () = f (g ()), F f (g ( + )) f (g ()) () = lim 0 g ( + ) g () f (g ( + )) f (g ()) = lim g () 0 g ( + ) g () = f (g ()) g () g ( + ) g () f f( +) f( ) f( ) f( ) ( ) = lim 0 = lim = f ( ). Cain Rule F () = f (g ()) g (), oter notation df d w/ y = f (g ()), u = g (). 6 = df d dg, = g() d d du du d
eg. Q Need a = e ln a and e ln a = a. eg7/hw2sec3.4 #6 F (t) = e sin t, F (t) eg8/hw2sec3.4 #7 f (θ) = sin (cos (tan θ)), df dθ =? eg4. f () = 3 2 ++, f () eg5. g (t) = ( t 2 9 2t+), bot notations & d. eg6. y = (2 + ) 5 ( 3 + ) 4 (later on ln differentiation) d HW2Sec3.4 #2 (cos 3) (55) =? Sec0.2 Differential calculus { w/ parametric curves (lengt/area in Cal II) = t 2 Q. Given curves eg0. y = t 4 or y = 2, eg. = t 2, y = t 3 3t, eg2. = 5 (θ sin θ), y = r ( cos θ), ow to find tangents? Find d Via limit y () y (a) d = lim a =a a Tus = d d = lim t ta (y () y (a)) / (t t a ) ( a) / (t t a ). Formally it makes sense, so does pysically. eg0 = 4t3, d = 2t,ten Anoter way, y = 2, Q. d2 y =? d 2 d 2 y d 2 = d d ( ) d = d d y = d = d d d = (2 ) = 2. = d = 4t3 2t = 2t 2 = 2. = d (y ) d = ( d d d ) = y t t. But d 2 y 2 d 2 2 = t=ta eg. Curve = t 2, y = t 3 3t a) sow C as two tangents at (3, 0), find teir equs b) find pts on C were tangents are orizontal or vertical eg2. = 5 (θ sin θ), y = 5 ( cos θ) a) find tangent line equ at θ = π/3. HW3Sec0.2 #7 Tangent lines troug (2, 9) to = 9t 2 + 3, y = 6t 3 + 3 Slope = 8t2 = t. d 8t Tangent point: (2, 9) or correspondingly t =. Tangent point: not at (2, 9), at different point (9t 2 + 3, 6t 3 + 3). Toug (2, 9) is still on te tangent line. Y (6t 3 + ) = t (X (9t 2 + 3)), plug X = 2 & Y = 9, solve ard equ for t, ten done in term X-Y equ. Sec3.5 Implicit differentiation Q. 2 + 4 y2 = = d y 5 + y 5 = 5y = y 7 + y 6 5y 2 + y + 3 d d =? No eplicit epression of y in terms of t=ta 7
sin ( y) = y y = ln e y = y = arcsin sin y = =? Eplicit epression doen t elp d y = arctan tan y = z Implicit differentiation Step. Tink of y as a function of Step2. Differentiate bot sides of te equation w.r.t.. Remember y still a fcn of Step3. Solve te resulting equ for y Rmk. y () in terms of( and y, not only in. eg. 2 + 4 y2 = at, ) 3 d 2 eg2. 5 + y 5 = 5y, y = y eg2 = y 7 + y 6 5y 2 + y + 3 Serious eamples eg. y = ln, = d y =? Q. y = log a, = d y =? eg. y = arctan, ( y =? HW5Sec3.5 #4 arctan + Two notations and d. Geometric meaning d d =? ) =? Geometric meaning: Te slopes of te tangent lines at symmetric points of te function and its inverse are reciprical of eac oter. eg. y = e, (0, ), (ln a, a), y = e =ln a = a. y = ln, (, 0), (a, ln a), y = =a =. a eg2 /HW5Sec3.5 #6#7 y = d2 y d 2 Sec3.6 Derivatives of logaritmic functions Q. y = log 0, y =? Review (a t ) = a t ln a = ( e t ln a) Implicit way: y = log a a y =... Cange base way: log a = ln sould skip ln a eg y = log 2 ( 3 + ), d d eg6. ln d ln + d 2 eg5 d Straigt way Simplify first way Logaritmic differentiation. ln bot sides 2. Differentiate bot sides 3. Solve for y eg7. y = 3/4 2 + (3+2) 5, y =? eg. e, [f ()] p e, a g() 8
, [f ()] g() Derivatives? HW6Sec3.5 #8 y = [sin (8)] ln, y =? Only wen bot power and base are canging, use log diff. (Oter cases 2, 2, no need.) Natural base e as a limit. Come back to tis log/ln conversion in Sec4.4. L Hospital s rule = ln ln(+) ln = lim 0 = lim 0 ln ( + ) e = e = e lim 0 ln(+) = lim 0 e ln(+) = lim 0 ( + ) ) n, = n ( eg. e = lim n + n Anoter way e = + + + + + +. In fact! 2! 3! n! e = + + 2! + 3 + 2! 3! + n n! + Calculus III Anoter way: area under te yperbola/pseudo-circle y = /, above -ais, from to e is, Calculus II Compare 2π to circumconference of unit circle, as a limit of lengt of circumscribed/subscribed polygons π = arctan = + + + Series in Calculus III. 4 3 5 7 9 Sec3.9 Related rates eg. Air is being pumped into a sperical balloon so tat its volume increases at a rate of 00 cm 3 /s. How fast is te radius of te balloon increasing wen te diameter is 50 cm? HW6Sec3.9 #2 similar eg2. A ladder 0 ft long rests against a vertical wall. If te bottom of te ladder slides away from te wall at a rate of ft s, ow fast is te top of te ladder sliding down te wall wen te bottom of te ladder is 6 ft from te wall? HW6Sec3.9 #3 similar; Little arder one HW7Sec3.9 #9. 2 +y 2 = 24 2 better tan 2 + y 2 = 24 eg3. A water tank as te sape of an inverted circular cone wit base radius 2 m and eigt 4 m. If water is being pumped into te tank at a rate of 2 m 3 per min, find te rate at wic te water level is rising wen te water is 3 m deep. HW7Sec3.9 #4 little arder tan eg3. eg4. Car A is traveling west at 50 mi and car B is traveling nort at 60 mi. Bot are eaded for te intersection of te two roads. At wat rate are te cars approacing eac oter wen car A is 0.3 mi and car B is 0.4 mi from te intersection? natural /common sense quantities V.S. coordinate/ mecanical quantities Sec3.0 Linear approimations and differentialspicture grap looks like tangent line near (a, f ()) function for te grap y = f () function for te tangent line y = L () = f (a) + f (a) ( a) Te linear function is called linear approimation, or linearization of (nonlinear) function f () at = a. RMK. Easy to get f (a) and f (a), ard/impossible to get nearby values around a. 9
Warm-up: not real for us in modern time, only for calculators/computers eg. Find te linearization of te fcn f () = at a = 4 and use it to approimate 3.98 and 4.05. Are tese approimation overestimates or underestimates? eg. sin (y ) = 3y, wen = 0.0 wat is y? Estimate/approimate y (0.0). Similar ones: HW8Sec3.0 #9 #. Sol Step (0, 0) on te curve, = 0.0 near 0 (picture) y = f (0.0) L (0.0) = f (0) + f (0) (0.0 0) = f (0) 0.0 Step 2. f (0) = d (0,0) cos (y ) ( ) = 3 plug in (0, 0), = 3 d d d Tus y (0.0) 0.0 = 0.005. 2 d d = /2 Sec4. Maimum and minimum values Def Absolute/Global maimum value at c if f (c) f () for all D minimum f (c) f () Global ma or min values are also called etreme values. Def. Local ma value near c, if f (c) f () for all near c min f (c) f () For ma/min eg. Regular one eg y = 2, eg y = For Etreme value t m: eg2 closed interval regular one, eg2 y = > 0; eg2 jump fcn no ma eg2 y = tan Etreme value t m. f () continuous on te closed interval [a, b], ten f attains a global maimum value f ( L ) for some input(s) L on [a, b] and a global minimum value f ( S ) for some input(s) S ; eg3 jump funtion ma/sup inside, not attained. eg. y = sin on [ 2, 4π] How to get loc ma/min eg. y = 2, y =, y = sin graps Fermat T m. If f () as local ma/min value at = c, ten f (c) = 0 or f (c) doesn t eist. Def. Critical pts/numbers c is tat eiter f (c) = 0 or f (c) doesn t eist. eg7. Find te critical numbers for f () = 3/5 (4 ). Split/power rule instead of product rule ( ) f () = 4 3/5 3 5 + = 4 3 5 2 5 8 3 5 = 4 3 2 5 5 5 2 How to get global ma/min of a continuous function on a closed interval [a, b]?. Critical pts c and f (c) 2. Boundary pts a, b and f (a), f (b) 3. Largest global ma Smallest global min eg8. Find te global ma/min for f () = 3 3 2 + on [, 4]. 2 Review for Mierm II Rules + - cain (sin ) = cos,, (arctan ) =, + 2 Tecniques: Implicit differentiation Logritmic differentiation 0
Applicationss: Related rates linear approimation, ma/min, (sape, later) Review problems: Sec3.4 eg2 d sin d 2 and d d sin2, eg4 d, eg7 d d 2 ++ d esin Sec3.5 HW5# Find tangent line equation to curve 2 + y + y 2 = 3 at (, ). #4 d arctan + d Sec3.6 eg7 y = 4/3 2 + (3+2) 5 y =? eg8 y =, y =? Sec3.9 eg4. Car A is traveling west at 50 mi and car B is traveling nort at 60 mi. Bot are eaded for te intersection of te two roads. At wat rate are te cars approacing eac oter wen car A is 0.3 mi and car B is 0.4 mi from te intersection? 2 + y 2 = 00 better tan 2 + y 2 = 0 in derivative calculation. Sec3.0 eg Use linear approimation to approimate 3.98 and 4.05. Sec4.3 How derivatives affect te sape of a grap Increasing/Decreasing property: Up/down grap Increasing/Decreasing test: a) f () > 0 on [a, b], ten f () on [a, b] b) f () < 0 on [a, b], ten f () on [a, b]] Reason: grap tangent line f () f (a) + f (a) ( a) near by >0 eg. Find te decreasing/increasing intervals for f () = 3 4 4 3 2 2 + 5 real line, critical pts, test sign, ten D/I property. Cart way: critical pts/ interval on "orizontal" real line, factors line up vertically st derivative test for loc ma/min: f (c) = 0 or f (c) doesn t eist a) f + to -, f, ten f (c) local ma b) f - to +, f, ten f (c) local min c) f doesn t cange sign, no local ma/min eg. 4, 2, 3 eg2. Find local ma/min in eg. Sape cancave up/down Def. up grap above tangent, down grap below tangent Obs. up: tangent up and up, slope ; down: tangent down and down, slope Concavity test: a) f > 0 on (a, b), ten f is concave up on (a, b) b) f < 0 on (a, b), ten f is concave down on (a, b) Def. i inflection pt, if te sape of grap f canges before and after i. eg. f () = 3, = 0 an inflection pt; sin, = kπ for all integers are inflection points/numbers/inputs 2nd derivative test for loc ma/min a) f (c) = 0, f (c) > 0 f - + f f (c) is local min. eg. y = 2 b) f (c) = 0, f (c) < 0 f + - f f (c) is local min. eg. y = 2 c) f (c) = 0, f (c) = 0 Noting eg. 3 or 4, 4.
eg6.discuss te curve y = 4 4 3 w.r.t.to concavity, points of inflection, and local maima and minima. Use tis information to sketc te curve. eg7. Sketc te grap of function f () = 2/3 (6 ) /3. eg5. Stetc a possible grap of a function f tat satisfies te following conditions: i) f () > 0 on (, ), f () < 0 on (, ) ii) f > 0 on (, 2) and (2, ), f < 0 on ( 2, 2) iii) lim f () = 2 and lim f () = 0 Sec4.4 Indeterminate forms and L Hospital s Rule Recall derivative limits f() f(a) lim a = f (a) eg. lim 2 a, lim ln ln sin, lim 2 sin 4 2 2 = n Sneak in: e = lim 0 ( + ) New questions lim 2, lim 2 9, lim 7 e lim, lim sin e 2 = lim n ( + n π /3 Quotient Law fails lim Bottom=0 Limit may eist lim Top =0 L Hopspital s Rule I 0 : 0 f() lim a = lim g() a f () ) n, provided lim f g () a f () = 0 = lim a g () and lim () a g () eists. Heuristic: Geometric: tangent line approimation Analytic: f () f (a) + f (a) ( a) = f (a) ( a) g () g (a) + g (a) ( a) = g (a) ( a) f() f (a) g() g (a) may not eist, go w/ limit e eg. lim, lim sin π /3 L Hopspital s Rule II f() lim a = lim g() a f (), provided lim g () a f () = ±, lim a g () = ± and f lim () a eists. g () 0, lim 0 a could be lim, lim + eg. lim e eg. lim 2 twice e e eg. lim twice 2 Oter forms of indeterminate limit f g 0 f g f g 0 0, 0, eg. lim 0 ( ln eg. lim π tan ) 2 cos eg. lim 0 +. Sec4.4 HW22#8 lim 0 ( 8) = lim 0 e ln( 8) =... sin Trap. eg. lim π two conditions invalid cos +sin eg. lim one condition invalid 2
Sec4.5 Summary of curve sketcing Real life, Hiking: were range points of interest road status Calculus A. Domain, B Symmetry: even/odd/periodic, C Intercepts, /y, D Asymptotes orizontal/vertical(/slant), E Decreasing/Increasing intervals, F Local ma/min, G Concavity, points of inflection, H wrap up. Sec4.3 eg5. Sketc a possible grap of a function f tat satisfies te following conditions: i) f () > 0 on (, ), f () < 0 on (, ) ii) f > 0 on (, 2) and (2, ), f < 0 on ( 2, 2) iii) lim f () = 2 and lim f () = 0 Sec4.3 eg7. Sketc te grap of function f () = 2/3 (6 ) /3. eg. Sketc te curve y = 22 2 Rmk. Picutre file large, Tet file small Sec4.7 Optimization problems Step. Read te problem Step2. Draw diagrams Step3. Notations calculus... eg2. A cylindrical can is to be made to old L of oil. Find te dimensions tat will minimize te cost of te metal to manufacture te can. eg5. Find te area of te largest rectangle tat can be inscribed in a semicircle of radius r. Sec4.7 HW24#2 A woman at a point A on te sore of a circular lake wit radius 2 mi wants to arrive at te point C diametrically opposite A on te oter side of te lake in te sortest possible time (see te figure). Se can walk at te rate of 4 mi/ and row a boat at 2 mi/. For wat value of te angle θ sown in te figure will se minimize er travel time? Sec4.7 HW24#4 A steel pipe is being carried down a allway 9 ft wide. At te end of te all tere is a rigt-angled turn into a narrower allway 6 ft wide. Wat is te lengt of te longest pipe tat can be carried orizontally around te corner? (Round your answer one decimal place.) Final review. Limit of functions Laws: + - composition Appls: continuity, asymptotes (Quotient form) tangent/velocity/oter rates of cange (Sum form integrals M25) Derivatives of functions Rules: + - composition/cain rule Tecniques: Implicit differentiation Logaritmic differentiation 3
parametric differentiation (derivatives of all elementary functions) Applications tangent line, velocity related rates linear approimation ma/min, optimization decreasing/increasing, sape L Hospital s rule for inderterminate forms 4