CHAPTER 10 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS. dy dx

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1 CHAPTER 0 PARAMETRIC, VECTOR, AND POLAR FUNCTIONS 0.. PARAMETRIC FUNCTIONS A) Recll tht for prmetric equtions,. B) If the equtions x f(t), nd y g(t) define y s twice-differentile function of x, then t ny point where 0, d y d C) A prmetric curve is differentile t point if the point. A prmetric curve is smooth if nd nd d d oth re differentile t re ech continuous nd not simultneously zero. D) Length (Arc Length) of Smooth Prmetrized Curve: If smooth curve is trversed exctly once s t increses from to, the curve s length is L + F) Assignment: P.58 { - 0 even, 4} 0.. VECTORS IN THE PLANE - All of which ws covered in Pre-Clculus A) Definition: A vector in the plne is represented y directed line segment. B) Definition: Two vectors re equl if they hve the sme length nd direction. C) Definition: If v is vector in the plne equl to the vector with initil point (0,0) nd terminl point (v, v ) then the component form of v is: v v, v D) The direction of v v v v, is θ v Tn (Possily djusting for qudrnts). E) The length (or mgnitude or norm) of v v, v, denoted v v + v. F) Definition: A unit vector hs length of one. Any non-zero vector cn e scled to unit vector y dividing it y its length.

2 G) Definition: The zero vector is 0 00,. It hs length of zero nd no direction. H) You should e le to operte on vectors (dd, sutrct, sclr multiples), for vectors in grphicl nd/or component form. I) Definition: The Dot Product of vectors u u, u, nd v v, v is the numer u v u v + u v J) Theorem: The ngle etween two vectors u nd v is: θ cos u v u v K) Definition: The slope of vector u u, u is u u L) Two vectors re orthogonl/norml if their dot product is zero. For ny non-zero vector there re two unit vectors nd two unit norml vectors. M) The irplne exmple for vector ddition. N) Assignment: P. 57 { - 6 even, 44, 45} 0.3. VECTOR-VALUED FUNCTIONS A) The stndrd unit vectors re i 0, nd j 0,. Any vector, v, i + j. B) A vector-vlued function tkes prmetric equtions nd writes ech of x f(t) nd y g(t) s components of the vector r(t) f(t)i + g(t)j. The grph is the pth of the vector. (To ctully grph vector-vlued, we just use prmetric equtions). C) Most of the Clculus of vector-vlued functions is tken cre of componentilly. If r(t) f(t)i + g(t)j, then I) Lim Lim Lim r(t) f(t)i + t c t c t c g(t)j. II) d r df i + dg j III) r (t) f ( t) i + g ( t) j + C IV) r (t) f ( t) i + g ( t) j D) Definitions for prticle movement for the function r(t) f(t)i + g(t)j I) v(t) d r is the prticle s velocity vector nd is tngent to the curve. II) v(t), the mgnitude of v, is the prticle s speed. III) (t) d v d r is the prticles ccelertion vector. IV) v, unit vector, is the direction of motion v E) Differentiility nd smoothiocity of vector-vlued functions mirror prmetric function.

3 F) Note tht velocity speed direction v v v G) Assignment: P.537 {6-6 even, 7, 33}

4 0.4. MODELING PROJECTILE MOTION A) Let g e the force of grvity. g 9.8 m/s 3 ft/s. Since grvity is only verticl component, (t) -gj B) If v 0 is the initil velocity nd θ is the lunch ngle, the initil velocity cn e roken down into its component prts s v(0) v 0 cos θ i + v 0 sin θ j. C) We cn solve for v(t) v 0 cos θ i + (- gt + v 0 sin θ) j D) If the initil position is r(0) hi + kj, then r(t) (v 0 cos θ t + h)i + ( gt + v 0 sin θ t + k) j E) The sell prolem nd other prolems. F) If the initil position is the origin then here re three hn formuls: ) Mximum height ( v sin ) 0 θ ) Flight time v0 sinθ g g G) Assignment: P. 549 { - 7,,, 6} 3) Rnge v 0 g sin( θ ) 0.5. POLAR COORDINATES AND POLAR GRAPHS - A Pre-Clculus Review A) We use (x,y) in the Crtesin Coordinte system nd (r, θ) in the Polr Coordinte system. B) Degenerte cses of on vrile in Crtesin nd in Polr. C) Converting etween coordinte systems: I) From Polr to Crtesin: x r cos θ, nd y r sin θ II) From Crtesin to Polr: r x + y, nd tn θ y x D) When grphing, mke sure your domin is lrge enough. E) Assignment: P. 558 { - 6 even, 8-56 multiples of 4} 0.6. CALCULUS OF POLAR CURVES A) We will e viewing lines in terms of x nd y. First we will write the rdius s function of θ. Since x r cos θ, nd y r sin θ, we will use x f(θ) cos θ, nd y f(θ) sin θ. As with prmetric equtions, d θ d θ f ( θ) sin( θ) + f ( θ) cos( θ) f ( θ)cos( θ) f ( θ)sin( θ) B) Using the slope eqution to find horizontl nd verticl symptotes. ) Potentil horizontl symptotes exist when the top 0. ) Potentil verticl symptotes when the ottom 0. 3) Actul horizontl symptotes exist when the top 0 nd the ottom 0. 4) Actul verticl symptotes exist when the ottom 0 nd the top 0. 5) If the top nd ottom re simultneously zero, we must use L Hopitl to find the vlue of the slope. It my e horizontl or verticl symptote nd it might e

5 something else. 6) We often wnt to write these tngent lines in Crtesin form nd must use the prmetric portion of x f(θ) cos θ, or y f(θ) sin θ to find the line. C) Tngent lines tht go through the origin (or pole) occur when the rdius, f(θ) 0. Once we find these vlues of θ, the tngent lines re of the form y tn(θ) x, unless tn(θ) is undefined. In this cse the line tngent to the pole is x 0. D) The good news is tht the clcultor cn find dr (nd ) for given grph. E) Assignment: p. 566 { - do 3, skip } F) Our first ppliction of integrtion requires the re of sector. A θ r. G) The re of the region etween the origin nd the curve r f(θ), α θ, is A α r H) To find the re etween two curves we must mke sure tht the rdii re lwys positive nd tht the rdii do not cross. The re of the region 0 < r (θ) < r (θ), α θ, is A r - r θ (r r ) d α α α I) The Length of Polr Curve: if r f(θ) hs continuous first derivtive for α θ nd if the point P(r, θ) trces the curve r f(θ) exctly once s θ runs from α to, then the length of the curve is: L α r dr + J) Assignment: P. 566 {7, 8-7 multiples of 3, 33, 36, 40} Mndtory Assignment: P. 569 {3-4 multiples of 3, 43-47, 5, 55, 58, 63-65, 67}