Vector Valued Functions

Transcription

1 SUGGESTED REFERENCE MATERIAL: Vector Valued Functions As you work throuh the problems listed below, you should reference Chapters. &. of the recommended textbook (or the equivalent chapter in your alternative textbook/online resource) and your lecture notes. EXPECTED SKILLS: Be able to find the domain of vector-valued functions. Be able to describe, sketch, and reconize raphs of vector-valued functions (parameterized curves). Know how to differentiate vector-valued functions. And, consequently, be able to find the tanent line to a curve (as a vector equation or as a set of parametric equations). Be able to determine anles between tanent lines. Know how to use differentiation formulas involvin cross-products and dot products. Be able to evaluate indefinite and definite interals of vector-valued functions as well as solve vector initial-value problems. PRACTICE PROBLEMS:. For each of the followin, determine the domain of the iven function. (a) r(t) = t i + t j t k (, 0) (0, ] (b) r(t) = ln (t + ), e t, t (, ln ) (ln, ) (c) r(t) = cos (t)i + sin (t)j + 5k (, )

2 . Consider the curve C : r(t) = 5 + t, 4 + t, shown below. (a) Sketch the followin position vectors: r( ), r(0), r(), r(), and r(3). (b) Indicate the orientation of the curve (i.e., the direction or increasin t). 3. Sketch the followin vector valued functions. Also, describe the curve in words. (a) r (t) = 4 cos t, 4 sin t, 5, 0 t 4π The curve is a circle in the z = 5 plane which has a radius of 4 and a center at (0, 0, 5), traversed twice counterclockwise.

3 (b) r (t) = 4 cos t, 4 sin t, t, 0 t 4π. The curve is a helix on the cylinder x + y = 6 which climbs from the point (4, 0, 0) to the point (4, 0, 4π) 4. Consider r(t) = t, t (a) Sketch r(t) and indicate the direction of increasin t. (b) On your sketch, draw r() and r (). 5. Consider r(t) = 3 cos t, sin t (a) Sketch r(t) and indicate the direction of increasin t. (b) On your sketch, draw r(π) and r (π). 3

4 6. For each of the followin, find an equation of the line which is tanent to the iven curve at the indicated point. (a) r(t) = ln t, t, t at (x, y, z) = (0,, ) l (t) = 0,, + t,, (b) r(t) = sin t, cos t, tan t when t = π l (t) = 0,, 0 + t, 0, 7. Find all points on the curve r(t) = ti + t j + t 3 k where its tanent line is parallel to the vector i + 8j + 4k. The tanent line will be parallel to the iven vector when t = which corresponds to the point (x, y, z) = (, 4, 8); Detailed Solution: Here 8. The followin vector valued functions describe the paths of two bus flyin in space. r (t) = t, t + 3, t r (t) = 5t 6, t, 9 At some moment in time, the two bus collide. (a) Determine the moment in time when the bus collide as well as the location in space where the bus collide. The bus intersect when t = 3. This corresponds to the point (x, y, z) = (9, 9, 9). Detailed Solution: Here (b) What is the anle between their paths at the point of collision? cos ( ); Detailed Solution: Here 9. Prove the followin theorem: Theorem: If r (t) is a differentiable vector valued function in -space or 3-space, and if r (t) is constant for all t, then r (t) r (t) = 0. That is, r (t) and r (t) are orthoonal vectors for all t. (Hint: r (t) = r (t) r (t)) 4

5 Suppose r (t) = k, where k is constant. Then: And, the result is proven. r (t) = k r (t) r (t) = k d dt [ r (t) r (t)] = d ( ) k dt r (t) r (t) + r (t) r (t) = 0 [ r (t) r (t)] = 0 r (t) r (t) = 0 0. Explain why the followin calculation is incorrect: d dt [r (t) r (t)] = r (t) r (t) + r (t) r (t) The order of the terms matters when dealin with cross products. The correct derivative statement is: d dt [r (t) r (t)] = r (t) r (t) + r (t) r (t). Evaluate the followin interals. (a) [(t + ) 5 i t ] j dt ( ) (t + )6 + c i (ln t + c )j; i.e., (b) sin t, cos t, tan t dt (t + )6, ln t + c (c) ( cos t + c )i + (sin t + c )j + (ln sec t + c 3 )k; or, equivalently, cos t, sin t, ln sec t + c ln 3 0, 4 [ e t i + e t j ] dt. Evaluate π 0 r (t) dt if r(t) = 3 cos t, 3 sin t. Interpret your answer eometrically. 6π. The iven curve represents a circle centered at the oriin with a radius of 3; this interal ives the arc lenth (circumference) of the circle. 5

6 dr 3. Solve the followin vector initial value problems: dt = e t i + 3t j r(0) = i 8j r(t) = e t + 3, t A particle moves throuh 3-space in such a way that its velocity is v(t) = ti+t j+t 3 k. If the particle s initial position at time t = 0 is (,, 3), what is the particle s position when t =? (Hint: set up an initial value problem.) ( 3 The position of the particle at time t = is (x, y, z) =, 7 3, 3 ). Detailed Solution: 4 Here 5. Suppose that C : r(t) is a smooth vector valued function in -space or 3-space defined for a t b. We define the arc lenth function by s(t) = t t 0 r (u) du This function ives the arc lenth for the part of C between r(t 0 ) and r(t). (a) Compute the arc lenth function for the helix r(t) = cos ti+sin tj+tk which ives the lenth of the curve from t 0 = 0 to an arbitrary t. s = t (b) Use your answer from part (a) to reparameterize the helix with respect to arc lenth. (In other words, express the curve C as r(s).) ( ) ( ) s s r(s) = cos i + sin j + s k (c) Compute r (s) and r (s) r (s) = sin ( s ) i + cos ( s ) j + k and r (s) = In fact, whenever a curve is parameterized in terms of arc lenth, it can be shown usin the chain rule that all tanent vectors will be unit tanent vectors. 6

7 6. ( From the round, a projectile is shot upward at an anle of α with the horizontal, 0 < α < π ), at an initial speed of v 0 meters/second, as demonstrated in the diaram below. You should make the followin assumptions: The mass of the object, m, is constant. The only force actin on the object after it is launched is the force of ravity,. Inore air resistance and assume that the force of ravity is constant. (a) Set up an initial value problem which can be used to find r(t), a vector valued function that ives the position of the particle at time t. a(t) = d r = 0, dt v(0) = r (0) = v 0 cos α, v 0 sin α r(0) = 0, 0 (b) Solve your initial value problem from part (a) to determine r(t). r(t) = v 0 (cos α)t, t + v 0 (sin α)t (c) Verify that the trajectory of the projectile is a parabola. 7

8 We can express the trajectory of the projectile (from part b) parametrically: x = v 0 (cos α)t y = t + v 0 (sin α)t x Notice that if we solve the first equation for t, we et t =. (It was OK to v 0 cos α do this division since we had some non-zero instantaneous speed v 0 and cos α 0 for 0 < α < π ). Then, pluin this into the second equation, we et: y = ( ) ( ) x x + v 0 (sin α) v 0 cos α v 0 cos α = (v 0 cos α) x + (tan α)x = Ax + Bx where A is the constant trajectory is parabolic. (d) What is the fliht time of the projectile? and B is the constant tan α. Thus, the (v 0 cos α) We can find the value of t for which the projectile returns to the round by settin y = 0 in the parametric representation of the trajectory. y = 0 t + v 0 (sin α)t = 0 ( ) t t v 0 sin α = 0 which happens when t = 0 and when t = v 0 sin α (e) What is the rane of the projectile? To find the rane, we need to determine the x coordinate at the time when the projectile returns to the round. Specifically, the rane is: ( ) ( ) v0 sin α v0 sin α x = v 0 (cos α) = v 0 sin (α) 8

9 (f) What anle α maximizes the rane? In part (e), we have already computed the rane to be v 0 sin (α). This is maximized when sin(α) = ; i.e., when α = π 4 7. Suppose that C : r(t) is a curve in -space or 3-space and that r (t) = 0. We define the followin vectors: The Unit Tanent Vector to C at t is T(t) = r (t) r (t). The Principal Unit Normal Vector to C at t is N(t) = T (t) T (t). The Unit Binormal Vector to C at t is B(t) = T(t) N(t). The coordinate system determined at the point t by T(t), N(t), and B(t) is called the Frenet Frame or the TNB Frame. (a) Explain why T(t), N(t), and B(t) are all mutually othoonal. N(t) T(t) by problem 9. B(t) N(t) and B(t) T(t) because for any vectors in three space v (v w) = 0 and w (v w) = 0 (b) Consider the helix described by r(t) = cos t, t, sin t. Compute the unit tanent, principal unit normal, and binormal vectors T(t), N(t), and B(t). NOTE: Here is a sketch of the helix from problems 7b with the TNB-Frame (Frenet Frame) represented at four different points. 9

10 T(t) = B(t) = sin t,, 5 5 sin t,, 5 5 cos t 5 5 cos t ; N(t) = cos t, 0, sin t ; (c) Definition: The plane determined by the unit tanent and normal vectors T and N at a point P on a curve C is called the osculatin plane of C at P. From the latin Osculum, meanin to kiss, this is the plane that comes closest to containin the part of the curve near P. Compute an equation of the osculatin plane of the helix from part (b) at the point which corresponds to t = π. y + z = π (d) Definition: The plane determined by the unit normal and binormal vectors N and B at a point P on a curve C is called the normal plane of C at P. It consists of all lines that are orthoonal to the tanent vector T. Compute an equation of the normal plane of the helix from part (b) at the point which corresponds to t = π. y z = π 0