Higher-Order DifferentiationIn Exercises 23–26, find (a)
Want to see the full answer?
Check out a sample textbook solutionChapter 12 Solutions
Multivariable Calculus (looseleaf)
- Find the unit tangent vector to the curve at the specified value of the parameter r(t) = 2sin(t)i + 4cos(t)j, t = pi / 6arrow_forwardEvaluate r(2) and r(-1) for r(t) = (sin 1,12, (1² + 1)~1). %3Darrow_forward4t Let (t) = ( - t +2, -3e-4, -5 sin( - 2t)) t¹ Find the unit tangent vector T (t) at the point t = 0 T(0) = <arrow_forward
- (a) Find the derivative of r(t) = (4 + t)i + te tj + sin(5t) k. (b) Find the unit tangent vector at the point t = 0. Solution (a) According to the theorem that states if r(t) = (f(t), g(t), h(t)) = f(t)i + g(t)j + h(t) k, where f, g, and h are differentiable functions, then r'(t) = (f'(t), g'(t), h'(t)) = f'(t)i + g'(t)j +h'(t) we differentiate each component of r. r' (t) = (b) Since r(0) T(0) = r'(0) Ir'(0)| and r'(0) = j + 5k, the unit tangent vector at the point (4, 0, 0) is j + 5karrow_forwardThe motion of a point on the circumference of a rolling wheel of radius 4 feet is described by the vector function r(t) = 4(12t - sin(12t))i + 4(1 − cos(12t))j Find the velocity vector of the point. v(t) = Find the acceleration vector of the point. a(t) = Find the speed of the point. s(t) = =arrow_forward(4) Consider the following vector-valued function: y: (π,0) x (-1, 1) → R³, y(u, v) = {cosh(v) cos(u), v, cosh(v) sin(u)}. (a) Sketch the values of y obtained by holding v = v₁ constant and varying u, where (i) Vo = ±1, (ii) vo = ±½, and (iii) vo = ±³ (you should draw six curves). (b) Sketch the values of y obtained by holding u = u。 constant and varying v, where (i) Uo = (ii) u -7, and (iii) uo 29 π 4' == (c) Sketch the full image of y. = 3πt 4arrow_forward
- Calculus: Early TranscendentalsCalculusISBN:9781285741550Author:James StewartPublisher:Cengage LearningThomas' Calculus (14th Edition)CalculusISBN:9780134438986Author:Joel R. Hass, Christopher E. Heil, Maurice D. WeirPublisher:PEARSONCalculus: Early Transcendentals (3rd Edition)CalculusISBN:9780134763644Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric SchulzPublisher:PEARSON
- Calculus: Early TranscendentalsCalculusISBN:9781319050740Author:Jon Rogawski, Colin Adams, Robert FranzosaPublisher:W. H. FreemanCalculus: Early Transcendental FunctionsCalculusISBN:9781337552516Author:Ron Larson, Bruce H. EdwardsPublisher:Cengage Learning