Finding the Domain In Exercises 3-10, find the domain of the
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Calculus: Early Transcendental Functions
- Differentiation of Vector-Valued Functions In Exercises 7 and 8, find r (t), r(t,), and r (t,) for the given value of t. Then sketch the space curve represented by the vector-valued function, and sketch the vectors r(t) and r'(t). 7. r(t) = 2 cos ti + 2 sin tj + tk, toarrow_forwardUse the model for projectile motion, assuming there is no air resistance. Find the vector-valued function for the path of a projectile launched at a height of 10 feet above the ground with an initial velocity of 72 feet per second and at an angle of 30° above the horizontal. r(t) - Use a graphing utility to graph the path of the projectile. r(t) r(t) 20 50 100 150 200 250 300 50 100 150 200 250 300 r(t) r(t) 50 100 150 200 250 300 50 100 150 200 250 300arrow_forwardFind the domain of the vector-valued function. (Enter your answer using interval notation.) r(t) = F) x G(t), where F(t) = ti - 5 + tk, G(t) = Vti + 1 j+(t+2)k t+ 5 (-00, - 5)u(-5, o) Your answer cannot be understood or graded. More Information Need Help? Read Itarrow_forward
- Derivatives of vector-valued functions Differentiate the following function. r(t) = ⟨cos t, t2, sin t⟩arrow_forwardSketch the plane curve represented by the vector-valued function and give the orientation of the curve. r(0) = cos(0)i + 6 sin(0)j O O -2 -2 y 5 -5 y -5 2 2 6 X X -6 -4 -2 -6 -4 -2 y 2 2 4 4 6 X Xarrow_forwardDerivatives of vector-valued functions Differentiate the following function. r(t) = ⟨(t + 1)-1, tan-1 t, ln (t + 1)⟩arrow_forward
- Determine the interval(s) on which the vector-valued function is continuous. (Enter your answer using interval notation.) 1 -i + 6t + 1 1 r(t) =arrow_forward(5) Let ß be the vector-valued function 3u ß: (-2,2) × (0, 2π) → R³, B(U₁₂ v) = { 3u² 4 B (0,7), 0₁B (0,7), 0₂B (0,7) u cos(v) VI+ u², sin(v), (a) Sketch the image of ß (i.e. plot all values ß(u, v), for (u, v) in the domain of ß). (b) On the sketch in part (a), indicate (i) the path obtained by holding v = π/2 and varying u, and (ii) the path obtained by holding u = O and varying v. (c) Compute the following quantities: (d) Draw the following tangent vectors on your sketch in part (a): X₁ = 0₁B (0₂7) B(0)¹ X₂ = 0₂ß (0,7) p(0.4)* ' cos(v) √1+u² +arrow_forwardLet F(t) = (31³-3, 4et, -sin(4t)) Find the unit tangent vector T(t) at the point t = 0 T(0) = < 0 Question Help: Add Work " Videoarrow_forward
- Determine the domain of the vector function r(t) = cos(4t) i + 7In(t - 5) j - 10 k Evaluate if the vector function is possible at the value of t=8, round to two tenths Find the derivative of the vector function r(t)arrow_forwardVector Calculus For scalar functions u and v, show that B = (Vu) × (Vv) is solenoidal and that A = - vVu) is a vector potential for B, i.e. B = V×Aarrow_forwardRepresent the plane curve by a vector-valued function. x² = y2 = r(t) = 2 cos t + sin t j Need Help? Read It Watch Itarrow_forward
- Algebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:CengageAlgebra and Trigonometry (MindTap Course List)AlgebraISBN:9781305071742Author:James Stewart, Lothar Redlin, Saleem WatsonPublisher:Cengage Learning