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Multivariable Calculus
- Derivatives of vector-valued functions Differentiate the following function. r(t) = ⟨(t + 1)-1, tan-1 t, ln (t + 1)⟩arrow_forwardTangent lines Suppose the vector-valued function r(t) = ⟨ƒ(t), g(t), h(t)⟩ is smooth on an interval containing the point t0. The line tangent to r(t) at t = t0 is the line parallel to the tangent vector r'(t0) that passes through (ƒ(t0), g(t0), h(t0)). For each of the following functions, find an equation of the line tangent to the curve at t = t0. Choose an orientation for the line that is the same as the direction of r'.arrow_forwardDetermine 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_forward
- The position vector r describes the path of an object moving in the xy-plane. Position Vector Point r(t) = 4 cos ti + 4 sin t (2V2,2V2) (a) Find the velocity vector v(t), speed s(t), and acceleration vector a(t) of the objeot. v(t) s(t) = a(t) = (b) Evaluate the velocity vector and acceleration vector of the object at the given point.arrow_forwardTangent lines Suppose the vector-valued function r(t) = ⟨ƒ(t), g(t), h(t)⟩ is smooth on an interval containing the point t0. The line tangent to r(t) at t = t0 is the line parallel to the tangent vector r'(t0) that passes through (ƒ(t0), g(t0), h(t0)). For each of the following functions, find an equation of the line tangent to the curve at t = t0. Choose an orientation for the line that is the same as the direction of r'. r(t) = ⟨3t - 1, 7t + 2, t2⟩; t0 = 1arrow_forwardVector-valued functions (VVF) VVF Point P t value (for point 1 r(t) = ½ t²i + √4 − t j + √t + 1k (0, 2, 1) Graph the tangent line together with the curve and the points. Copy your code and the resulting picture. Use MATLAB (Octave) code for graphing vector valued functions. A) 2arrow_forward
- Find the directional derivative of the function at the point Pin the direction the unit vector u = cos ôi + sin 0j. Sketch each the graph of the function, t point P, and the unit vector u. 2. f(x,y) = sin(2x + y), P(0, n), 0 = -.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_forwardMotion around a circle of radius a is described by the 2D vector-valued function r(t) = ⟨a cos(t), a sin(t)⟩. Find the derivative r′ (t) and the unit tangent vector T(t), and verify that the tangent vector to r(t) is always perpendicular to r(t).arrow_forward
- The position vector r describes the path of an object moving in the xy-plane. Position Vector Point r(t) = 2 cos ti + 2 sin tj (VZ, V2) (a) Find the velocity vector, speed, and acceleration vector of the object. v(t) = s(t) a(t) = (b) Evaluate the velocity vector and acceleration vector of the object at the given point. a(#) =arrow_forwardInterpreting directional derivatives Consider the functionƒ(x, y) = 3x2 - 2y2.a. Compute ∇ƒ(x, y) and ∇ƒ(2, 3).b. Let u = ⟨cos θ, sin θ⟩ be a unit vector. At (2, 3), for what values of θ (measured relative to the positive x-axis), with 0 ≤ θ < 2π, does the directional derivative have its maximum and minimum values? What are those values?arrow_forwardActivity (calculus of vector function) 1. An object moves with velocity vector (t,t²,-t) starting at (1,2,3) when t=0. Find the function r giving its location. 2. Show, using the rules of cross products and differentiation, that d/dt (r(t) x r'(t)) = r(t) × r"(t). 3. Find a vector function for the line tangent to ( cost , sint , cos(6t) ) when t = t/7. 4. Find the cosine of the angle between the curves (cost, -sin(t)/5, sint) and (cos(tt/3), sin(t/3), t) where they intersect.arrow_forward
- Trigonometry (MindTap Course List)TrigonometryISBN:9781337278461Author:Ron LarsonPublisher:Cengage LearningAlgebra & Trigonometry with Analytic GeometryAlgebraISBN:9781133382119Author:SwokowskiPublisher:Cengage