(9) (a) If f(t) = x1e −kt cos(ωt) + x2e −kt sin(ωt), then f'(t) has the...

(9) (a) If f(t) = x₁e ^−kt cos(ωt) + x₂e ^−kt sin(ωt), then f'(t) has the same form with coefficients y₁, y₂.

(i) Find the matrix A such that y = Ax.

(ii) Find the characteristic polynomial of A.

(iii) What can you say about eigenvalues of A? (iv) Interpret your answer to

(iii) as a calculus statement. That is, explain how your answer to (iii) could have been predicted from a basic fact of calculus.

(b) If f(t) = x₁ sin(t) + x₂ cos(t) + x₃tsin(t) + x₄t cos(t), then f'(t) has the same form with coefficients y1, ..., y4. Same questions (a)-(d) as previously.

Expert Solution

b)

Colby Messinger answered 3 months ago

If u(t) = < sin(8t), cos(4t), t > and v(t) = < t, cos(4t), sin(8t) >,...

If u(t) = < sin(8t), cos(4t), t > and v(t) = < t, cos(4t), sin(8t) >, use the formula below to find the given derivative. d/(dt)[u(t)* v(t)] = u'(t)* v(t) + u(t)* v'(t) d/(dt)[u(t) x v(t)] = <.______ , _________ , _______>

If u(t) = < sin(5t), cos(5t), t > and v(t) = < t, cos(5t), sin(5t) >,...

If u(t) = < sin(5t), cos(5t), t > and v(t) = < t, cos(5t), sin(5t) >, use the formula below to find the given derivative. d/dt[ u(t) * v(t)] = u'(t) * v(t) + u(t)* v'(t) d/dt [ u(t) x v(t)] = ?

The cycloid has parametric equations x = a(t + sin t), y = a(1 - cos...

The cycloid has parametric equations x = a(t + sin t), y = a(1 - cos t). Find the length of the arc from t = 0 to t = pi. [ Hint: 1 + cosA = 2 cos2 A/2 ]. and the arc length of a parametric

The equation x(t) = A sin(ωt + φ) describes the simple harmonic motion of a block...

The equation x(t) = A sin(ωt + φ) describes the simple harmonic motion of a block attached to a spring with spring constant k = 20 N/m. At t = 0 s, x = 0.5 m and the block is at rest. a (5 points) After 0.5 s, the potential energy stored in the spring first reaches zero and the velocity of the block is in the negative x direction. What is the period of the oscillation? b (5 points)...

Convert x=cos(3t)+sin(3t) & y=cos(t)-sin(t) into an equation of x-y form (cartesian equation). Thank you

Consider the helix r(t)=(cos(2t),sin(2t),−3t)r(t)=(cos(2t),sin(2t),−3t). Compute, at t=π/6 A. The unit tangent vector T=T= ( , ,...

Consider the helix r(t)=(cos(2t),sin(2t),−3t)r(t)=(cos(2t),sin(2t),−3t). Compute, at t=π/6 A. The unit tangent vector T=T= ( , , ) B. The unit normal vector N=N= ( , , ) C. The unit binormal vector B=B= ( , , ) D. The curvature κ=κ=

The plane curve represented by x(t) = t − sin(t), y(t) = 7 − cos(t), is...

The plane curve represented by x(t) = t − sin(t), y(t) = 7 − cos(t), is a cycloid. (a) Find the slope of the tangent line to the cycloid for 0 < t < 2π. dy dx (b) Find an equation of the tangent line to the cycloid at t = π 3 (c) Find the length of the cycloid from t = 0 to t = π 2

Question

(9) (a) If f(t) = x1e −kt cos(ωt) + x2e −kt sin(ωt), then f'(t) has the...

Solutions

Expert Solution

Related Solutions