F128 Exercicios Resolvidos Cap 8

m l T P R P⊥ P₁ k x mg α N F_at N P_⊥ P_∥ P h D d x 1. Determine and classify the critical points of the following functions. (i) f (x) = x4 − 4x3 −2 −1 0 1 2 (ii) g(x) = 4x3 + 3x2 − 6x − 5 (ii) h(x) = x4 − 4x3 −2 −1 0 1 Sec 4.7. 1. Find the equation of the plane through the points (1, 1, 0), (−1, −1, 0), and (0, 1, 1). 2. Determine whether the following vectors are linearly independent. (a) (1, 0, 1), (0, 1, 1), (1, 1, 1) (b) (1, 2, 3), (4, 5, 6), (6, 8, 10) 3. For what values of t are the following matrices invertible? (a) A = [1, 0, t; 0, 1, 0; t, 0, 1] (b) B = [1, t, 0; 0, 1, 0; 0, 0, 1] 4. Determine whether the following functions are linearly independent. (a) f (x) = x2 , g(x) = x, h(x) = 1 (b) f (x) = x3 , g(x) = x2 , h(x) = x 5. Determine whether the following sets form a basis for R^3. (a) (1, 0, 0), (0, 1, 0), (0, 0, 1) (b) (2, 0, 0), (0, 2, 0), (0, 0, 2) 6. For what values of λ do the following systems have a unique solution? (a) x + y + z = λ x + 2y − z = λ 2x + 3y + 2z = λ x Figure 6. Time representation of the rotation matrix A defined by the quaternion q (blue) and the angular velocity matrix B (yellow), both in units of the body frame. β α β Figure 7. Representation over the time of the control torque τc (blue) and its derivative with respect to time τ̇c (yellow), both in units of the body frame. γ α δ Figure 8. Evolution of the control torque τc in the space of the quaternion alone (blue) and considering both quaternion and its time derivative (yellow). β