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- The linearization of ex at x = 0 Derive the linear approximation ex = 1 + x at x = 0.7. Find the linear approximation of the function f (x, y) = у - 1 at (0,0)Verify the given linear approximation at a = 0. Then determine the values of x for which the linear approximation is accurate to within 0.1. (Enter your answer using interval notation. Round your answers to three decimal places.) ln(1 + x) ≈ x
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- Evalauete the integration of function f(x)=(sin(2x))/(a^2+b^2sin^2x)A spherical dust particle, with a radius a=0.11mm and density of 1030kg/m3, is sedimenting in the quiescent air that has a viscosity of μ=1.81×10−5 kg/m⋅s. The particle experiences the gravitational force, with g=9.81m/s2, as well as a friction force from the viscous air that can be described by F=−6πμaU, where U is the instantaneous velocity of the particle. The motion of the particle is governed by the Newton's second law. At the time t=0, the dust particle has a downward speed of 0.17m/s. Use the Euler's method, and a time step of h=0.1s, calculate: When t=0.1s, the downward speed of the dust particle is m/sUsing linear approximation, estimate e^0.1 cos(0.1)
- Integration by substitution -Find the antiderivative of the given derivative ds/dt= 8(3t^2-5)^3The Hubble Space Telescope was deployed on April 24, 1990, by the space shuttle Discovery. A model for the velocity of the shuttle during this mission, from liftoff at t = 0 until the solid rocket boosters were jettisoned at t = 125 seconds, is given byv(t) = 0.00146t3 – 0.11553t2 + 24.98169t – 21.26872 (in feet per second). Using this model, estimate the absolute maximum and minimum values of acceleration of the shuttle between liftoff and the jettisoning of the boosters.Compute the second derivative of f(x)=ex+x at x = 1 using a step size h = 0.1 The forward difference approximation of Order-h is ? The central difference approximation of O(h2) is ?