Chat with us, powered by LiveChat 5. Problems 2.5 The height of a ball that is thrown straight up with a certain force is a function of the time (t) from which it released given by At)–0.5 g40r (where g is a positive constant determined by gravity) The time at which the ball is at its maximum height depends on the parameter g, and is given by * Substituting t* back into the function f(t) yields f(t*) – dfíi*) ,the derivative (with respect to g) of which is equal to This shows how the maximum height of the ball changes as the parameter g changes Holding t constant at t*, the time at which the ball reaches its maximum height, yields the function f t*) =-0 5gtg2 + 40t*. Taking the derivative of this function with respect to g yields noyields . After plugging in the formula for t into result for aft*) than what you previously calculated. If g 32, as it does on Earth, the height at which the ball reaches its maximum height is t* ball is and the maximum height reached by the If g increases by a small amount, up to g 32.1, then the maximum height reached by the ball changes to approximately This result been predicted by the envelope theorem | Writedemy

5. Problems 2.5 The height of a ball that is thrown straight up with a certain force is a function of the time (t) from which it released given by At)–0.5 g40r (where g is a positive constant determined by gravity) The time at which the ball is at its maximum height depends on the parameter g, and is given by * Substituting t* back into the function f(t) yields f(t*) – dfíi*) ,the derivative (with respect to g) of which is equal to This shows how the maximum height of the ball changes as the parameter g changes Holding t constant at t*, the time at which the ball reaches its maximum height, yields the function f t*) =-0 5gtg2 + 40t*. Taking the derivative of this function with respect to g yields noyields . After plugging in the formula for t into result for aft*) than what you previously calculated. If g 32, as it does on Earth, the height at which the ball reaches its maximum height is t* ball is and the maximum height reached by the If g increases by a small amount, up to g 32.1, then the maximum height reached by the ball changes to approximately This result been predicted by the envelope theorem

02 May 5. Problems 2.5 The height of a ball that is thrown straight up with a certain force is a function of the time (t) from which it released given by At)–0.5 g40r (where g is a positive constant determined by gravity) The time at which the ball is at its maximum height depends on the parameter g, and is given by * Substituting t* back into the function f(t) yields f(t*) – dfíi*) ,the derivative (with respect to g) of which is equal to This shows how the maximum height of the ball changes as the parameter g changes Holding t constant at t*, the time at which the ball reaches its maximum height, yields the function f t*) =-0 5gtg2 + 40t*. Taking the derivative of this function with respect to g yields noyields . After plugging in the formula for t into result for aft*) than what you previously calculated. If g 32, as it does on Earth, the height at which the ball reaches its maximum height is t* ball is and the maximum height reached by the If g increases by a small amount, up to g 32.1, then the maximum height reached by the ball changes to approximately This result been predicted by the envelope theorem

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4. Problems 2.4 Suppose you would like to maximizefxy)-xy, subject to the constraint that x and y are constrained to sum to 1. That is, xy 1 Instead of working through this maximization problem, however, you could also work through the dual problem to this, which is: Minimize x+ y subject to xy = 0.25 Hint: Assume x and y take only positive values The Lagrangian for this constrained minimization in the dual problem is Which of the following are the first order conditions for a critical point for the Lagrangian function L? Check all that apply. dl dL dx dy dL dL dr = 1 + x = 0 ▼ andy”= Solving the first order conditions for this Lagrangian yields x*- multiplier is i-Finally, this implies the minimum value offx*y*)-x*+y*- . This implies that the value of the Lagrangian The Lagrangian multiplier λ for this constrained minimization dual problem is constrained maximization, or “primal,” problem. the Lagrangian multiplier in the corresponding

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