Exam 8: Numerical Methods

Consider the initial value problem This question is related to using the Runge-Kutta method for approximating the solution y at t= 1.1 with a step size of h = 0.05. How many approximations Y_n are needed to estimate a solution at y(1.1) if h = 0.05? n = ________

Consider the system of initial value problems given by This problem can be expressed using matrix notation as When using the Euler method with h = 0.1, = ________

Consider the initial value problem (Note: The exact solution is The size of the error e_K is large because

Consider the following initial value problem The following question pertains to the various computational steps and identifications involved in applying Runge-Kutta's 4th order method to evaluate y(0.7).

Consider the system of initial value problems given by This problem can be expressed using matrix notation as where When using the Euler method with h = 0.1, t₂ = ________

The velocity (measured in meters per second) of an air-dropped container of food and supplies affixed to a parachute is described by the function v(t) = 57.2 tanh(0.12t), t $\ge$ 3.0, where t is measured in seconds. You are interested in approximating the vertical distance s(t) traveled by the package by time t = 3.05 seconds.Using a step size of h = 0.05 seconds, compute the vertical distance traveled in the first 3.05 seconds.

Consider the initial value problem (Note: The exact solution is Which of the following expressions represents the error in the estimation for the improved Euler method? E₄ = ________

Given the initial value problem how many steps n are needed for the Euler method to find an approximation for y(4.6) using a step size of h = 0.06? n = ________

Consider the system of initial value problems given by This problem can be expressed using matrix notation as where When using the improved Euler method with h = 0.05, how many iterations n do you need in order to estimate the solution y_n at t = 0.10? n = ________

Consider the following initial value problem How would you need to rewrite this problem in order to apply the Runge-Kutta 4th order method to approximate the solution of this problem at a value of x?

Consider the system of initial value problems given by This problem can be expressed using matrix notation as Where When using the Euler method with h = 0.05, what are the values of when using the Euler method?

Consider the initial value problem This question relates to using the Euler method to approximate the solution of this problem at t = 0.2 using a step size of h = Y1 = ________

Consider the initial value problem This question relates to using the backward Euler method to approximate the solution at t = 0.4, namely = y(0.4), using a step size of h = 0.02. Which of these equations is the result of applying the backward Euler method to solve for

Consider the following initial value problem on the interval [0, 1] Approximate the solution y(t) at t = 2.5 using the fourth-order backward differentiation formula with step size h = 0.05. Using the Runge-Kutta method you are provided with some initial Y_n values to start the fourth-order backward differentiation formula. Y₁ = 2.157777 Y₂= 2.3322791 Y₃ = 2.5253944 Use the fourth-order backward Euler differentiation formula to compute the following approximation. Express your answer accurate to 6 decimal places. Y₅ = ________

Consider the following initial value problem

Consider the initial value problem This question is related to using the fourth-order backward differentiation formula to estimate the solution y(0.2) using a step size of . Which of these is the correct formula for

Consider the initial value problem This question is related to using the predictor-corrector method to estimate the solution y(0.2) using a step size of h = 0.05. For this problem, you will need these values to carry out the computations: Which of the following show a portion of the formula for the corrected value of

The velocity (measured in meters per second) of an air-dropped container of food and supplies affixed to a parachute is described by the function v(t) = 65.6 tanh(0.19t), t $\ge$ 3.0, where t is measured in seconds. You are interested in approximating the vertical distance s(t) traveled by the package by time t = 3.10 seconds.Using a step size of h = 0.10 seconds, compute the vertical distance traveled in the first 3.10 seconds.

Consider the initial value problem The following table provides the estimation for the solution at different t values using the Euler and Improved Euler methods with a step size of 0.25. Based on the table above, which of the following is true regarding the given initial value problem?

Consider the following initial value problem How would you need to rewrite this problem in order to apply Euler's method to approximate the solution of this problem at a value of x?