25. Falling parachutist As in Problem 25 of Section 2.4, you bail out of a helicopter and immediately open your parachute, so your downward velocity satisfies the initial value problem dv dt = 32-1.6v, v (0) = 0 (with t in seconds and v in ft/s). Use the improved Euler method with a programmable calculator or computer to approximate the solution for 0 ≤ t ≤2, first with step size h = 0.01 and then with h = 0.005, rounding off approx- imate v-values to three decimal places. What percentage of the limiting velocity 20 ft/s has been attained after 1 second? After 2 seconds?

Do not use a calculator for this problem. Find the x-value where the relative maximum of ye² y = e² - x² closest to x = 0 occurs. When you need to solve for an expression equal to zero, use Newton's method with x = 0 as your first guess and enter your second guess as the answer. Enter your answer as a fraction.

In the previous Problem Set question, we started looking at the position functions (t), the position of an object at time t. Two important physics concepts are the velocity and the acceleration. If the current position of the object at time is as (t), then the position at time h later is a (t+h). The average velocity (speed) during that additional time his (s(t+h)-s(t)) If we want to analyze the instantaneous velocity at time t, this can be made into a mathematical model by taking the limit as h→0. i.e. the derivative a' (t). Use this function in the model below for the velocity function (). h The acceleration is the rate of change of velocity, so using the same logic, the acceleration function a(t) can be modeled with the derivative of the velocity function, or the second derivative of the position function a(t) = ✔ (t) =" (t). Problem set question: A particle moves according to the position functions (t) = etsin (2). Enclose arguments of functions in parentheses. For example, sin…

12. A stone is thrown vertically upward from the edge of a cliff with initial velocity 32 ft/s from a height of 48 ft above the ground. The height of the stone above the ground t seconds after it is thrown in given by s(t) = -16t2 + 32t + 48. With what velocity does the stone hit the ground?