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7:24107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .
b. Find the position of the object for all relevant times.
A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
c. Find the time at which the object passes the rest position for the second time.
{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.
c. By experimentation, determine the approximate value of a (3 < a < 4)at which the graph separates into two curves.
Cylinder in a cone A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone.
b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).
{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).
b. Consider the polynomial g(x) = f(f(x)). Write g in terms of a and powers of x. What is its degree?
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. If f(x) = mx + b, then the linear approximation to f at any point is L(x) = f(x).
