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05:21107–110. {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} 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.
Population models The population of a species is given by the function P(t) = Kt²/(t² + b) , where t ≥ 0 is measured in years and K and b are positive real numbers.
c. For arbitrary positive values of K and b, when does the maximum growth rate occur (in terms of K and b)?
107–110. {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 .
c. Find the time when the object reaches its highest point. What is the height?
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} 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).
