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11:41{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).
d. Find the number and location of the fixed points of g for a = 2, 3, and 4 on the interval 0 ≤ x ≤ 1.
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for 8 < x < 10
Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>
Plot a possible graph of f.
if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.
g. Use your work in parts (a) through (f) to sketch a graph of ƒ .
{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>
d. Find the time and the displacement when the object reaches its high point for the second time.
Let ƒ(x) = (x - 3) (x + 3)²
g. Use your work in parts (a) through (f) to sketch a graph of ƒ.
