1. Suppose that f(x) and g(x) are differentiable functions and that the following information about them is known:
|
x
|
f(x)
|
f'(x)
|
g(x)
|
g'(x)
|
|
-1
|
2
|
-5
|
-3
|
4
|
|
2
|
-3
|
4
|
-1
|
2
|
If C(x) is a function given by the formula f(g(x)), determine C'(2). In addition, if D(x) is the function f(f(x)), find D'(-l).
2. Consider the basic functions f(x) = X3 and g(x) = sin(x).
a. Let h(x) = f(g(x)). Find the exact instantaneous rate of change of h at the point where x = π / 4.
b. Which function is changing most rapidly at x = 0.25: h(x) = f(g(x)) or r(x) = g(f(x))? Why?
c. Let h(x) = f(g(x)) and r(x) = g(f(x)). Which of these functions has a derivative that is periodic? Why?
3. Let functions p and q be the piecewise linear functions given by their respective graphs in Figure. Use the graphs to answer the following questions.
a. Let C(x) = p(q(x)). Determine C'(0) and C'(3).
b. Find a value of x for which C'(x) does not exist. Explain your thinking.
Figure: The graphs of p (in blue) and q (in green).
Let Y(x) = q(q(x)) and Z(x) = q(p(x)). Determine Y'(-2) and Z'(0).