Verified step by step guidance
05:14
04:16
09:07Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.
x ƒ(x) g(x) ƒ'(x) g'(x)
0 1 1 -3 1/2
1 3 5 1/2 -4
Find the first derivatives of the following combinations at the given value of x.
d. ƒ(g(x)), x = 0
Analyzing Motion Using Graphs
[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:
d. When does it speed up and slow down?
s = t³ - 6t² + 7t, 0 ≤ t ≤ 4
Theory and Examples
In Exercises 51–54,
c. For what values of x, if any, is f' positive? Zero? Negative?
y = −x²
Theory and Examples
In Exercises 51–54,
d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)
y = −1/x
Theory and Examples
In Exercises 51–54,
d. Over what intervals of x-values, if any, does the function y = f(x) increase as x increases? Decrease as x increases? How is this related to what you found in part (c)? (We will say more about this relationship in Section 4.3.)
y = x³/3
Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.
d. During what year did home prices drop most rapidly and what is an estimate of this rate?
