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Q9)
For the function graphed in the figure, are the following nonzero quantities positive or negative? (a) f(3) positive negative (b) f'(3) positive negative (c) f"( 3) positive negative Give the signs of the first and second derivatives for the following function. Each derivative is either positive everywhere, zero everywhere, or negative everywhere. (a) f'(x) positive zero negative (b) f"(x) positive zero negative Give the signs of the first and second derivatives for the following function. Each derivative is either positive everywhere, zero everywhere, or negative everywhere. (a) f'(x) positive zero negative (b) f"(x) positive zero negative The table gives the number of passenger cars, C = f(t), in millions, in a country in the year t. (a) Do f'(t) and f"(t) appear to be positive or negative during the period 1975-1990? f"(t) appears to be negative and f"(t) appears to be positive. Both f'(t) and f"(t) appear to be negative. Both f'(t) and f"(t) appear to be positive. Both f'(t) and f"(t) appear to be exactly zero. f'(t) appears to be positive and f"(t) appears to be negative. (b) Do f'(t) and f"(t) appear to be positive or negative during the period 1990-2000? f'(t) appears to be positive from 1990 to 2000, and f"(t) appears to be negative from 1990 to 1995 and positive from 1995 to 2000. f'(t) appears to be negative from 1990 to 1995 and positive from 1995 to 2000, and f"(t) appears to be positive from 1990 to 2000. f'(t) appears to be negative from 1990 to 2000, and f"(t) appears to be positive from 1990 to 1995 and negative from 1995 to 2000. f(t) appears to be positive from 1990 to 2000, and f"(t) appears to be positive from 1990 to 1995 and negative from 1995 to 2000. f'(t) appears to be positive from 1990 to 1995 and negative from 1995 to 2000, and f"(t) appears to be negative from 1990 to 2000. (c) Estimate f'(2005). Using units, interpret your answer in terms of passenger cars. negative zero Let P(t) represent the price of a share of stock of a corporation at time t. What does each of the following statements tell us about the signs of the first and second derivatives of P(t)? (a) The price of the stock is rising faster and faster. The sign of P '(t) is positive negative zero The sign of P "(t) is positive negative zero (b) The price of the stock is close to bottoming out. The sign of P '(t) is positive negative zero The sign of P "(t) is O positive negative zero The function shown below gives the position of a particle at time t. At what labeled points is the position of the particle positive? (Select all that apply.) t1 t2 t3 t4 t5 none of these At what labeled points is the velocity of the particle negative? (Select all that apply.) t1 t2 ts t4 t5 none of these At what labeled points is the acceleration of the particle negative? (Select all that apply.) t1 t2 t3 t4 t5 none of these At what labeled points is the position of the particle increasing? (Select all that apply.) t1 t2 t3 t4 t5 none of these At what labeled points is the velocity of the particle increasing? (Select all that apply.) t1 t2 t3 t4 t5 none of these The graph of f' (not f) is given in the figure. At which of the marked values of x are the following true? Use lower case x and the subscript in your answer. For example, enter x 2. (a) f(x) is greatest. (b) f(x) is least. (c) f(x) is greatest. (d) f'(x) is least. (e) f"(x) is greatest. (f) f"(x) is least. A function f has f(2) = 23, f'(2) = 3, and f"(x)
Explanation / Answer
1.
f(3) = negative
f'(3) = positive
f''(3) = positive
2.
f'(x) = negative
f''(x) = positive
3.
f'(x) = positive
f''(x) = negative
4.
a)
f'(t) appears to be positive and f''(t) appears to be negative.
b)
f'(t) appears to be negative from 1990 to 1995 and positive from 1995 to 2000, and f''(t) appears to be positive from 1990 to 2000.
c)
f'(2005) = (136.5 - 133.2) / (2005 - 2000) = 0.66 millions / year
increasing, 0.66 million cars per year.
5.
a)
P'(t) = positive
P''(t) = positive
b)
P'(t) = negative
P''(t) = positive
6.
Position = positive at t3, t4, t5.
Velocity = negative at t1, t4, t5
Acceleration = negative at t3, t4
Position = increasing at t2, t3
Velocity = increasing at t1, t2, t5
7.
f(x) is greatest at x6.
f(x) is least at x1.
f'(x) is greatest at x3.
f'(x) is least at x2.
f''(x) is greatest at x6.
f''(x) is least at x1.
8.
In the imiting case, f''(x) = 0 or f'(x) = constant = f'(2) = 3.
Therefore, f(4) = f(2) + f'(2)*(4-2) = 23 + 3*(4-2) = 29.
Thus f(4) can never be greater than or equal to 29.
Hence,
a) 32 = impossible
b) 29 = impossible
c) 26 = possible.
9.
a)
constant velocity = particle iv
b)
Greatest initial velocity = particle iii
c)
Greatest average velocity = particle iii
d)
Zero average velocity = particle i
e)
Zero acceleration = particle iv
f)
Positive acceleration throughout = particle ii
