Chapter 4 Worksheet

Worksheet Chapter 4
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Find the location of the indicated absolute extremum for the function.
1) Maximum
A) x = 5
B) x = 0
C) No maximum
1)
D) x = 3
Find the extreme values of the function and where they occur.
2) f(x) = -3x4 + 20x3 - 36x2 + 9
2)
Find the extreme values of the function on the interval and where they occur.
3) y = 10 - 8x2 on [-2, 4]
3)
A) The maximum is 9 at x = 0.
C) The maximum is 0 at x = 0.
B) The minimum is 9 at x = 0.
D) There are none.
A) Maximum at (0, 20); minimum at (4, -22)
B) Maximum at (0, 8); minimum at (4, -138)
C) Maximum at (0, 10); minimum at (4, -118)
D) Maximum at (0, 80); minimum at (-2, -22)
Use analytic methods to find those values of x for which the given function is increasing and those values of x for
which it is decreasing.
4) f(x) = 27x - x3
4)
A) Increasing on (-3, 3), decreasing on (- , -3) and (3, )
B) Increasing on (- , -3), decreasing on (-3, 3)
C) Increasing on (- , 3), decreasing on (3, )
D) Increasing on (-9, 9), decreasing on (- , -9) and (9, )
1
Sketch a graph of a function y = f(x) that has the given properties.
5) a) Differentiable everywhere except x = 0
5)
b) Continuous for all real numbers
c) f'(x) < 0 on (- , 0)
d) f'(x) > 0 on (0, )
e) f(-2) = f(2) = 5
f) y-intercept and x-intercept at (0, 0)
Solve the problem.
6) Find the number of units that must be produced and sold in order to yield the maximum
profit, given the following equations for revenue and cost:
R(x) = 70x - 0.5x 2
C(x) = 4x + 2.
A) 67 units
B) 68 units
C) 74 units
D) 66 units
7) A ladder is slipping down a vertical wall. If the ladder is 20 ft long and the top of it is
slipping at the constant rate of 2 ft/s, how fast is the bottom of the ladder moving along the
ground when the bottom is 16 ft from the wall?
A) 1.5 ft/s
B) 0.8 ft/s
C) 0.13 ft/s
6)
7)
D) 2.5 ft/s
Use the First Derivative Test to determine the local extrema of the function, and identify any absolute extrema.
8) y = 4x 3 + 7
8)
A) Local maximum at (0, 4)
C) Local minimum at (0, 0)
B) Absolute maximum at (0, 7)
D) None
Find the linearization L(x) of f(x) at x = a.
9) f(x) = 4x + 36, a = 0
A) L(x) = 1 x - 6
3
B) L(x) = 2 x + 6
C) L(x) = 1 x + 6
3
3
D) L(x) = 2 x - 6
3
Solve the problem.
10) One airplane is approaching an airport from the north at 139 km/hr. A second airplane
approaches from the east at 290 km/hr. Find the rate at which the distance between the planes
changes when the southbound plane is 36 km away from the airport and the westbound
plane is 18 km from the airport.
A) 1780 km/hr
B) 108 km/hr
C) 1609 km/hr
2
9)
D) 402 km/hr
10)
Use the graph of f to estimate where f' is 0, positive, and negative.
11)
A) Zero: x = ±1; positive: x = (B) Zero: x = ±1; positive: x = (C) Zero: x = ±1; positive: x = (D) Zero: x = ±1; positive: x = (1,
11)
, -1); negative: x = (-1, 1)
, -1) and (1, ); negative: x = (0, 1)
, -1) and (1, ); negative: x = (-1, 1)
); negative: x = (-1, 1)
Sketch a graph of a single function that has these properties.
12) a) Continuous and differentiable for all real numbers
b) f (x) > 0 on (-3 , -1) and ( 2 , )
c) f (x) < 0 on (- , -3) and ( -1 , 2)
d) f (x) > 0 on (- , -2) and ( 1 , )
e) f (x) < 0 on (-2 , 1)
f) f (-3) = f (-1) = f (2) = 0
g) f (x) = 0 at (-2 , 0) and (1, 1)
12)
Solve the problem analytically.
13) Of all numbers whose difference is 16, find the two that have the minimum product.
A) 32 and 16
B) 8 and -8
C) 0 and 16
D) 1 and 17
Solve the problem.
14) A company is constructing an open-top, square-based, rectangular metal tank that will have
a volume of 42.5 ft3 . What dimensions yield the minimum surface area? Round to the
nearest tenth, if necessary.
A) 9.2 ft × 9.2 ft. × 0.5 ft
C) 4.4 ft × 4.4 ft. × 2.2 ft
B) 5 ft × 5 ft. × 1.7 ft
D) 3.5 ft × 3.5 ft. × 3.5 ft
3
13)
14)

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