MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) dc. D) dr dt = 2πdC dt

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1 MAC 3 Chapter Review Materials (Part III) Topics Include Related Rates, Differentials, and Linear Approximations MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. ) Suppose that the radius r and the circumference C = πr of a circle are differentiable functions of t. Write an equation that relates dc/ to dr/. A) dc = πrdr B) dc = dr C) dc = πdr D) dr = πdc ) ) The area of the base B and the height h of a pyramid are related to the pyramid's volume V by the formula V = Bh. How is dv/ related to dh/ if B is constant? 3 ) A) dv = Bdh B) dv = dh C) dv = dh 3 D) dv = B dh 3 3) If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, c = a + b. How is dc/ related to da/ and db/? A) dc = ada + bdb C) dc = ada + bdb B) dc = ada + bdb D) dc = c ada + bdb 3) 4) The range R of a projectile is related to the initial velocity v and projection angle θ by the equation R = v sin θ, where g is a constant. How is dr/ related to dθ/ if v is constant? g A) dr = v cos θ dθ C) dr = v cos θ dθ B) dr = - v cos θ dθ D) dr = v sin θ dθ 4) 5) The range R of a projectile is related to the initial velocity v and projection angle θ by the equation R = v sin θ, where g is a constant. How is dr/ related to dv/ and dθ/ if neither v nor θ is g constant? A) dr = dv 4v cos θdθ C) dr = v cos θdv + sin θdθ B) dr = v v cos θdθ + sin θdv D) dr = v g v cos θdθ + sin θdv 5) Provide an appropriate response. 6) If xy + x = and / = -3, then what is dy/ when x = and y = 5? A) -3 B) -9 C)9 D) 3 6)

2 7) If xy = 4 and / = -5, then what is dy/ when x = 4 and y =? 7) A) 5 8 B) 8 5 C) D) ) If y x + = and / = 8, then what is dy/ when x = 5 and y = 3? 8) A) 4 3 B) C) 3 4 D) ) A company knows that the unit cost C and the unit revenue R from the production and sale of x R units are related by C = Find the rate of change of unit revenue when the unit cost 58,000 is changing by $/unit and the unit revenue is $000. A) $474.00/unit B) $0.00/unit C) $340.50/unit D) $407.5/unit 0) A wheel with radius 3 m rolls at 9 rad/s. How fast is a point on the rim of the wheel rising when the point is π/3 radians above the horizontal (and rising)? (Round your answer to one decimal place.) A) 57.0 m/s B) 8.5 m/s C)4.3 m/s D) 4.0 m/s 9) 0) ) A piece of land is shaped like a right triangle. Two people start at the right angle of the triangle at the same time, and walk at the same speed along different legs of the triangle. If the area formed by the positions of the two people and their starting point (the right angle) is changing at 3 m/s, then how fast are the people moving when they are 3 m from the right angle? (Round your answer to two decimal places.) A) 3.00 m/s B).00 m/s C).00 m/s D) 0.50 m/s ) Round your answer, if appropriate. ) Water is discharged from a pipeline at a velocity v (in ft/sec) given by v = 054p(/), where p is the pressure (in psi). If the water pressure is changing at a rate of psi/sec, find the acceleration (dv/) of the water when p = 57.0 psi. A) 39.8 ft/sec B) 350 ft/sec C) 3.7 ft/sec D) 69.8 ft/sec ) 3) A container, in the shape of an inverted right circular cone, has a radius of 4 inches at the top and a height of 5 inches. At the instant when the water in the container is inches deep, the surface level is falling at the rate of -.3 in./s. Find the rate at which water is being drained. A) in.3/s B) in.3/s C) in.3/s D) in.3/s 3) Round your answer, if appropriate. 4) One airplane is approaching an airport from the north at 50 km/hr. A second airplane approaches from the east at 97 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 7 km away from the airport and the westbound plane is 9 km from the airport. A)-588 km/hr B)-47 km/hr C)-44 km/hr D)-94 km/hr 4)

3 5) A man 6 ft tall walks at a rate of 3 ft/sec away from a lamppost that is 3 ft high. At what rate is the length of his shadow changing when he is 50 ft away from the lamppost? (Do not round your answer) A) 9 9 ft/sec B) ft/sec C) ft/sec D)5 ft/sec 9 5) 6) Electrical systems are governed by Ohm's law, which states that V = IR, where V = voltage, I = current, and R = resistance. If the current in an electrical system is decreasing at a rate of 4 A/s while the voltage remains constant at 4 V, at what rate is the resistance increasing (in Ω/sec) when the current is 44 A? (Do not round your answer.) A) 56 4 Ω/sec B) Ω/sec C) 4 Ω/sec D) 7 4 Ω/sec 6) 7) The volume of a rectangular box with a square base remains constant at 000 cm3 as the area of the base increases at a rate of 3 cm/sec. Find the rate at which the height of the box is decreasing when each side of the base is cm long. (Do not round your answer.) A) 000 cm/sec B) cm/sec C) cm/sec D) cm/sec 7) 8) The radius of a right circular cylinder is increasing at the rate of 4 in./sec, while the height is decreasing at the rate of 0 in./sec. At what rate is the volume of the cylinder changing when the radius is 9 in. and the height is 9 in.? A)-96 in.3/sec B)-4π in.3/sec C) 8 in.3/sec D)-96π in.3/sec 8) Find the linearization L(x) of f(x) at x = a. 9) f(x) = 3x + 3x + 4, a = A) L(x) = 3x + 7 B) L(x) = 3x + C)L(x) = 9x + 7 D) L(x) = 9x + 0) f(x) = 6x +, a = 0 9) 0) A) L(x) = 3 x + B) L(x) = - 3 x + 4 C)L(x) = 3 x + 4 D) L(x) = - 3 x + ) f(x) = x + x, a = 4 ) A) L(x) = 7 6 x + B) L(x) = 5 6 x + C)L(x) = 5 6 x + 5 D) L(x) = 7 6 x + 5 ) f(x) = tan x, a = π A) L(x) = x - 3π B) L(x) = x - π C)L(x) = x + π D) L(x) = 3x - π ) You want a linearization that will replace the function over an interval that includes the point x0. To make your subsequent work as simple as possible, you want to center the linearization not at x0 but at a nearby integer x = a at which the function and its derivative are easy to evaluate. What linearization do you use? 3) f(x) = -3x - 4x + 3, x0 =. 3) A)-4 B)-6-0x C)7 - x D) 6-0x 3

4 4) f(x) = x, x0 = 9. 4) A) x B) + x C) x D) 3-6 x Use the linear approximation ( + x) k + kx, as specified. 5) Find an approximation for the function f(x) = + x for values of x near zero. 5) A) f(x) - x 4 B) f(x) - x C)f(x) - x 4 D) f(x) - x 6) Find an approximation for the function f(x) = ( - x) 4 for values of x near zero. A) f(x) + 4x B) f(x) 4 + 4x C)f(x) - 4x D) f(x) + 5x 6) 7) Estimate (.0005) 50. A).005 B).05 C).05 D).0 7) 8) Estimate A).0 B).00 C).003 D).03 8) Find dy. 9) y = 3x + 6x + 8 A) 6x B) 6x + 8 C)(6x + 6) D) 6x + 9) 30) y = x 9x - 6 7x - A) 9x - 6 B) 7x - 9x - 6 7x + C) 9x - 6 7x + D) 9x ) 3) 6y / - 3xy + x = 0 3y - A) 3y -/ + 3x B) 3y - 6y - 3x C) 3y - 3y -/ - 3x D) - 3y -/ - 3x 3) 3) y = cos(9 x) A) C) -9 x sin(9 x) 9 x sin(9 x) B) D) -9 sin(9 x) x 9 sin(9 x) x 3) 4

5 The function f(x) changes value when x changes from x0 to x0 +. Find the approximation error f - df. Round your answer, if appropriate. 33) f(x) = x, x0 = 6, = ) A) B) C) D) ) f(x) = x 3, x0 = 4, = 0.05 A) B) C) D) ) 35) f(x) = x - x, x0 = 5, = 0.03 A) B) C) 0.08 D) ) Write a differential formula that estimates the given change in volume or surface area. 36) The change in the surface area S = 4πr of a sphere when the radius changes from r0 to r0 + A) ds = πr0 dr B) ds = 4πr0 dr C)dS = 8πr0 dr D) ds = 4πr0 dr 36) 37) A cube 4 inches on an edge is given a protective coating 0.3 inches thick. About how much coating should a production manager order for 700 cubes? A) About 3360 in. B) About 3,440 in.3 C) About 0,60 in.3 D) About 0,080 in. 37) 38) A = πr, where r is the radius, in centimeters. By approximately how much does the area of a circle decrease when the radius is decreased from 3.0 cm to.8 cm? (Use 3.4 for π.) A).9 cm B) 4.0 cm C)3.6 cm D) 3.8 cm 38) 39) The diameter of a tree was in. During the following year, the circumference increased in. About how much did the tree's diameter increase? (Leave your answer in terms of π.) A) π in. B) π in. C) 3 π in. D) π in. 39) 40) About how accurately must the interior diameter of a cylindrical storage tank that is 4 m high be measured in order to calculate the tank's volume within 4% of its true value? A) Within 4 meters B) Within 4% C)Within meters D) Within % 40) Use the differential to approximate the quantity to four decimal places. 4) 45 A).0833 B).047 C) D) ) 4) e0.38 A).463 B).3800 C).600 D) ) Find dy for the given values of x and Δx. 43) y = x3 + x; x =, Δx = 0.0 A) 0.04 B) 0.4 C) D) ) 5

6 44) y = x ; x = 0, Δx = 0. x + 44) A) 46 B) 48 C) 44 D) 4 45) y = x + 3; x = 8, Δx = 0.5 A) B) 0.5 C)0. D) 5 45) 46) A tumor is approximately spherical in shape. If the radius of the tumor changes from 8 mm to 0 mm, find the approximate change in volume. Round your answer to the nearest hundred. A) 900 mm 3 B) 00 mm 3 C)00 mm 3 D) 600 mm 3 46) 6

7 Answer Key Testname: CHAPTER (PART III) RELATED RATES, DIFFERENTIALS, AND LINEAR APPROXIMATIONS ) C ) D 3) D 4) A 5) D 6) C 7) A 8) B 9) A 0) B ) C ) C 3) A 4) D 5) B 6) D 7) D 8) B 9) D 0) D ) B ) B 3) D 4) C 5) C 6) C 7) B 8) B 9) C 30) B 3) C 3) B 33) A 34) B 35) D 36) C 37) C 38) D 39) D 40) D 4) B 4) A 43) B 44) D 45) A 46) D 7

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. 6 C) - 12 (6x - 7)3

Part B- Pre-Test 2 for Cal (2.4, 2.5, 2.6) Test 2 will be on Oct 4th, chapter 2 (except 2.6) Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.