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

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1 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. Given y = f(u) and u = g(x), find dy/dx = f (g(x))g (x). ) y = u2, u = 6x - 7 ) A) - 2 6x - 7 B) - 6 6x (6x - 7)3 2x 6x - 7 2) y = tan u, u = -4x + 8 A) -4 sec2(-4x + 8) B) -4 sec (-4x + 8) tan (-4x + 8) sec2(-4x + 8) - sec2(-4x + 8) 2) Write the function in the form y = f(u) and u = g(x). Then find dy/dx as a function of x. 3) y = 6x2-4 x - x 0 3) A) y = u0; u = 6x2-4 x B) y = 6u2-4 u - u; u = x 0; dy dx = 2x 20 - y = u0; u = 6x2-4 x y = u0; u = 6x2-4 x - x; dy dx = 0 6x 2-4 x - x 9 2x + 4 x2-4 x0 - x 0 - x; dy dx = 0 6x 2-4 x - x 9 - x; dy dx = 0 2x + 4 x2-9 4) y = tan π - 3 x 4) A) y = tan u; u = π - 3 x ; dy dx = 3 x2 sec 2 π - 3 x B) y = tan u; u = π - 3 x ; dy dx = sec 2 3 x2 y = tan u; u = π - 3 x ; dy dx = 3 x2 sec π - 3 x tan π - 3 x y = tan u; u = π - 3 x ; dy dx = sec 2 π - 3 x Find the derivative of the function. 5) q = 5r - r7 A) 2 5-7r6 B) 5-7r6 2 5r - r7 2 5r - r7-7r6 5r - r7 5)

2 6) r = (sec θ + tan θ)-5 A) -5(sec θ tan θ + sec2θ) -6 B) -5(sec θ + tan θ)-6(tan2 θ + sec θ tan θ) 6) -5 sec θ (sec θ + tan θ)5-5(sec θ + tan θ)-6 Find dy/. 7) y = cos6(πt - 9) A) 6 cos5(πt - 9) B) -6 cos5(πt - 9) sin(πt - 9) -6π cos5(πt - 9) sin(πt - 9) -6π sin5(πt - 9) 7) Find y. 8) y = ( x - 5) -5 A) 20( x - 5) -7 B) - 5 2x ( x - 5)-7 5x - 5 8) 5 4x ( x - 5)-7-5x x ( x - 5)-6 9) y = 4 tan(6x - 4) 9) A) 3 2 sec 2(6x - 4) B) 2 sec 2(6x - 4) tan(6x - 4) 8 sec2(6x - 4) tan(6x - 4) 2 sec(6x - 4) Find the value of (f g) at the given value of x. 0) f(u) = u + 2, u = g(x) = x2-2, x = 7 0) A) -26 B) ) f(u) = - u, u = g(x) = πx, x = 6 cos3 u ) A) 6π B) 3 - π -π -6π Suppose that the functions f and g and their derivatives with respect to x have the following values at the given values of x. Find the derivative with respect to x of the given combination at the given value of x. x f(x) g(x) f (x) g (x) 2) ) f(x) + g(x), x = 3 A) B)

3 Use implicit differentiation to find dy/dx. 3) xy + x = 2 A) - + x y B) + x y - + y x + y x 3) Find dy/. 4) y = t5(t4 + 8) 5 A) t5(t4 + 8) 4 (25t3 + 40) B) 5t4(t4 + 8) 4 (20t4 + 8) t4(t4 + 8) 4 (25t4 + 40) 00t8(t4 + 8) 4 4) Use implicit differentiation to find dy/dx. 5) y cos y = 8x + 8y 5) A) 8 - y sin 8y y sin B) y + y cos y - 8y cos y - 8 8y2 sin y - 8y 2 8 sin y + y cos y - 8 Use implicit differentiation to find dy/dx and d2y/dx2. 6) 2y - x + xy = 8 A) dy dx = y + x + 2 ; d 2y dx2 = 2y + 2 (x + 2)2 dy dx = - y 2 + x ; d 2y dx2 = 2y - 2 (2 + x)2 B) dy dx = - + y x + 2 ; d 2y dx2 = 2y - 2 (x + 2)2 dy dx = - + y x + 2 ; d 2y dx2 = y + (2 + x)2 6) 7) x3/5 + y3/5 = 2 A) dy dx = x 2/5 y2/5 ; d 2y dx2 = - 2x 3/5 + 2y3/5 5x/5y7/5 dy dx = - y 2/5 x2/5 ; d 2y dx2 = 2y /5-2x 5x7/5y3/5 B) dy dx = y 2/5 x2/5 ; d 2y dx2 = - 2x 3/5 + 2y3/5 5x7/5y/5 dy dx = - y 2/5 x2/5 ; d 2y dx2 = 2x 3/5 + 2y3/5 5x7/5y/5 7) At the given point, find the slope of the curve, the line that is tangent to the curve, or the line that is normal to the curve, as requested. 8) x4y4 = 6, tangent at (2, ) 8) A) y = -8x + B) y = 2 x y = 8x - y = - 2 x + 2 3

4 9) x3y3 = 8, slope at (2, ) 9) A) - 2 B) Solve the problem. 20) Suppose that the radius r and the circumference C = 2πr of a circle are differentiable functions of t. Write an equation that relates dc/ to dr/. A) dc = 2π dr B) dc = 2πr dr dc = dr dr dc = 2π 20) 2) If a and b are the lengths of the legs of a right triangle and c is the length of the hypotenuse, c2 = a2 + b2. How is dc/ related to da/ and db/? A) dc = a 2 da + b 2 db B) dc = a da + bdb dc da db = 2a + 2b dc = c a da + bdb 2) Provide an appropriate response. 22) If x3 + y3 = 9 and dx/ = -3, then what is dy/ when x = and y = 2? 22) A) 4 3 B) Solve the problem. 23) Water is falling on a surface, wetting a circular area that is expanding at a rate of 8 mm2/s. How fast is the radius of the wetted area expanding when the radius is 72 mm? (Round your answer to four decimal places.) 23) A) mm/s B) mm/s mm/s mm/s Solve the problem. Round your answer, if appropriate. 24) One airplane is approaching an airport from the north at 44 km/hr. A second airplane approaches from the east at 90 km/hr. Find the rate at which the distance between the planes changes when the southbound plane is 33 km away from the airport and the westbound plane is 7 km from the airport. 24) A) -07 km/hr B) -25 km/hr -430 km/hr -322 km/hr 25) Water is being drained from a container which has the shape of an inverted right circular cone. The container has a radius of 6.00 inches at the top and a height of 8.00 inches. At the instant when the water in the container is 6.00 inches deep, the surface level is falling at a rate of 0.9 in./sec. Find the rate at which water is being drained from the container. 25) A) 54.7 in.3/s B) 55. in.3s 57.3 in.3/s 70.0 in.3/s 4

5 26) A man 6 ft tall walks at a rate of 7 ft/sec away from a lamppost that is 20 ft high. At what rate is the length of his shadow changing when he is 35 ft away from the lamppost? (Do not round your answer) A) 3 ft/sec B) ft/sec ft/sec 26 3 ft/sec 26) 27) Boyleʹs law states that if the temperature of a gas remains constant, then PV = c, where P = pressure, V = volume, and c is a constant. Given a quantity of gas at constant temperature, if V is decreasing at a rate of 4 in. 3/sec, at what rate is P increasing when P = 70 lb/in.2 and V = 90 in.3? (Do not round your answer.) A) 98 9 lb/in. 2 per sec B) 49 8 lb/in. 2 per sec 27) 450 lb/in.2 per sec 8 lb/in.2 per sec 28) The radius of a right circular cylinder is increasing at the rate of 2 in./sec, while the height is decreasing at the rate of 8 in./sec. At what rate is the volume of the cylinder changing when the radius is 8 in. and the height is 5 in.? 28) A) 24 in.3/sec B) -242π in.3/sec -2232π in.3/sec -242 in.3/sec Use implicit differentiation to find dy/dx. 29) x + y x - y = x 2 + y2 29) A) x(x - y) 2 - y x - y(x - y)2 B) x(x - y) 2 + y x + y(x - y)2 x(x - y) 2 - y x + y(x - y)2 x(x - y) 2 + y x - y(x - y)2 5

6 Answer Key Testname: PART B CAL PRE TEST 2 ) C 2) A 3) A 4) A 5) B 6) C 7) C 8) C 9) C 0) B ) C 2) D 3) C 4) C 5) A 6) C 7) D 8) D 9) A 20) A 2) D 22) C 23) C 24) B 25) C 26) A 27) A 28) C 29) D 6