( f + g ) (3) = ( fg ) (3) = g(x) = x 7 cos x. s = 200t 10t 2. sin x cos x cos2x. lim. f (x) = 7 x 5. y = 1+ 4sin x, (0,1) f (x) = x 2 g(x)

Transcription

1 Stewart - Calculus ET 6e Chapter Form A 1. If f ( ) =, g() =, f () =, g () = 6, find the following numbers. ( f + g ) () = ( fg ) () = ( f / g) () = f f g ( ) =. Find the points on the curve y = where the tangent is horizontal.. Find the equation of the tangent to the curve at the given point. y = 1+ sin, (0,1). Differentiate. g() = 7 cos. Find f in terms of g. f () = g() 6. If a ball is thrown vertically upward with a velocity of 00 ft/s, then its height after t seconds is s = 00t 10t. What is the maimum height reached by the ball? 7. Find the limit. lim π / sin cos cos 8. Calculate y. y = cos 9. A spherical balloon is being inflated. Find the rate of increase of the surface area respect to the radius r when r = 1 ft. S = πr with 10. Differentiate the function. f () = 7

2 Stewart - Calculus ET 6e Chapter Form A 11. Find an equation of the tangent line to the curve. y = + 6 at (, 0.) 1. A plane flying horizontally at an altitude of mi and a speed of 6 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 mi away from the station. Round the result to the nearest integer. 1. Find the limit if g() =. g() g() lim 1. A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 16 mm. The area is A(). Find A (16). 1. Calculate y. y + y = + y 16. Find the first and the second derivatives of the function. y = 17. If f (t) = t + 1, find f (). 18. If y d = + and =, dt dy find when =. dt 19. The volume of a cube is increasing at a rate of 10 cm / min. How fast is the surface area increasing when the length of an edge is 0cm If f ( t) = find f (t). + t

3 ANSWER KEY Stewart - Calculus ET 6e Chapter Form A 1. 1, 1, -8., 8.. (1, -6), (-, 1). y=+ 1 dg( ) 6 7. = 7 cos( ) sin( ) d. fgg ()()() = sincos y = 9. 8π 10. df 7 = d 6 1 y = ( ) yy y = y ( ) ( ), f () = /min cm 6t ( + t )

4 Stewart - Calculus ET 6e Chapter Form B 1. If f() = -, f '() =, g() = and g'() = 0, find (f + g)'().. Find the limit if g () =. gg ()(1) lim 1 1. Calculate y. y= cos. The position function of a particle is given by s = t 10.t t, t 0 When does the particle reach a velocity of m/s?. Calculate y. 1/ e y = 6. Use the table to estimate the value of h (10.), where h ( ) = f ( g( )) f () g() A spherical balloon is being inflated. Find the rate of increase of the surface area respect to the radius r when r = 1 ft. S = πr with 8. If h ( ) = 7 and h ( ) =, find 9. Differentiate. sin y = 7 + cos 10. Differentiate. tan y = sec d d h( ) = 11. Find an equation of the tangent line to the curve y = tan at the point( π /, ).

5 Stewart - Calculus ET 6e Chapter Form B 1. Find the derivative of the following function and calculate it for = 6 to the nearest tenth. y ( ) = Find the differential of the function. + y = 1. Find all points at which the tangent line is horizontal on the graph of the function. y( ) = 6 sin + sin 1. Calculate y. y ey = If ftt ()1=+, find f (). 17. Find the equation of the tangent to the curve at the given point. y=+ 1sin,(0,1) 18. Find y, if y = Find a formula for f (n) ( ) ( + ) 1 f ( ) = 0. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 8 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth if necessary.

6 ANSWER KEY Stewart - Calculus ET 6e Chapter Form B sincos y =. 9. 1/ 1+ ye = π dy d = 7 cos + 1 ( 7 + cos ) dy 10. = cos( ) + sin( ) d π 11. y = dy = ( + )d π π 1. + πn, 7, + πn, y e y = y e 1 f () = y=+ 1 / 18. ( ) + 1 n 19. ( ) ( ) ( n+ 1) 1 n!

7 Stewart - Calculus ET 6e Chapter Form C 1. Calculate y. e y = a. b. c. d. e. e y = 1 ye = ye = + ye = ye =. Find the equation of the tangent to the curve at the given point. y=+ 1sin,(0,1) a. y = + b. y = + c. y= 1 d. y=+ 1 e. y=+ 1. Compute Δy and dy for the given values of and y =, = 1, Δ = 0. a. Δy = 1., dy = 1 b. Δy = 0., dy = 1 c. Δy = 0., dy = 0 d. Δy = 1., dy = 0 e. Δy = 1., dy = 0. d = Δ.

8 Stewart - Calculus ET 6e Chapter Form C. Find f in terms of g. fg= ()() a. fgg ()()() =+ b. fg ()() =+ c. fgg ()()() =+ d. fg ()() = e. ffg ()()() =+. Find equations of the tangent lines to the curve 8 y = that are parallel to the line y = Select all that apply. a. - y = - 17 b. - y = - 1 c. - y = - 1 d. - y = - 1 e. - y = - 6. The position function of a particle is given by s = t 1.t t, t 0. When does the particle reach a velocity of 166 m/s? a. t = 7 sec b. t = 8 sec c. t = sec d. t = sec e. t = 1 sec 7. The mass of the part of a metal rod that lies between its left end and a point meters to the right is S =. Find the linear density when is 1 m. a. b. 16 c. 8 d. 1 e. 18

9 Stewart - Calculus ET 6e Chapter Form C 8. If f is the focal length of a conve lens and an object is placed at a distance v from the lens, then its image will be at a distance u from the lens, where f, v, and u are related by the lens equation = +. f v u Find the rate of change of v with respect to u. a. b. c. d. e. dv du dv du dv du dv du dv du f = ( u f ) f = ( u f ) f = u f f = ( u f ) f = ( u f ) 9. The gas law for an ideal gas at absolute temperature T (in kelvins), pressure P (in atmospheres), and volume V (in liters) is PV = nrt, where n is the number of moles of the gas and R = is the gas constant. Suppose that, at a certain instant, P = 7 atm and is increasing at a rate of 0.10 atm/min and V = 10L and is decreasing at a rate of 0.1 L/min. Find the rate of change of T with respect to time at that instant if n = 10 moles. a..97 b c d e Find the limit. lim 0 θ cos(cosθ ) secθ a. 1 b. sin 1 c. cos 1 d. 0 e. 11. Find the derivative of the following function and calculate it for = to the nearest tenth. y ( ) = + + a. 1.1 b c. 0.1 d. 0. e. 0.

10 Stewart - Calculus ET 6e Chapter Form C 1. Suppose that F ( ) = f ( g( )) and g ( 1) =, g (1) =, f (1) = 1, and f ( ) = 1. Find F (1). a. 60 b. 10 c. d. 17 e Calculate y. yyy +=+ a. b. c. d. yy y = y + 1 yy y = y + y + y = y y = ( 1 ) 1 y y + e. none of these 1. If ftt ()1=+, find f (). a. b. c. 7 d. e Evaluate. lim sinh e a. b. 1 / c. d. 0 e. 1/ 16. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.18 cm thick to a hemispherical dome with diameter 60 m. a..π b..π c..8π d..8π e..11π

11 Stewart - Calculus ET 6e Chapter Form C 17. Gravel is being dumped from a conveyor belt at a rate of ft /min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 1 ft high? Round the result to the nearest hundredth. a. 0.7 ft/min b. 1. ft/min c. 0.1 ft/min d. 0. ft/min e. 0.6 ft/min 18. Two sides of a triangle are 6 m and 7 m in length and the angle between them is increasing at a rate of 0.07 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fied length is π. a. 1.0 m /s b. 1.8 m /s c m /s d. 0.7 m /s e..7 m /s 19. The table lists the amount of U.S. cash per capita in circulation as of June 0 in the given year. Use a linear approimation to estimate the amount of cash per capita in circulation in the year 000. t C(t) $176 $70 $7 $1,068 Choose the correct answer from the following. a. $1,91 b. $1, c. $1,87 d. $1,6 e. $1,7 0. Two carts, A and B, are connected by a rope ft long that passes over a pulley (see the figure below). The point Q is on the floor 16 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at the instant when cart A is ft from Q? Round the result to the nearest hundredth. a..9 ft/s b..9 ft/s c.. ft/s d..7 ft/s e..7 ft/s

12 ANSWER KEY Stewart - Calculus ET 6e Chapter Form C 1. c. e. a. a. a, d 6. b 7. c 8. b 9. c 10. c 11. c 1. a 1. b 1. a 1. b 16. b 17. d 18. d 19. d 0. b

13 Stewart - Calculus ET 6e Chapter Form D 1. Calculate y. y= cos a. b. c. d. e. sin y = cos sincos y = 1cos1 y = 1sin1 y = 1sincos y =. If f ( ) = cos + sin, find f ' () and f '' (). Select the correct answers. a. f ( ) = cos( ) + cos() b. f ( ) = cos() + cos( ) c. f ( ) = sin() + sin( ) d. f ( ) = sin( ) + sin() e. f ( ) = cos() + cos( ). Use logarithmic differentiation to find the derivative of the function. y = ( + 1) ( 6) a. b. c. d. e. y = ( 0 + 9)( + 1) ( 6) y = 9( + 1) ( 6) + 0 ( + 1) ( 6) y = 9( + 1) ( 6) + 0( + 1) ( 6) y = 9( + 1) ( 6) + ( + 1) ( 6) y = ( + 1) ( 6) + 0( + 1) ( 6)

14 Stewart - Calculus ET 6e Chapter Form D. The equation of motion is given for a particle, where s is in meters and t is in seconds. Find the acceleration after. seconds. s = sin πt a. 9π m/ s b. 9π m/ s c. 0 m/ s d.. Differentiate the function. 81π m/ s e. 81π m/ s G ( u) = ln u + 6 u 6 a. b. c. d. e. 0 G ( u) = u 6 11 G ( u) = u 6 u 6 G ( u) = u + 6 u 6 G ( u) = (u + 6) u G ( u) = (u + 6) 6. Use logarithmic differentiation to find the derivative of the function. y = a. y = ( 1) b. y = ( 1) c. 0 y = 1 0 d. y = 1 e. y =

15 Stewart - Calculus ET 6e Chapter Form D 7. Use logarithmic differentiation to find the derivative of the function. y = 6 a. y = 6 6 (6ln + 1) b. y = 6 (ln + 1) c. y = 6 6 (ln + 1) d. y = 6 6 (ln + 6) e. y = (ln 6 + 1) 8. Use the linear approimation of the function f ( ) = 9 at a = 0 to approimate the number a..0 b. 0.1 c. 7. d. 7. e.. 9. Determine the values of for which the given linear approimation is accurate to within 0.07 at a = 0. tan a < < 0.8 b < < 0.68 c < < 1. d < < 0.7 e < < The turkey is removed from the oven when its temperature reaches 17 F and is placed on a table in a room where the temperature is 70 F. After 10 minutes the temperature of the turkey is 160 F and after 0 minutes it is 10 F. Use a linear approimation to predict the temperature of the turkey after half an hour. a. 16 b. 10 c. 1 d. 10 e Two cars start moving from the same point. One travels south at 8 mi/h and the other travels west at 70 mi/h. At hat rate is the distance between the cars increasing hours later? Round the result to the nearest hundredth. a. 7. mi/h b. 7.9 mi/h c. 76. mi/h d. 7.9 mi/h e. 7.8 mi/h

16 Stewart - Calculus ET 6e Chapter Form D 1. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 6 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base? 1 a. ft / s 90 b. ft / s 6 c. ft / s d. ft / s 6 e. ft / s 1. A water trough is 10 m long and a cross-section has the shape of an isosceles trapezoid that is 0 cm wide at the bottom, 7 cm wide at the top, and has height cm. If the trough is being filled with water at the rate of 0.7 m /min, how fast is the water level rising when the water is cm deep? Round the result to the nearest hundredth. a. 10. cm/min b. 10 cm/min c. 10. cm/min d. 10. cm/min e. 11. cm/min 1. A plane flying horizontally at an altitude of mi and a speed of 90 mi/h passes directly over a radar station. Find the rate at which the distance from the plane to the station is increasing when it is 10 mi away from the station. a. 9 mi/h b. 8 mi/h c. mi/h d. 970 mi/h e. 870 mi/h 1. Two sides of a triangle are m and m in length and the angle between them is increasing at a rate of 0.0 rad/s. Find the rate at which the area of the triangle is increasing when the angle between the sides of fied length is π. a..0 m /s b m /s c. 0.0 m /s d m /s e. 1.1 m /s

17 Stewart - Calculus ET 6e Chapter Form D 16. If two resistors with resistances R 1 and R are connected in parallel, as in the figure, then the total resistance R measured in ohms ( Ω ), is given by = +. If R 1 and R are increasing at rates R R R of 0.1 Ω /s and 0. Ω /s respectively, how fast is R changing when R1 = 7 and R = 100? Round the result to the nearest thousandth. 1 a Ω /s b. 0.1 Ω /s c Ω /s d Ω /s e Ω /s 17. Use logarithmic differentiation to find the derivative of the function. y = / a. b. c. d. e. y = y = y = y = y = / / / / 1/ (1 ln ) (1 ln ) (1 + ln ) (1 ln ) (1 + ln ) Determine the values of for which the linear approimation 1 6 (1 + ) is accurate to within a. -0. < < 0.9 b. -0. < < 0.9 c < < 0.09 d. -0. < < 0.8 e. -0. < < 0.19

18 Stewart - Calculus ET 6e Chapter Form D d 19. Find ( ln ). d a. b. c. d. e. d d d d ( ( 6 ln ) = ln ) = d ( ln ) = d d 6 ( ln ) = d d d ( 6 ln ) = 0. Gravel is being dumped from a conveyor belt at a rate of 0 ft /min and its coarseness is such that it forms a pile in the shape of a cone whose base diameter and height are always equal. How fast is the height of the pile increasing when the pile is 1 ft high? Round the result to the nearest hundredth. a. 0.1 ft/min b. 1.1 ft/min c. 0.7 ft/min d. 0. ft/min e. 0. ft/min

19 ANSWER KEY Stewart - Calculus ET 6e Chapter Form D 1. e. a, d. b. c. a 6. a 7. c 8. a 9. d 10. d 11. d 1. c 1. c 1. b 1. c 16. d 17. b 18. c 19. d 0 c

20 Stewart - Calculus ET 6e Chapter Form E 1. Find f in terms of g. fg ()() = [ ] fg '()() = a. [ ] b. fgg '()()() = c. fggg '()[][] =+ d. fg '()() = e. fg '()() =. Evaluate. lim Calculate y. y ey = 1. Find y by implicit differentiation. 8 cos sin y = 7. 1 The curve y = 1+ is called a witch of Maria Agnesi. Find an equation of the tangent line to this curve at the point (, 1/ ).

21 Stewart - Calculus ET 6e Chapter Form E 6. Given a diagram with two tangent lines to the parabola y = that pass through the point (0, - ). Find the coordinates of the points where these tangent lines intersect the parabola. a. (±., 6.) b. (±, ) c. (±1.,.) d. (±, 9) e. (±, 6.) 7. Find the equation of the tangent to the curve at the given point. y=+ 1sin,(0,1) 8. Differentiate. f ( ) = e 9. The mass of part of a wire is (1) + kilograms, where is measured in meters from one end of the wire. Find the linear density of one the wire when m= 16. a. 6/kgm b. /kgm c. 7/kgm d. 1./kgm e. none of these 10. Find an equation of the tangent line to the curve + y = 7 at the point (1, 1). 11. If a tank holds 000 gallons of water, and that water can drain from the tank in 0 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as 000 t V = 1. 0 Find the rate at which water is draining from the tank after 6 minutes.

22 Stewart - Calculus ET 6e Chapter Form E 1. The quantity Q of charge in coulombs C that has passed through a point in a wire up to time t (measured in seconds), is given by Q ( t) = t t + t Find the current when t = s. 1. The volume of a cube is increasing at a rate of when the length of an edge is 0cm. 10/min cm. How fast is the surface area increasing 1. Find the derivative of f (). f ( ) = cosh 1. Evaluate lim sinh e 16. Differentiate. g( ) = sec + tan 17. Find the derivative of the function. 1 G( ) = (7 + 10) (8 + 6) The displacement of a particle on a vibrating string is given by the equation s( t) = 8 + sin(πt ) 7 where s is measured in centimeter and t in seconds. Find the velocity of the particle after t seconds. d 19. Find (sin 8 cos 8). d 0. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 0 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth.

23 ANSWER KEY Stewart - Calculus ET 6e Chapter Form E 1. b y = e y e y 1. tan( ) tan( y). y = b 7. y= f ( ) = ( + ) e 9. c 10. y = /min cm 1. cosh( ) + sinh( ) dg( ) = sec( ) tan( ) + sec d (7 + 10) (8 + 6) + 1(7 + 10) (8 + 6) (16 + ) π 18. cos(πt ) sin ( ) cos(9)

24 Stewart - Calculus ET 6e Chapter Form F 1. Differentiate the function. Bycy () = 6 a. b. c. d. e. 7c By () = 6 y c By () = 6y By () = 7 7 6c y By () = 6c 7 y By () = 7c 6 y. Differentiate the function. V ( r) = πr. Differentiate the function. fttt ()() =+ 1. Find an equation of the tangent line to the curve at the given point. y = + 7e, (0, 7). A television camera is positioned,600 ft from the base of a rocket launching pad. The angle of elevation of the camera has to change at the correct rate in order to keep the rocket in sight. Also, the mechanism for focusing the camera has to take into account the increasing distance from the camera to the rising rocket. Let's assume the rocket rises vertically and its speed is 680 ft/s when it has risen,600 ft. If the television camera is always kept aimed at the rocket, how fast is the camera's angle of elevation changing at this moment? Round the result to the nearest thousandth. 6. Evaluate. lim Find f in terms of g. fg= ()()

25 Stewart - Calculus ET 6e Chapter Form F 8. Differentiate. y = The table lists the amount of U.S. cash per capita in circulation as of June 0 in the given year. Use a linear approimation to estimate the amount of cash per capita in circulation in the year 000. t C(t) $17 $ $71 $1, Use the linear approimation of the function f ( ) = 7 at a = 0 to approimate the number Find an equation of the tangent line to the curve. y = at (, 0.) The position function of a particle is given by s = t t t, t 0. When does the particle reach a velocity of m/s? 1. A company makes computer chips from square wafers of silicon. It wants to keep the side length of a wafer very close to 19 mm. The area is A (). Find A (19). 1. The volume of a cube is increasing at a rate of 10/min cm when the length of an edge is 0cm.. How fast is the surface area increasing 1. If ftt ()1=+, find f (). a. b. c. 7 d. e The quantity Q of charge in coulombs C that has passed through a point in a wire up to time t (measured in seconds) is given by Q ( t) = t t + t +. Find the current when t = s. a. 1 b. c. d. e Newton's Law of Gravitation says that the magnitude F of the force eerted by a body of mass m on a GmM body of mass M is F =. r df Find (6). dr

26 Stewart - Calculus ET 6e Chapter Form F 18. Calculate y. yyy +=+ a. b. c. d. yy y = y + 1 yy y = y + y + y = y y = ( 1 ) 1 y y + e. none of these 19. Use differentials to estimate the amount of paint needed to apply a coat of paint 0.18 cm thick to a hemispherical dome with diameter 60 m. a..π b..π c..8π d..8π e..11π 0. Two carts, A and B, are connected by a rope 0 ft long that passes over a pulley (see the figure below). The point Q is on the floor 10 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at the instant when cart A is 8 ft from Q?

27 ANSWER KEY Stewart - Calculus ET 6e Chapter Form F 1. d. V ( r) = πr t. ( ) ftt ()1=+. y = t fgg ()()() =+ (6 + ) 8. y = 6 ( + + 1) y = ( ) /min cm 1. a 16. b 17. GmM b 19. b 0..6

28 Stewart - Calculus ET 6e Chapter Form G 1. Differentiate the function. S( r) = πr. Differentiate the function. Bycy () = 6 a. 7c By () = b. 6 y c By () = c. 7 6y 6c By () = d. y 7 6c By () = e. 7 y By () = 7c y 6. Find the points on the curve y = where the tangent is horizontal. a. (, 71), (, 9) b. (, 88), (, 9) c. (, 71), (, 7) d. (, 88), (, 7) e. (, 7), (, 7). Evaluate. lim Differentiate. Y ( u) = ( u + u )(u 6. Differentiate the function. u ) fttt ()() = The position function of a particle is given by s = t t t, t 0. When does the particle reach a velocity of 19 m/s? 8. If a ball is thrown vertically upward with a velocity of 7 ft/s, then its height after t seconds is s = 7t 6t. What is the maimum height reached by the ball? a. 6 ft b. 16 ft c. 6 ft d. ft e. 81 ft

29 Stewart - Calculus ET 6e Chapter Form G 9. The mass of part of a wire is (1) + kilograms, where is measured in meters from one end of the wire. Find the linear density of one the wire when m=. 10. A telephone line hangs between two poles at 1 m apart in the shape of the catenary y = 0 cosh( / 0), where and y are measured in meters. Find the slope of this curve where it meets the right pole. a b c d e Find the equation of the tangent to the curve at the given point. y=+ 1sin,(0,1) 1. If a snowball melts so that its surface area decreases at a rate of cm /min, find the rate at which the diameter decreases when the diameter is 9 cm. 1. Two cars start moving from the same point. One travels south at 7 mi/h and the other travels west at 0 mi/h. At what rate is the distance between the cars increasing hours later? Round the result to the nearest hundredth. 1. A baseball diamond is a square with side 90 ft. A batter hits the ball and runs toward first base with a speed of 8 ft/s. At what rate is his distance from second base decreasing when he is halfway to first base? Round the result to the nearest hundredth if necessary. 1. The altitude of a triangle is increasing at a rate of cm/min while the area of the triangle is increasing at a rate of cm /min. At what rate is the base of the triangle changing when the altitude is 10 cm and the area is 90 cm.

30 Stewart - Calculus ET 6e Chapter Form G 16. A water trough is 0 m long and a cross-section has the shape of an isosceles trapezoid that is 0 cm wide at the bottom, 60 cm wide at the top, and has height 0 cm. If the trough is being filled with water at the rate of 0.7 m /min, how fast is the water level rising when the water is cm deep? Round the result to the nearest hundredth. 17. A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of m/s how fast is the boat approaching the dock when it is m from the dock? Round the result to the nearest hundredth if necessary. 18. The volume of a cube is increasing at a rate of 10/min cm when the length of an edge is 0cm.. How fast is the surface area increasing 19. Two carts, A and B, are connected by a rope 6 ft long that passes over a pulley (see the figure below). The point Q is on the floor 1 ft directly beneath and between the carts. Cart A is being pulled away from Q at a speed of ft/s. How fast is cart B moving toward Q at the instant when cart A is 8 ft from Q? Round the result to the nearest hundredth. 0. The circumference of a sphere was measured to be 90 cm with a possible error of 0. cm. Use differentials to estimate the maimum error in the calculated volume.

31 ANSWER KEY Stewart - Calculus ET 6e Chapter Form G 1. S ( r) = 8πr. d. d Y ( u) = 6u + u 1 t 6. ( ) 7. 8 ftt ()1=+ t 1 8. b 9. /kgm 10. d 11. y= π /min cm

32 Stewart - Calculus ET 6e Chapter Form H 1. Find the limit if g () =. gg ()() lim. Differentiate the function. 1 6 f ( t) = t t + t. Find an equation of the tangent line to the curve 1( + y ) = 89( y ) at the point (, 1). a. y = b. y = c. y = d. y = e. none of these. In this eercise we estimate the rate at which the total personal income is rising in the Richmond- Petersburg, Virginia, metropolitan area. In 1999, the population of this area was 961,600, and the population was increasing at roughly 9,00 people per year. The average annual income was $0,91 per capita, and this average was increasing at about $1,00 per year (a little above the national average of about $1, yearly). Use the Product Rule and these figures to estimate the rate at which total personal income was rising in the Richmond-Petersburg area in The mass of the part of a metal rod that lies between its left end and a point meters to the right is S =. Find the linear density when is 1 m. a. 9 b. c. 6 d. e Find an equation of the tangent line to the parabola y= at (,). 7. Refer to the law of laminar flow. Consider a blood vessel with radius 0.01 cm, length cm, pressure difference,00 dynes/ cm and viscosity η = Find the velocity of the blood at radius r = Differentiate. g( ) = 8sec + tan 9. Find an equation of the tangent line to the curve y = sec 9cos at the point (/,.) π.

33 Stewart - Calculus ET 6e Chapter Form H 10. Find the limit. lim 0 θ sin(sinθ ) secθ 11. Use logarithmic differentiation to find the derivative of the function Calculate y. ye= ln ( ) 1. Find, correct to three decimal places, the area of the region above the hyperbola y = /( ), below the -ais, and between the lines = - 6 and = The curve with equation y = 6 is called a kampyle of Eudous. Find an equation of the tangent line to this curve at the point (1, ). 1. If ftt ()1=+, find f (). a. /7 b. c. / d. / e. /7 16. Find the first and the second derivatives of the function. G ( r) = r + r 17. Find the equation of the tangent line to the given curve at the specified point. y = e, (0, 0) 18. Find a third-degree polynomial Q such that Q ( 1 ) =, Q ' ( 1 ) = 7, Q '' ( 1 ) = 1, and Q ''' ( 1 ) = Differentiate the function. G( u) = ln u + 6 u 6 0. If f ( ) =, find f ( e ). ln

34 ANSWER KEY Stewart - Calculus ET 6e Chapter Form H f ( t) = t 8t + 1. c. $1,7,6,00. c 6. y = + () dg( ) = 8sec( ) tan( ) + sec d 9. y = 6. ( π / ) ( 1) ln 1. + ye = y = a / 1 / 1 / 9 / r + r, r r 17. y = 18. Q = u