( ) 8 5 t ( ) e x. y = 7 + 4x. R ' ( t ) = e + 4x e. 29e t 6 t 14 t e t. 24e t 15 t 15 t e t d. 15e t 2 t. 3e x. + 4x e x x.

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1 Name: Class: Date: e x 1 Differentiate y = x 3e x + 4x e x 3e x + 4x e x ( 7 + 4x ) ( 7 + 4x ) 2 x x 7 + 4x 3e + 4x e 3e x + 4x e x ( 7 + 4x ) 2 ( ) 8 5 t ( ) 3 Differentiate R ( t ) = 2t + 3e t. R ' ( t ) = e t 15 t 15 t e t 15e t 2 t R ' ( t ) = e t 6 t 14 t e t 7e t t 3e x 4x e x ( 7 + 4x ) 2 y = 8x 5 2 Differentiate. 4x + 7 y' = 36 ( 4x + 7 ) 2 y' = 76 ( 4x + 7 ) 2 y' = 36 4x + 7 y' = 36 4x 7 R ' ( t ) = e t 16 t 17 t e t 17e t 2 t R ' ( t ) = e t 29 t 2 16 t e t 8e t t R ' ( t ) = e t 14 t 13 t e t 13e t 2 t y' = ( 4x + 7 ) PAGE 1

2 Name: Class: Date: y = t 3 + 8t 4 Differentiate t 4. 6 t t t ( t 4 6) 2 t t t ( t 4 6) 2 6 The curve y = x 1 + x 2 is called a serpentin Find an equation of the tangent line to this curve at the point 2. (, ) y = 533 x + 89 y = 529 x + 83 y = 527 x + 77 y = 531 x + 86 t t t ( t 4 6) 2 t t t ( t 4 6) 2 y = 525 x + 80 t t t ( t 4 6) 2 5 Find the equation of the tangent line to the curve at the point ( 0, 0 ). y = y = 7xe x PAGE 2

3 Name: Class: Date: 7 If f (5) = 4, g (5) = 2, f' (5) = 6 and 9 Let P(x) = F(x)G (x) and Q(x) = F(x), where g' (5) = 2, find the following numbers. G (x) F and G are the functions whose graphs are shown. (a) Find ( f + g )' (5). ( f + g )' (5) = (b) Find ( fg)' (5). ( fg)' (5) = (a) Find P' (0). (c) Find f g ' (5). P' (0) = f g ' (5) = (b) Find Q' (5). (d) Find f f g f f g ' (5) = ' (5). 10 A manufacturer produces bolts of a fabric with a fixed width. The quantity q of this fabric (measured in yards) that is sold is a function of the selling price p (in dollars per yard), so we can write q = f ( p ). Then the total revenue earned with selling price p is R ( p ) = p f ( p ). Given f ( 16 ) = 8000 and f ' ( 16 ) = 320, find R ' ( 16 ) If h ( 8 ) = 7 and h ' ( 8 ) = 4, find d h ( x ). dx x x = PAGE 3

4 Name: Class: Date: 11 At which of the following points are lines passing through the P ( 1, 10 ) y = x point tangent to the curve? x A particle moves according to the law of motion s = f (t ) = t 3 15t t + 29 t 0 measured in seconds and s in feet.,, where t is 10 11, (a) Find the velocity at time t , , (b) What is the velocity after 3 s? ft/s 12 57, (c) When is the particle at rest? s , (d) When is the particle moving in the positive direction? f , (e) Find the total distance traveled during the first 14 s. Total distance = ft (f) Draw a diagram to illustrate the motion of the particl..to be continued PAGE 4

5 Name: Class: Date: continuation 13 If a ball is given a push so that it has an initial velocity of 4 m/s down a certain inclined plane, then the distance it has rolled after t seconds is s = 4t + 2t 2. (a) Find the velocity after 3 s. v(3) = m/s (b) How long does it take for the velocity to reach 32 m/s? t = s 14 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 and it wants to know how the area A ( x ) of a wafer changes when the side length x changes. Find A ' ( 19 ). What does this quantity represent? A ' ( 19 ) represents the rate at which the side length is increasing with respect to the area as x reaches 4 mm. A ' ( 19 ) represents the rate at which the area is increasing with respect to the side length as x reaches 19 mm. A ' ( 19 ) represents the rate at which the area is increasing with respect to the side length as A reaches 38 mm 2. A ' ( 19 ) = 40 mm 2 /mm A ' ( 19 ) = A ' ( 19 ) = 38 mm 2 f. /mm 46 mm 2 /mm 15 A spherical balloon is being inflate Find the rate of increase of the surface area ( S = 4 r 2 ) with respect to the radius r when r = 6 m. 48 m 2 /m 12 m 2 /m 24 m 2 /m 48 m 2 /m 24 m 2 /m PAGE 5

6 Name: Class: Date: 16 If a tank holds 6000 gallons of water, which drains from the bottom of the tank in 40 minutes, then Torricelli's Law gives the volume V of water remaining in the tank after t minutes as V = 6000 V ' (5) = V ' (10) = 1 t t 40 (a) Find the rate at which water is draining from the tank after 5 min. gal/min (b) Find the rate at which water is draining from the tank after 10 min. gal/min (c) Find the rate at which water is draining from the tank after 20 min. 17 Newton's Law of Gravitation says that the magnitude F of the force exerted by a body of mass m on a body of mass M is df dr. F = GmM r 2 where G is the gravitational constant and r is the distance between the bodies. (a) Find (b) Suppose it is known that Earth attracts an object with a force that decreases at the rate of 4 N/km when r = km. How fast does this force change when r = 5000 km? V ' (20) = gal/min (d) Find the rate at which water is draining from the tank after 40 min. F ' (5000) = N/km V ' (40) = gal/min (e) At what time between t = 0 and t = 40 is the water flowing out the most quickly? t = min (f) At what time between t = 0 and t = 40 is the water flowing out the most slowly? t = min 18 The data in the table concern the lactonization of hydroxyvaleric acid at 25 C. They give the concentration C ( t ) of this acid in moles per liter after t minutes. t C ( t ) Plot the points from the table and draw a smooth curve through them as an approximation to the graph of the concentration function. Then draw the tangent at t = 2 and use it to estimate the instantaneous rate of reaction (conventionally expressed as a positive quantity) when t = (moles/ (moles/ (moles/ L)/min L)/min L)/min (moles/ (moles/ L)/min L)/min PAGE 6

7 Name: Class: Date: 19 The frequency of vibrations of a vibrating violin string is given by f = 1 2L T where L is the length of the string, T is its tension, and linear density. (a) Find the rate of change of the frequency with respect to length (when T and are constant). is its 20 Suppose that the cost, in dollars, for a company to produce x pairs of a new line of jeans is C ( x ) = x x x 3. Find the marginal cost function. 3x + 3x x x x x x x x x 2 (b) Find the rate of change of the frequency with respect to tension (when L and are constant) x x 3 (c) Find the rate of change of the frequency with respect to linear density (when L and T are constant). 21 Suppose that the cost, in dollars, for a company to produce x pairs of a new line of jeans is C ( x ) = x x x 3. Find C' ( 150 ). What does this quantity predict? (d) The pitch of a note (how high or low the note sounds) is determined by the frequency f. (The higher the frequency, the higher the pitch.) Use the signs of the derivatives in parts (a), (b) and (c) to determine what happens to the pitch of a note (i) when the effective length of a string is decreased by placing a finger on the string so a shorter portion of the string vibrates. C' 150 predicts the cost of producing the 151st pair. ( ) C' ( 150 ) = C' ( 150 ) = $40.00/ pair $57.18/ pair C' ( 150 ) predicts the cost of producing the first pair. pitch (ii) when the tension is increased by turning a tuning peg. pitch C' ( 150 ) predicts the cost of producing the 150th through the 450th pairs. f. C' ( 150 ) = $22.82/ pair (iii) when the linear density is decreased by switching to another string. pitch 22 Suppose that the cost, in dollars, for a company to produce x pairs of a new line of jeans is C ( x ) = x x x 3. Find the cost of producing the 121 st pair of jeans. $29.31 $32.71 $31.01 $30.16 $31.86 PAGE 7

8 Name: Class: Date: 23 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 0.13 atm/min and V = 12 L and is decreasing at a rate of 0.17 L/min. Find the rate of change of T with respect to time at that instant if n = 10 moles. dt K/min dt dt K/min dt y ( x ) = 6 + sin x 25 Differentiate. 6x + cos x 6x cos x 35 ( 6x + cos x ) 2 x sin x 36 x + cos x 6 cos x x cos x ( x + cos x ) 2 ( 6x + cos x ) 2 36x sin x ( x + 6 cos x ) 2 dt K/min dt dt K/min dt 26 Find an equation of the tangent line to the curve cot(x) at the point 4, 1. y = dt K/min dt 24 Differentiate the function y = e u ( sin u + c u ) with respect to u. e u ( cos u + sin u c u c ) e u ( cos u sin u c u c ) e u ( cos u sin u c u + c ) e u ( cos u + sin u c u c ) e u ( cos u + sin u + c u + c ) PAGE 8

9 Name: Class: Date: 27 A mass on a spring vibrates horizontally on a smooth level surface (see the figure). Its equation of motion is x(t ) = 4 sin t, where t is in seconds and x in centimeters. 29 An object with weight W is dragged along a horizontal plane by a force acting along a rope attached to the object. If the rope makes an angle with the plane, then the magnitude of the force is where k F = kw k sin + cos is a constant called the coefficient of friction. (a) Find the rate of change of F with respect to. (b) When is this rate of change equal to 0? (a) Find the velocity at time t. (b) Find the position of the mass at time t = 2 3. df d W = 60 lb k = 0.5, (c) If and draw the graph of F as a function of and use it to locate the value of for which = 0. (c) Find the velocity of the mass at time t = 2 3. v = cm/s In what direction is it moving at that time? 28 A ladder 22 ft long rests against a vertical wall. Let be the angle between the top of the ladder and the wall and let x be the distance from the bottom of the ladder to the wall. If the bottom of the ladder slides away from the wall, how fast does x change with respect to when =? 3 11 ft/rad 17 ft/rad 15 ft/rad..to be continued 13 ft/rad 19 ft/rad PAGE 9

10 Name: Class: Date: continuation 33 A semicircle with diameter TQ sits on an isosceles triangle TQF to form a region shaped like an ice cream cone, as shown in the figur If B( ) is the area of the semicircle and S( ) is the area of the triangle, find lim B( ) 0 + S( ). lim B( ) 0 + S( ) = lim sin5x 30 Find. sin10x x 0 lim sin 2 31 Find 4t. t 0 2t 2 lim sinx cosx 32 Find. cos2x x 5 /4 PAGE 10