8/6/2010 Assignment Previewer

Transcription

1 8//2010 Assignment Previewer Week 8 Friday Homework ( ) Question Question Detailsscalcet 3.9.ae.05.nva [129124] EXAMPLE 5 A man walks along a straight path at a speed of 4 ft/s. A searchlight is located on the ground 4 ft from the path and is kept focused on the man. At what rate is the searchlight rotating when the man is 3 ft from the point on the path closest to the searchlight? SOLUTION We draw the figure to the left and let x be the distance from the man to the point on the path closest to the searchlight. We let A be the angle between the beam of the searchlight and the perpendicular to the path. We are given that dx / dt 4 ft/s and are asked to find da / dt when x 3. The equation that relates x and A can be written from the figure: x tan A x tan A Differentiating each side with respect to t, we get Video Example Online Textbook dx dt 4 so da 1 dx dt 4 dt cos(a) (4) When x 3, the length of the beam is 5, so cos(a) 4 / 5 and da dt ( ) 2 The searchlight is rotating at a rate of rad/s. webassign.net/ /control.pl 1/

2 8//2010 Assignment Previewer 2. Question DetailsSCalcET 3.9.Tut.01. [57130] 3. Question DetailsSCalcET 3.9.Tut.04. [5940] webassign.net/ /control.pl 2/

3 8//2010 Assignment Previewer 4. Question DetailsSCalcET 3.9.Tut.05. [5838] 5. Question DetailsSCalcET [3275] Each side of a square is increasing at a rate of 5 cm/s. At what rate is the area of the square increasing when the area of the square is 4 cm 2? cm 2 /s. Question DetailsSCalcET MI. [ ] The length of a rectangle is increasing at a rate of 5 cm/s and its width is increasing at a rate of cm/s. When the length is 13 cm and the width is 8 cm, how fast is the area of the rectangle increasing? cm 2 /s Tutorial 7. Question DetailsSCalcET MI. [138953] An upright cylindrical tank with radius 5 m is being filled with water at a rate of 4 m 3 /min. How fast is the height of the water increasing? (Round the answer to four decimal places.) m/min Tutorial 8. Question DetailsSCalcET [ ] If y 3x 3 3x and dx 5, find dy when x 2. dt dt dy dt 9. Question DetailsSCalcET [ ] A plane flying horizontally at an altitude of 5 mi and a speed of 40 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 to the nearest whole number.) mi/h 10. Question DetailsSCalcET [807085] A street light is mounted at the top of a 15 foot tall pole. A man ft tall walks away from the pole with a speed of 4 ft/s along a straight path. How fast is the tip of his shadow moving when he is 40 ft from the pole? ft/s webassign.net/ /control.pl 3/

4 8//2010 Assignment Previewer 11. Question DetailsSCalcET [32582] At 2 P.M., ship A is 150 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance between the ships changing at P.M.? km/h 12. Question DetailsSCalcET 3.10.AE.01. [ ] EXAMPLE 1 Find the linearization of the function x 7 at a 2 and use it to approximate the numbers 8.98 and Are these approximations overestimates or underestimates? SOLUTION The derivative of f(x) (x 7) 1/2 is f '(x) (x 7) 1 and so we have f(2) and f '(2). Putting these values into Equation 2, we see that the linearization is Video Example Online Textbook L(x) f( ) f '( )(x - ) 1 (x - ) The corresponding linear approximation is x 7 In particular, we have 8.98 and The linear approximation is illustrated in the figure to the left. We see that, indeed, that tangent line approximation is a good approximation to the given function when it is near 2. We also see that our approximations are overestimates because the tangent line lies above the curve. Of course, a calculator could give us approximations for 8.98 and 9.01, but the linear approximation gives an approximation over an entire interval. 13. Question DetailsSCalcET 3.10.AE.04. [57044] Video Example Online Textbook EXAMPLE 4 The radius of a sphere was measured and found to be 1 cm with a possible error in measurement of at most 0.05 cm. What is the maximum error in using this value of the radius to compute the volume of the sphere? SOLUTION If the radius of the sphere is r then its volume is V ( 4 / 3 )πr 3. If the error in the measured value of V is V, which can be approximated by the differential dv πr dr When r 1 and dr 0.05, this becomes dv 4π( ) 2 (Round to the nearest integer.) The maximum error in the calculated volume is about cm 3. webassign.net/ /control.pl 4/

5 8//2010 Assignment Previewer 14. Question DetailsSCalcET 3.10.Tut.01. [57171] 15. Question DetailsSCalcET MI. [13899] Find the linearization L(x) of the function at a -1. f(x) x 4 4x 2 L(x) Tutorial 1. Question DetailsSCalcET [ ] Find the linearization of the function given below at the specified point. f(x) ln(x) at a 1 L(x) 17. Question DetailsSCalcET [128882] Find the linearization L(x) of the function at a 3π/2. f(x) cos(x) L(x) 18. Question DetailsSCalcET [128879] Find the linear approximation of the function f(x) 1-x at a 0 and use it to approximate the numbers 0.9 and (Round your answers to four decimal places.) L(x) webassign.net/ /control.pl 5/

6 8//2010 Assignment Previewer 19. Question DetailsSCalcET [128833] Find the differential dy. y e x/2 dy Evaluate dy for x 0 and dx 0.05 dy 20. Question DetailsSCalcET [7847] Compute y and dy for x 4 and dx x 0.2. (Round the answers to three decimal places.) y 4x - x 2 dy y Assignment Details Name (AID): Week 8 Friday Homework ( ) Submissions Allow ed: 5 Category: Homework Code: Locked: No Author: Jernigan, John ( jjernigan@ccp.edu ) Last Saved: Jul 27, :20 PM EDT Permission: Protected Randomization: Person Which graded: Last Feedback Settings Before due date Question Score Assignment Score Publish Essay Scores Question Part Score Mark Add Practice Button Help/Hints Response Save Work After due date Question Score Assignment Score Publish Essay Scores Key Question Part Score Solution Mark Add Practice Button Help/Hints Response webassign.net/ /control.pl /