Calculus I Notes on Related Rates.

<strong>Calculus</strong> I<strong>Notes</strong> on Related Rates.Quite often in an application you can have 2 or more related quantities that are in turn changing with time. Thus, their rates of changeare related or we have related rates.Finding the Equation of Related Rates:To find the relation of the rates of change, we use implicit differentiation, taking the derivative with respect to time, t.Example #1: Find an equation of the related rate for►□3x cot(y) 7y .3x cot(y) 7ydddt dt3x csc (y) 73x cot(y) 7y2 dx 2 dy dydt dt dt2 dx dy 2 dy3x 7 csc (y)dt dt dt2 dx2 dy3x 7 csc (y) dtdt2 6Example #2: Find an equation of the related rate for y sec(x) 5z .►2 6y sec(x) 5z2y sec(x) y sec(x) tan(x) 30z□Solving Problems Involving Related Rates:Example #3: If the radius of a circle is growing atr 24in ?►a) Using the circumference formula for a circle.b) Using the area formula for a circle.□dy 2 dx5 dzdt dtdtdy 2 dx 5 dz2ysec(x) y sec(x) tan(x) 30zdt dt dtinsSeminole State:Rickman <strong>Notes</strong> on Related Rates. Page #1 of 27 , how fast is the a) circumference growing, b) the area growing whendAdtC 2r 2 2 (7 )14dCdtdrdtinsins2A rdr2rdtin2 (24in)(7 )s336Two notes on the previous example. First, notice that while the circumference, C, was growing at a constant rate, the area wasgrowing at a rate proportional to r. Thus, the area was growing at an increasingly faster rate.Also notice that the units for say dAdtwere the units for A over the units for t.3Example #4: If a spherical balloon is inflating at a rate of 3 , at what rate is the radius increasing when r 5cm.cms►3First note that the inflation rate, 3 , is a rate of change in the volume, dV . Thus, we start with the volume formula for a sphere.□cms43dVdt3cm s 3 cm dr100s dt2ins3V r2 dr4rdt3 4 (5cm)dt2 drdt

Applications:Example #5: A rectangular pool is 20ft by 50ft. The depth changes from 5ft to 12ft along the 50ft side. The water is filling the pool at2 . How fast is the depth of the water, measured at the deep end, rising when the depth is a) 1ft, b) 4ft, and c) 6ft? Round answersft 3minto the 4 th decimal place.►First, notice that all 3 given depths are before the bottom of the pool is covered with water since the shallow end is 7ft above the deepend. Thus, we can ignore the change in the formula for the volume of water at a depth of 7ft.1V hL(20)2 10hLh depth 50 10h h□Thus,L length of water at the toph L72hh5007dV 1000 dh7dt 7 dt50 1000 dh50h L2h7 dt7 7 dh h500 dt7 dh500h dta) .0140 b)dh 7 ftdt 500(1) min .0035 c)dh 7 ftdt 500(4) min .0023dh 7 ftdt 500(6) minmExample #6: One end of a 5m ramp is being lowered at a rate of 2 . The other end is on wheels and moves freely on the ground.minAssuming the end being lowered is moving vertically only, at what rate is the bottom end moving when it’s 4m horizontally from thetop end.►Since the top end is being lowered at2 , dy m -2 .mmindtmin□2 2x y 252 24 y 25216 y 252y 9y 3m2 2x y 25dx dy dt dtdx dy dt dtdy dxy -xdt dtdxdy dtdt2x 2y 0x y 0-x -(4)(-2)y 38 m3 minExample #7: If a spotlight is rotating atlight.►□y tan 6y 6 tan 6sec 6sec 2 12secdy 2 d2dtdt2rad2s, what is the slowest rate the light is moving across a wall which is 6ft away from theSince the minimum positive value of sec is 1.2Min rate 12 (1)12ftsExample #8: If a truck with a camera in the back is speeding away from a launch pad at 50mph, and a rocket is moving up at 40mph,how fast will the camera have to rotate to keep centered on the rocket at the time that the truck is 0.4miles away from the launch padand the rocket is 0.3miles above the ground? Give answer in degrees per min. Round to the hundredths.►□y tan xy x tan tan x sec dy dx2 ddt dt dt0.3 0.52d40 (50) 0.4 (0.4) 0.4 dtd40 37.5 0.625dt4037.5 d0.625 dtd4 dt 1hr 43.82per mindrad 180dt hr rad 60minSeminole State:Rickman <strong>Notes</strong> on Related Rates. Page #2 of 2