Note for MATH 124 with Professor Long at UA
Note for MATH 124 with Professor Long at UA
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Date Created: 02/06/15
A Chapter 4 Using the Derivatve Section 45 Optimization and Modeling Section 46 Rates amp Related Rates Section 47 L Hopitals Rule Section 48 Parametric Equations Math 124 Section 023 Fall 2007 Instructor Paul Dostert p1 A 45 Optimization and Modeling We wish to use the methods from absolute minmax in applications The general idea is that we wish to 1 Draw and label a diagram 2 Use relations and constraints to come up with a function of only one variable 3 Find the absolute minmax with respect to this variable 4 Determine other unknown values if needed Ex The efficiency of a screw E is given by 9 92 u 6 where 6 is the angle of pitch of the thread and u is the coefficient of friction of the material a positive constant What value of 6 maximizes E E 6gtO p2 A 45 Optimization and Modeling Ex A rectangular storage container with an open top is to have a volume of 10m3 The length of its base is twice the width Material for the base costs 10 per square meter Material for the sides costs 6 per square meter Find the cost of materials for the cheapest such container Ex A box with an open top is to be constructed from a square piece of cardboard 3 ft wide by cutting out a square from each of the four corners and bending up the sides Find the largest volume that such a box can have Ex Which point on the parabola y 32 is nearest to 1 O p3 A 45 Optimization and Modeling Ex A pipeline is to be constructed to connect a station on the shore of a straight section of coastline to a drilling rig that lies 5 km down the coast and 2 km out at sea Find the minimum cost to construct the pipeline given that pipeline costs 4 million dollars per km to lay under water and 2 million dollars per km to lay along shore Ex If you have 100 ft of fencing and want to enclose a rectangular area up against a long straight wall what is the largest area you can enclose Ex Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 3 cm and 4 cm if two sides of the rectangle lie along the legs p4 A 46 Related Rates We wish to determine the rate of change of some quantity We are given a value and the rate of change of a different quantity To solve these problems 1 Plug in given values to determine Write your equation as yt fzt or zt fzt yt if more than 1 variable d where y dt is the quantity you wish to find Take the derivative of y using the chain rule on the left side You should have a 2 term on the left side dy Ex If a snowball melts so that its surface area decreases at a rate of 1 cmZmz39n find the rate at which the radius decreases when the radius is 10 cm p5 A 46 Related Rates Ex The gravitational force F on a rocket at a distance r from the center of the earth is given by k F 72 where k 1013 newtonkm2 When the rocket is 104 km from the center of the earth it is moving away at 02 kmsec How fast is the gravitational force changing at that moment Ex The radius of a spherical balloon is increasing by 2 cmsec At what rate is air being blown into the balloon at the moment when the radius is 10 cm Ex Two cars start moving from the same point One travels south at 60 mih and the other travels west at 25 mih At what rate is the distance between the cars increasing two hours later p6 A 47 Indeterminate Limits Suppose we cannot evaluate a limit 69 1 O lim 00 gt0 13 0 How can we find this limit if it exists There are four situations in which we have a limit that is indeterminate the limit cannot be determined Indeterminate quotient Ooo oooo oo 7 77 7 Ooooo oo oo Indeterminate product ioo O Indeterminate difference 00 oo Indeterminate power 00 000 100 1 00 p8 A 47 L Hopital s Rule L Hopital s Rule Let f and g be differentiable with g y O on an open interval I except perhaps at a a If lim is an indeterminate quotient then gta 9 LC lim amp lim f H1 9 v Ha 9 93 provided the second limit exists Note 0 Do NOT use unless you have an indeterminate quotient The rule holds for a gt ai or a gt ioo For products and differences use algebra to transform limit into an indeterminate quo ent For indeterminate powers use logarithms to make an indeterminate quotient p9 A 47 L Hopital s Rule Ex Find each of the following limits Use L Hopital s rule only when necessary em 1 a 11m 00 gt0 81D 13 332 1 b11m 00 gt0Ij 1 13 a a13 c11m 00 gta 139 CL sin2 a d 139 90131 tan 32 4001 e lim ac gt0 3082 00 gt00 5 A 47 L Hopital s Rule Ex Find each of the following limits Use L Hopital s rule only when necessary a rm 8600 gt OO 1 b lim osoa 00 gtO QC 1 a 33 c 1m 00 gt00 332 332 1 3 CL CC 1 bar tan 00 cl lim 1 gtOO I 1 e 361S1I133 A 48 Parametric Equations We wish to determine the equations of the motion of a particle in the myplane We have a function of two variables each a function of one variable Essentially we want to represent a function in a and y as only a function of t We represent motion in time t in the a direction by a ft and motion in the y direction as y gt Thus the particle at time t is given by ft gt We call these parametric equations with parameter 25 Ex Describe the motion of the particle given by a sint y 2 cost What about a sin3t y 008375 HOW are they different Ex Describe the motion of the particle given by a 21 ZSint y 2 3 boost A 48 Parametric Equations Derivative of parametric equations Suppose we have a ft y gt We have d d a y t t dt f 7 dt 9 What do these describe How do we find dydm dy dydt dZy d9 dajdt7 d332 dajdt Parametric equation of a line How can we find parametric equations representing a line Suppose we have a 1 05 9 9t Suppose ym93 930yo There are infinitely many ways to choose f t and gt Which way is best If we have a point 90 yo and slope m ba then 3375 2 330at yt y0bt 00 lt t lt co A 48 Parametric Equations Ex What curve does 3375 2 yt 1 t represent Ex Find the parametric equations for the line between 3 4 and 2 8 Ex Describe the motion of a particle with position 3375 2 ZSint yt Boost 0 g t g 27r Ex Describe the motion of a particle with position 3375 7500875 yt tsint Ex At time t the position of a particle is given by At what 25 does the curve has a horizontal tangent A vertical tangent A 48 Parametric Equations If the position of a particle is given by zt yt then the instantaneous velocity is given by d3 dy U9 E 9 E in the a and y directions respectively How do we defined the speed The instantaneous speed is given by da 2 dy 2 l a Ex Find the speed for a particle with the given motion Find any times when the particle comes to stop a a cos 275 y sint bIt2 y2t3 A 48 Parametric Equations Ex Determine whether the lines L122 t L2210t8y2t2 are parallel perpendicular or neither If the lines are not parallel find their point of intersection Ex A position of a particle at time t is given by et3 y262t6et9 a Find dyda in terms of t b Find dZyd932 c Eliminate the parameter and write y in terms of 23 d Find dyda and dZyd332 using part c Compare them with parts a and b
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