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All Textbook Solutions for Calculus
44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59E60E61E62E63EMotion of a Particle A particle moves along the plane curve C described by r(t)=ti+t2j. (a) Find the length of C on the interval 0t2. (b) Find the curvature C at t=0,t=1, and t=2. (c) Describe the curvature of C as t changes from t=0tot=2.65E66E67ESpeed The smaller the curvature of a bend in a road, the faster a car can travel. Assume that the maximum speed around a turn is inversely proportional to the square root of the curvature. A car moving on the path y=13x3, where x and y are measured in miles, can safely go 30 miles per hour at (1,13). How fast can it go at (32,98)?69E70E71E72ECurvature Given the polar curve r=ea,a0, use the result of Exercise 69 to find the curvature K and determine the limit of K as (a) and (b) a74E75E76E77E78E79E80E81E82E83E84ETrue or False? In Exercises 83-86, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The arc length of a space curve depends on the parametrization.86E87E88E89E90E91E92EKepler's Laws In Exercises 87-94, you are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector-valued function r. Let r= r , let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of the planet. Prove Kepler's First Law: Each planet moves in an elliptical orbit with the sun as a focus.94E95EProve Keplers Third Law: The square of the period of a planets orbit is proportional to the cube of the mean distance between the planet and the sun.Domain and Continuity In Exercises 1-4, (a) find the domain of r, and (b) determine the interval(s) on which the function is continuous. r(t)=tanti+j+tk2RE3RE4RE5RE6RE7REWriting a Vector-Valued Function In Exercises 7 and 8, represent the line segment from P to Q by a vector-valued function and by a set of parametric equations. P(2, 3, 8), Q(5, 1, 2)9RE10RE11RE12RE13RE14RE15RE16RE17RE18RE19RE20RE21RE22RE23RE24RE25RE26RE27RE28RE29REEvaluating a Definite Integral In Exercises 31-34, evaluate the definite integral. 03(ti+tj+4tk)dt31RE32RE33RE34RE35RE36RE37RE38RE39RE57095-12-40RE-Question-Digital.docx Projectile Motion In Exercises 3942, use the model for projectile motion, assuming there is no air resistance. [a(t) = 32 feet per second per second or a(t) = 9.8 meters per second per second] A baseball is hit from a height of 3.5 feet above the ground with an initial velocity of 120 feet per second and at an angle of 30 above the horizontal. Find the maximum height reached by the baseball. Determine whether it will clear an 8-foot-high fence located 375 feet from home plate.41RE42RE43RE44RE45RE46RE47RE48RE49RE50RE51RE52RE53RE54RE55RE56RE57RE58RE59RE60RE61RE62RE63RE64RE65RE66RE67RE68RE69RE70RE71RE1PS2PS3PS4PSCycloid Consider one arch of the cycloid r()=(sin)i+(1cos)j,02 as shown in the figure. Let s() be the arc length from the highest point on the arch to the point (x(),y()), and let ()=1/K be the radius of curvature at the point (x(),y()). Show that s and are related by the equation s2+2=16. (This equation is called a natural equation for the curve.)6PS7PS8PS9PS10PS11PS12PS13PSFerris Wheel You want to toss an object to a friend who is riding a Ferris wheel (see figure). The following parametric equation give the path of the friend r1(t) and the path of the object r2(t). Distance is measured in meters, and time is measured in seconds. r1(t)=15(sint10)i+(1615cost10)jr2(t)=[ 228.03(tt0) ]i+[1+11.47(tt0)4.9(tt0)2]jDetermining Whether a Graph Is a Function In Exercises 1 and 2, use the graph to determine whether z is a function of x and y. Explain.
Determine whether graph is a function. Use the graph to determine whether z is a function of x and y. Explain.3EDetermining Whether an Equation Is a Function In Exercises 5-8, determine whether z is a function of x and y. xz2+2xyy2=4Determining Whether an Equation Is a Function In Exercises 5-8, determine whether z is a function of x and y. x24+y29+x2=16E7E8E57095-13.1-9E-Question-Digital.docx Evaluating a Function In Exercises 718, find and simplify the function values. f(x,y)=xey(a)(5,0)(b)(0,1)(c)(2,3)(d)(5,y)(e)(x,2)(f)(t,t)10E11E12E13E14EEvaluating a Function In Exercises 9-20, evaluate the function at the given values of the independent variables. Simplify the results. g(x,y)ty(2t3)dt (a) g(4,0) (b) g(4,1) (c) g(4,12) (d) g(32,0)16E17E18E19E20E21E22E23E24E25E26E27EFinding the Domain and Range of a Function In Exercises 21-32, find the domain and range of the function. f(x,y)=arcsin(y/x)29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51E52E53ESketching a Contour Map In Exercises 51-58, describe the level curves of the function. Sketch a contour map of the surface using level curves for the given c-values. f(x,y)=exy/2,c=2,3,4,12,36,1455E56E57E58ESraphing Level Curves Using Technology In Exercises 59- 62, use a graphing utility to graph six level curves of the function. g(x,y)=81+x2+y260E61EUsing Level Curves All of the level curves of the surface given by z=f(x,y) are concentric circles. Does this imply that the graph of f is a hemisphere? Illustrate your answer with an example.63EConjecture Consider the function f(x,y)=xy, for ran and x0andy0. (a) Sketch the graph of the surface given by f. (b) Make a conjecture about the relationship between the graphs of f and g(x,y)=f(x,y)3. Explain your reasoning. (c) Repeat part (b) for g(x,y)=f(x,y) (d) Repeat part (b) for g(x,y)=12f(x,y). (e) On the surface in part (a), sketch the graph of z=f(x,y).65E66E67EInvestment A principal of $5000 is deposited in a savings account that earns interest at a rate of r (written as a decimal), compounded continuously. The amount A(r, t) after t years is A(r,t)=5000ert Use this function of two variables to complete the table. Number of Years Rate 5 10 15 20 0.02 0.03 0.04 0.0569E70E71E72E73E74E75EQueuing Model The average length of time that a customer waits in line for service is W(x,y)=1xy,xy where is die average arrival rate, written as the number of customers per unit of time, and x is the average service rate, written in the same units. Evaluate each of the following. (a) W(15, 9) (b) W(15, 13) (c) W(12, 7) (d) W(5, 2)77. Temperature Distribution The temperature T (in degrees Celsius) at any point (x, y) in a circular steel plate of radius 10 meters is
T = 600 – 0.75x2 –0.75y2
where x and y are measured in meters. Sketch some of the isothermal curves.
Electric Potential The electric potential V at any point (x. y) is V(x,y)=525+x2+y2 Sketch the equipotential curves for V=12,V=13,andV=14.79E80E81E82E83E84E85E86E87E88E89E90E91E1E2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26EFinding a Limit In Exercises 25-36, find the limit (if it exists). If the limit does not exist, explain why. lim(x,y)(0,0)xyxy28EFinding a Limit In Exercises 25-36, find the limit (if it exists). If the limit does not exist, explain why. lim(x,y)(0,0)x+yx2+yFinding a Limit In Exercises 25-36, find the limit (if it exists). If the limit does not exist, explain why. lim(x,y)(0,0)xx2y231E32E33E34E83E84E35E36E37E38E39E40ELimit Consider lim(x,y)(0,0)x2+y2xy (see figure). (a) Determine (if possible) the limit along any line of the form y=ax. (b) Determine (if possible) the limit along the parabola y=x1. (c) Does the limit exist? Explain.74E41EComparing Continuity In Exercises 49 and 50, discuss the continuity of the functions f and g. Explain any differences. f(x,y)={x2+2xy2=y2x2+y2,(x,y)(0,0)0,(x,y)=(0,0)g(x,y)={x2+2xy2+y2x2+y2,(x,y)(0,0)1,(x,y)=(0,0)43E44E45E46E47E48E49E50E51E52E53E54E55E56E57E58E59EContinuity of a Composite Function In Exercises 67-70, discuss the continuity of the composite function fg. f(t)=1tg(x,y)=x2+y261E62E63E64E65E66E67E