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All Textbook Solutions for Calculus (MindTap Course List)
73E74E75E76E77E78E79E80E81E82E83E84E85E86E87E88E89E90E91E92E93E94E95E96E97E98E99E100E101E102E103EHeat Equation In Exercises 103 and 104, show that the function satisfies the heat equation z/t=c2(2z/x2). z=etsinxc105E106E107E108E109E110E111E112EThink About It The price P (in dollars) of q used car is a function of its initial cost C (in dollars) and its age A (in years). What are the units of P/A ? Is P/A positive or negative? Explain.114E115E116E117E118E119E120EThink About It Let V be the number of applicants to a university, p the charge for food and housing at the university, and r the tuition. Suppose that .N is a function of p and t such that N/p0 and N/t0. What information is gained by noticing that both partials are negative?Investment The value of an investment of $1000 earning 6% compounded annually is V(I,R)=10001+0.06(1R)1+t10 where t is the annual rate of inflation and R is the tax rate for the person making the investment. Calculate Vt(0.03,0.28) and VR(0.03,0.28). Determine whether the tax rate or the rate of inflation is the greater negative factor in the growth of the investment123EApparent Temperature A measure of how hot weather feels to an average person is the Apparent Temperature Index. A model for this index is A=0.885t22.4h+1.20th0.544 where A is the apparent temperature in degrees Celsius. t is the air temperature, and h is the relative humidity in decimal form. (Source: The UMAP Journal) (a) Find At and Ah when t=30 and h= 0.80. (b) Which has a greater effect on A. air temperature or humidity? Explain.125E126E127E128E129EUsing a Function Consider die function f(x,y)=(x3+y3)1/3. (a) Find fx(0,0) and fy(0,0). (b) Determine the points (if any) at which fx(x,y) or fy(x,y) fails to exist131E1E2E3E4E5E6E7E8E9E10EUsing a Differential as an Approximation In Exercises (a) find f (2,1) and f (2.1,1.05) and calculate z , and (b) use the total differential d z to approximate z. f(x,y)=16x2y212E13E14E15E16E17E18E19E20E21E22E23EVolume The possible error involved in measuring each dimension of a right circular cylinder is 0.05 centimeter. The radius is 3 centimeters and the height is 10 centimeters. Approximate the propagated error and the relative error in the calculated volume of the cylinder.25E26E27EResistance The total resistance R (in ohms) of two resistors connected in parallel is given by 1R=1R1+1R2 Approximate the change in R as R1 is increased from 10 ohms to 10.5 ohms and R2 is decreased from 15 ohms to 13 ohms.Power Electrical power P is given by P=E2R where F. is voltage and R is resistance. Approximate the maximum percent error in calculating power when 120 volts is applied to a 2000-ohm resistor and the possible percent errors in measuring F. and R are 3% and 4%, respectively.30EVolume A trough is 16 feet long (see figure). Its cross sections are isosceles triangles with each of the two equal sides measuring 18 inches. The angle between the two equal sides is (a) Write the volume of the trough as a function of and determine die value of such that die volume is a maximum. (b) The maximum error in the linear measurements is one-half inch and die maximum error in the angle measure is? Approximate the change in the maximum volume.Sports A baseball player in center field is playing approximately 330 feet from a television camera that is behind home plate. A batter hits a fly ball that goes to the wall 420 feet from the camera (see figure). (a) The camera turns 9 to follow the play. Approximate the number of feet that the center fielder has to run to make the catch. (b) The position of the center fielder could be in error by as much as 6 feet and the maximum error in measuring the rotation of the camera is 1 Approximate the maximum possible error in the result of part (a).33E34E35EDifferentiability In Exercises 35-38, show that: the function is differentiable by finding values 1 and 2 as designated in the definition of differentiability, and verify that both 1, and 2 approach 0 as (x,y)(0,0). f(x,y)=x2+y237EDifferentiability In Exercises 35-38, show that: the function is differentiable by finding values 1 and 2 as designated in the definition of differentiability, and verify that both 1, and 2 approach 0 as (x,y)(0,0). f(x,y)=5x10y+y3Differentiability In Exercises 39 and 40, use the function to show that fx(0,0) and fy(0,0) both exist but that f is not differentiable at (0, 0). f(x,y)=(3x2yx4+y2(x,y)(0,0)0,(x,y)=(0,0)Differentiability In Exercises 39 and 40, use the function to show that fx(0,0) and fy(0,0) both exist but that f is not differentiable at (0, 0). f(x,y)=(5x2yx3+y3,(x,y)(0,0)0,(x,y)=(0,0)1E2EUsing the Chain Rule In Exercises 3-6, find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t Function Value w=x2+5y t=2 x=2t,y=t4E5E6E7E8EUsing Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=x2+y2+z2,x=cost,y=sint,z=etUsing Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=xycosz,x=t,y=t2,z=arccostUsing Different Methods In Exercises 7-12, find dw/dt (a) by using the appropriate Chain Rule and (b) by converting w to a function of t before differentiating. w=xy+xz+yz,x=t1,y=t21,z=t12E13E14E15E16E17E18E19E20E21EUsing Different Methods In Exercises 19-22, find w/ s and w/ t (a) by using the appropriate Chain Rule and (b) by converting w to a function of s and t before differentiating. w=xcosyz,x=s2,y=t2,z=s2t23EFinding a Derivative Implicitly In Exercises 23-26. differentiate implicitly to Find dy/dx. secxy+tanxy+5=0Finding a Derivative Implicitly In Exercises 23-26. differentiate implicitly to Find dy/dx. lnx2+y2+x+y=4Finding a Derivative Implicitly In Exercises 23-26. differentiate implicitly to Find dy/dx. xx2+y2y2=627E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46EUsing the Chain Rule Let F(u,v) be a function of two variables. Find a formula for f(x) when (a) f(x)=F(4x,4) and (b) f(x)=F(2x,x2)HOW DO YOU SEE IT? The path of an object represented by w=f(x,y) is shown, where r and v are functions of t. The point on the graph represents the position of the object. Determine whether each of the following is positive, negative, or zero. (a) dxdt (b) dydt49E50EMoment of Inertia An annular cylinder has an inside radius of r1, and an outside radius of r2 (see figure). Its moment of inertia is 1=12m(r12+r22) where m is the mass. The two radii are increasing at a rate of 2 centimeters per second. Find the rate at which I is changing at the instant the radii are 6 centimeters and 8 centimeters. (Assume mass is a constant.)52ECauchy-Riemann Equations Given the functions u(x, y) and v(x, y), verify that the Cauchy-Riemann equations ux=vy and uy=vx can be written in polar coordinate form as ur=1rv and vr=1ru54EHomogeneous Function Show that if f(x, y) is homogeneous of degree n, then xfx(x,y)+yfy(x,y)=nf(x,y) [Hint: Let g(t)=f(tx,ty)=tnf(x,y) Find g (t) and then let t=1.]1E2EFinding a Directional DerivativeIn Exercises 36, use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u=cosi+sinj. f(x,y)=x2+y2,P(1,2),=4Finding a Directional DerivativeIn Exercises 36, use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u=cosi+sinj. f(x,y)=yx+y,P(3,0),=65E6E7EFinding a Directional DerivativeIn Exercises 710, use Theorem 13.9 to find the directional derivative of the function at P in the direction of v. f(x,y)=x3y3,P(4,3),v=22(i+j)9E10E11E12E13E14E15E16EFinding the Gradient of a FunctionIn Exercises 1520, find the gradient of the function at given point. z=ln(x2y)x4,(2,3)18E19EFinding the Gradient of a FunctionIn Exercises 1520, find the gradient of the function at given point. w=xtan(y+z),(4,3,1)21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46EUsing a Function In Exercises 37-42, consider the function f(x,y)=3x3y2. Find a unit vector u orthogonal to f(3,2) and calculate Duf(3,2). Discuss the geometric meaning of the result.Using a Function Consider the function f(x,y)=9x2y2. (a) Sketch the graph of f in the first octant and plot the point (1, 2, 4) on the surface. (b) Find Duf(1,2), where u=cosi+sinj, using each given value of 0. (i) =4 (ii) =3 =34 (iv) =2 (c) Find Duf(1,2), where u=vv licintr pqpH i (i) v=3i+j (ii) v=8i6j (iii) v is the vector from (1,1) to (3, 5). (iv) v is the vector from (2,0) to (1, 3). (d) Find f(1,2). (e) Find the maximum value of the directional derivative at (1, 2). (f) Find a unit vector u orthogonal to f(1,2) and calculate Duf(1,2). Discuss the geometric meaning of the result.49E50E51E52E53E54E55E56E57E58EFinding the Path of a Heat-Seeking ParticleIn Exercises 59 and 60, find the path of a heat-seeking particle placed at point P on a metal plate whose temperature at (x,y) is T(x,y). T(x,y)=4002x2y2,P(10,10)60E61E62ETrue or False? In Exercises 6164, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If Duf(x,y) exists, then Duf(x,y)=Duf(x,y).64E65EOcean Floor A team of oceanographers is mapping the ocean floor to assist in the recovery of a sunken ship. Using sonar, they develop the model D=250+30x2+50siny2,0x2,0y2 where D is the depth in meters, and x and y are the distances in kilometers. (a) Use a computer algebra system to graph D. (b) Because the graph in part (a) is showing depth, it is not a map of the ocean floor. How could the model be changed so that the graph of the ocean floor could be obtained? (c) What is the depth of the ship if it is located at the coordinates x=1 and y=0.5? (d) Determine the steepness of the ocean floor in the positive x-direction from the position of the ship. (e) Determine the steepness of the ocean floor in the positive y-direction from the position of the ship. (f) Determine the direction of the greatest rate of change of depth from the position of the ship.67EDirectional DerivativeConsider the function f(x,y)={4xyx2+y2,(x,y)k(0,0)0,(x,y)(0,0) and the unit vector u=12(i+j). Does the directional derivative of f at P(0,0) in the direction of u exist? If f(0,0) were defines as 2 instead of 0, would the directional derivative exist? Explain.CONCEPT CHECK Tangent VectorConsider a point (x0,y0,z0) on a surface given by F(x,y,z)=0. What is the relationship between F(x0,y0,z0) and any tangent vector v at (x0,y0,z0) ? How do you represent this relationship mathematically?2E3E4E5E6E7E8E9E10E11E12E13E14E15E16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E34E35E36E37E38E39E40E41E42E43E44E45E46E47E48E49E50E51EHOW DO YOU SEE IT? The graph shows the ellipsoid x2+4y2+z2=16. Use the graph to determine the equation of the tangent plane at each of the given points. (a) (4,0,0) (b) (0,2,0) (c) (0,0,4)53E54E55E56EWriting a Tangent PlaneIn Exercises 57 and 58, show that the tangent plane to the quadric at the point (x0,y0,z0) can be written in the given form. Ellipsoid: x2a2+y2b2+z2c2=1 Tangent Plane: x0xa2+y0yb2+z0zc2=1Writing a Tangent PlaneIn Exercises 57 and 58, show that the tangent plane to the quadric at the point (x0,y0,z0) can be written in the given form. Hyperboloid: x2a2+y2b2z2c2=1 Tangent Plane: x0xa2+y0yb2+z0zc2=159E60EApproximation Consider the following approximations for a function f(x, y) centered at (0, 0). Linear Approximation: P1(x,y)=f(0,0)+fx(0,0)x+fy(0,0)y Quadratic Approximation: P2(x,y)=f(0,0)+fx(0,0)x+fy(0,0)y+12fxx(0,0)x2+fxy(0,0)xy+12fyy(0,0)y2 Note that the linear approximation is the tangent plane to the surface at (0, 0, f (0, 0)).] (a) Find the linear approximation of f(x,y)=exy centered at (0, 0). (b) Find the quadratic approximation of f(x,y)=exy centered at (0, 0). (c) When x=0 in the quadratic approximation, you obtain the second-degree Taylor polynomial for what function? Answer the same question for y=0. (d) Complete the table. x y f(x, y) P1(x,y) P2(x,y) 0 0 0 0.1 0.2 0.1 0.2 0.5 1 0.5 (e) Use a computer algebra system to graph the surfaces z=f(x,y),z=P1(x,y) and z=P2(x,y)62E63E64ECONCEPT CHECK Function of Two VariablesFor a function of two variables, describe (a) relative minimum, (b) relative maximum, (c) critical point, and (d) saddle point.2E3E4E5E6E7E8E9E10E11E12E13E14E