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All Textbook Solutions for Calculus

62E 63E 64E 65E 66E 67E 68E 69E 70E 71E 72E 73E 74E 75E 76E 77E 78E 79E 80E 81E HOW DO YOU SEE IT? The four figures below are graphs of the vector-valued function r(t)=4costi+4sintj+t4k Match each of the four graphs with the point in space from which the helix is viewed. (i) (0, 0, 20) (ii) (20, 0, 0) (iii) (-20, 0, 0) (iv) (10, 20, 10)83E 84E 85E 86E 87E 88E 89E 90E 91E 92E 93E 94E 1E 2E 3E 4E 5E 6E 7E 8E Finding a Derivative In Exercises 9–20, find r'(t). 9. 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E Finding a Derivative In Exercises 11-18, find r(t). r(t)=arcsint,arccost,0 21E 22E Higher-Order DifferentiationIn Exercises 1922, find (a) r(t), (b) r(t), and (c) r(t)r(t). r(t)=4costi+4sintj 24E 25E 26E 27E 28E 29E 30E Finding Intervals on Which a Curve Is Smooth In Exercises 2734, find the open interval(s) on which the curve given by the vector-valued function is smooth. r()=2cos3i+3sin3j,02 32E 33E 34E 35E 36E 37E 38E Using Properties of the Derivative In Exercises 35 and 36, use the properties of the derivative to find the following. (a) r(t) (b) ddt[3r(t)-u(t)] (c) ddt[(5t)u(t)] (d) ddt[r(t)u(t)] (e) ddt[r(t)u(t)] (f) ddt[r(2t)] r(t)=ti+3tj+t2k,u(t)=4ti+t2j+t3k Using Properties of the DerivativeIn Exercises 35 and 36, use the properties of the derivative to find the following. (a) r(t) (b) ddt[3r(t)u(t)] (c) ddt[(5t)u(t)] (d) ddt[r(t)u(t)] (e) ddt[r(t)u(t)] (f) ddt[r(2t)] r(t)=t,2sint,2cost,u(t)=1t,2sint,2cost Using Two MethodsIn Exercises 37 and 38, find (a) ddt[r(t)u(t)] and (b) ddt[r(t)u(t)] in two different ways. (i) Find the product first, then differentiate. (ii) Apply the properties of Theorem 12.2. r(t)=ti+2t2j+t3k,u(t)=t4k 42E 43E 44E 45E 46E 47E 49E 48E Finding an Indefinite Integral In Exercises 39-46, find the indefinite integral. (etsinti+cottj)dt 51E Evaluating a Definite Integral In Exercises 47-52, evaluate the definite integral. 11(ti+t3j+t3k)dt 53E 54E Evaluating a Definite Integral In Exercises 47-52, evaluate the definite integral. 02(ti+etjtetk)dt 56E 57E 58E 59E 60E 61E Finding an Antiderivative In Exercises 53-58, find r(t) that satisfies the initial condition(s). r(t)=11+t2i+1t2j+1tk,r(1)=2i 65E 64E 63. Differentiation State the definition of the derivative of a vector-valued function. Describe how to find the derivative of a vector-valued function and give its geometric interpretation. 66E 67E 68E 69E 70E 71E 72E 73E 74E Particle MotionA particle moves in the xy-plane along the curve represented by the vector-valued function r(t)=(tsint)i+(1cost)j. (a) Use a graphing utility to graph r. Describe the curve. (b) Find the minimum and maximum values of r and r.Particle MotionA particle moves in the yz-plane along the curve represented by the vector-valued function r(t)=(2cost)j+(3sint)k. (a) Describe the curve. (b) Find the minimum and maximum values of r and r.77E 78E 79E True or False? In Exercises 73-76, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The definite integral of a vector-valued function is a real number.81E 82E 1E 2E 3E 4E 5E 6E 7E 8E Finding Velocity and Acceleration Vectors in Space In Exercises 11-20, the position vector r describes the path or an object moving in space. (a) Kind the velocity vector, speed, and acceleration vector of the object. (b) Evaluate the velocity vector and acceleration vector of the object at the given value of t. Position Vector Time r(t)=ti+5tj+3tk t=1 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 21E 22E 23E 24E 25E 26E 27E A baseball player at second base throws a ball 90 feet to the player at first base. The ball is released at a point 5 feet above the ground with an initial speed of 50 miles per hour and at an angle of 15 above the horizontal. At what height does the player at first base catch the ball?29E 30E 31E 32E 33E A bomber is flying horizontally at an altitude of 30.000 feet with a speed of 540 miles per hour (see figure). Wien should the bomb be released for it to hit the target? (Give your answer in terms of the angle of depression from the plane to the target.) What is the speed of the bomb at the time of impact?35E 36E 37E 38E Projectile Motion In Exercises 41 and 42, use the model for projectile motion, assuming there is no air resistance and g=9.8 meters per second per second. Determine the maximum height and range of a projectile tired at a height of 1.5 meters above tire ground with an initial speed of 100 meters per second and at an angle of 30 above the horizontal.40E 41E 42E 43E 44E 45E 46E 47E 48E 57095-12.3-49E-Question-Digital.docx Circular Motion In Exercises 49 and 50, use the results of Exercises 4548. A stone weighing 1 pound is attached to a two-foot string and is whirled horizontally (see figure). The string will break under a force of 10 pounds. Find the maximum speed the stone can attain without breaking the string. (Use F = ma, where m=132.)50E 51E 52E 53E 54E Particle Motion Consider a particle moving on an elliptical path described by r(t)=acosti+bsintj where =d/dt is the constant angular speed. (a) Find the velocity vector. What is the speed of the particle? (b) Find the acceleration vector and show that its direction is always toward the center of the ellipse.57E 55E 58E 59E 60E 61E 62E 44E 1E 2E 3E 4E Finding the Unit Tangent Vector In Exercises 3-8, find the unit tangent vector to the curve at the specified value of the parameter. r(t)=3tilntj,t=e 6E 7E 8E 9E 10E 11E 12E 13E 14E 15E 16E 17E 18E 19E 20E 33E 34E 35E 36E 21E 22E 23E 24E 25E 26E 27E 28E 29E 30E 31E 32E 37E 38E 39E 40E 41E Finding Vectors In Exercises 37–42, find T(t), N(t), , and at the given time t for the space curve r(t). [Hint: Find a(t), T(t), , and . Solve for N in the equation a(t) = T + N.] Vector-Valued Function Time 42. 43E 45E 46E 47E 48E Cycloidal Motion The figure shows the path of a particle modeled by the vector-valued function r(t)=(tsint,1cost). The figure also shows the vectors v(t)/v(t)anda(t)/a(t) at the indicated values of t (a) Find a1 and a2 at t=12,t=1,andt=32 (b) Determine whether the speed of the particle is increasing or decreasing at each of the indicated values of t Give reasons for your answers.Motion Along an Involute of a Circle The figure shows a particle moving along a path modeled by r(t)=cost+tsint,sinttcost. The figure also shows the vectors v(t) and a(t) for t=1 and t=2. (a) Find aT and aN at t=1 and t=2. (b) Determine whether the speed of the particle is increasing or decreasing at each of the indicated values of t. Give reasons for your answers.51E 52E 53E 54E 55E 56E 57E 58E 59E 60E Projectile Motion Find the tangential and normal components of acceleration for a projectile fired at an angle with the horizontal at an initial speed of v0. What are the components when the projectile is at its maximum height?62E 63E 64E Air Traffic ControlBecause of a storm, ground controllers instruct the pilot of a plane flying at an altitude of 4 miles to make a 90 turn and climb to an altitude of 4.2 miles. The model for the path of the plane during this maneuver is r(t)=10cos10t,10sin10t,4+4t,0t120 where t is the time in hours and r is the distance in miles. (a) Determine the speed of the plane. (b) Calculate aT and aN. Why is one of these equal to 0?Projectile Motion A plane flying at an altitude of 36,000 feet at a speed of 600 miles per hour releases a bomb. Find the tangential and normal components of acceleration acting on the bomb.67E 68E 69E 70E 71E 72E 73E 74E 75E Proof Prove that the principal unit normal vector N points toward the concave side of a plane curve.77E 78E 79E 80E 1E 2E 3E 4E 5E 6E 57095-12.5-7E-Question-Digital.docx Projectile Motion A baseball is hit 3 feet above the ground at 100 feet per second and at an angle of 45 with respect to the ground. Find the vector-valued function for the path of the baseball. Find the maximum height. Find the range. Find the arc length of the trajectory.8E 9E 10E 11E 12E 13E 14E 15E Investigation Consider the helix represented by the vector-valued function r(t)=2cost,2sint,t. (a) Write the length of the arc s on the helix as a function of t by evaluating the integral s=0t[x(u)]2+[y(u)]2+[z(u)]2du (b) Solve for t in the relationship derived in part (a), and substitute the result into the original set of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter. s. (c) Find the coordinates of the point on the helix for arc lengths s=5ands=4. (d) Verify that r(s)=1.16E 18E 19E 20E Finding Curvature In Exercises 19–22, find the curvature K of the curve, where s is the arc length parameter. 21. Helix in Exercise 17: 22E 23E 24E 25E 26E 27E 28E Finding CurvatureIn Exercises 2936, find the curvature of the curve. r(t)=4cos2ti+4sin2tj 30E 31E 32E 33E 34E 35E 36E 37E 38E 39E 40E 41E 42E Finding Curvature in Rectangular Coordinates In Exercises 41–48, find the curvature and radius of curvature of the plane curve at the given value of x. 43.

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