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mathematics
calculus 4th

Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions

Find a parametrization of the tangent line at the point indicated.r(t) = (cos t, sin 2t), t = π/3
Find a parametrization of the tangent line at the point indicated.r(t) = (1 − t2, 5t, 2t3), t = 2
Find a parametrization of the tangent line at the point indicated.r(t) =(6t, 4t2, 2t3), t = −2
Find a parametrization of the tangent line at the point indicated.r(s) = 4s−1i − 8/3s−3k, s = 2
Use Example 4 to calculate d/dt (r × r'), where r(t) = (t, t2, et). d EXAMPLE 4 Prove that (r(1) x r'(1)) = r(t) x r' (1).
Find a parametrization of the tangent line at the point indicated.r(s) = (ln s)i + s−1j + 9sk, s = 1
Let r(t) = (3 cos t, 5 sin t, 4 cos t). Show that r(t) is constant and conclude, using Example 7, that r(t) and r(t) are orthogonal. Then compute r(t) and verify directly that r' (t) is orthogonal to r(t). EXAMPLE 7 Orthogonality of r and r' when r Has Constant Length Prove that if r(1) and r' (1)
Show that the derivative of the norm is not equal to the norm of the derivative by verifying that ΙΙr(t)'ΙΙ r(t) for r(t) = (t, 1, 1).
Evaluate the integrals. S ( + 41,41 - 1) dt 4t -2
Show that d/dt (a × r) = a × r' for any constant vector a.
Evaluate the integrals. S J (1471.2) 1 + $ 1+s ds
Evaluate the integrals. 1-2 (ui+uj) du
Evaluate the integrals. S (- sint, 6t, 2t + cos 2t) dt
Evaluate the integrals. So (e + + (te-i+tln(t + 1)j) dt
Evaluate the integrals. 1 1 1 u4' us S (= 1/2 du
Evaluate the integrals. S* (tr- i +4rj - 81/2 k) dt
Evaluate the integrals. So (3si + 6s2j+9k) ds
Find both the general solution of the differential equation and the solution with the given initial condition. dr dt = (1-2t, 4t), r(0) = (3, 1)
Find both the general solution of the differential equation and the solution with the given initial condition.r '(t) = (sin 3t, sin 3t, t), () = (2.4.)
Find both the general solution of the differential equation and the solution with the given initial condition.r '(t) = i − j, r(0) = 2i + 3k
Find both the general solution of the differential equation and the solution with the given initial condition. r"(t) = 16k, r(0)= (1,0, 0), r'(0) = (0, 1,0)
Find both the general solution of the differential equation and the solution with the given initial condition.r '(t) = t2i + 5tj + k, r(1) = j + 2k
Find both the general solution of the differential equation and the solution with the given initial condition. r"(t) = (e, sint, cost), r(0) = (1,0,1), r'(0) = (0, 2,2)
Find both the general solution of the differential equation and the solution with the given initial condition. r"(t) = (-2,-1, 1), r(1) = (0,0,1), r' (1) = (2,0,0)
Find both the general solution of the differential equation and the solution with the given initial condition. r"(t) = (0,2,0), r(3)= (1, 1,0), r'(3) = (0,0,1)
Find the location at t = 3 of a particle whose path (Figure 8) satisfies dr = (2-(+1221-4). dt 1) 10- 5 (3,8) t=0 5 t = 3 + + 10 15 20 r(0) = (3,8) 25
Find the location and velocity at t = 4 of a particle whose path satisfies dr | dt = (2t-1/, 6, 8t), r(1) = (4,9, 2
A fighter plane, which can shoot a laser beam straight ahead, travels along the path r(t) = (5 − t, 21 − t2, 3 − t3/27). Show that there is precisely one time t at which the pilot can hit a target located at the origin.
The fighter plane of Exercise 59 travels along the path r(t) = (t − t3, 12 − t2, 3 − t). Show that the pilot cannot hit any target on the x-axis.Data From Exercise 59A fighter plane, which can shoot a laser beam straight ahead, travels along the path r(t) = (5 − t, 21 − t2, 3 − t3/27).
Find all solutions to r '(t) = v with initial condition r(1) = w, where v and w are constant vectors in R3.
Let u be a constant vector in R3. Find the solution of the equation r '(t) = (sin t)u satisfying r '(0) = 0.
Prove that the Bernoulli spiral (Figure 9) with parametrization r(t) = et cos 4t, et sin 4t has the property that the angle ψ between the position vector and the tangent vector is constant. Find the angle ψ in degrees. -10- t=0 1=2 20 -X
Find all solutions to r '(t) = 2r(t), where r(t) is a vector-valued function in 3-space.
A curve in polar form r = ƒ(θ) has parametrizationCompute r(θ) × r(θ) and r(θ) · r(θ). r(0) = f(0) (cos , sin ) Let be the angle between the radial and tangent vectors (Figure 10). Prove that r f(0) dr/de f'(0) tan = =
Show that w(t) =(sin(3t + 4), sin(3t − 2), cos 3t) satisfies the differential equation w"(t) = −9w(t).
Prove that if ΙΙr(t)ΙΙ takes on a local minimum or maximum value at t0, then r(t0) is orthogonal to r'(t0). Explain how this result is related to Figure 11. Observe that if r(t0) is a minimum, then r(t) is tangent at t0 to the sphere of radius ΙΙr(t0)ΙΙcentered at the origin. N r'(to) r(to)
Use FTC II from single-variable calculus to prove the second part of the Fundamental Theorem of Calculus for Vector-Valued Functions. Fundamental Theorem of Calculus for Vector-Valued Functions Part I: If r(t) is continuous on [a, b], and R(1) is an antiderivative of r(t), then S." r(t) dt = R(b) -
Newton’s Second Law of Motion in vector form states that F = dp/dt, where F is the force acting on an object of mass m and p = mr'(t) is the object’s momentum. The analogs of force and momentum for rotational motion are the torque τ = r × F and angular momentum J = r(t) × p(t) Use the Second
Let r(t) = (x(t), y(t)) trace a plane curve C. Assume that x'(t0) ≠ 0. Show that the slope of the tangent vector r(t0) is equal to the slope dy/dx of the curve at r(t0).
Prove the Cross Product Rule [Eq. (5)]. Cross Product Rule: d (r(1) r(t)) = (r{(1) r(t)) + (r1(1) r(t)) dt
Prove that d/dt (r · (r' × r")) = r · (r' × r"').
Prove the Sum and Constant Multiple Rules for derivatives of vector-valued functions.
Prove the linearity properties fer(1) dt = c fr(1) dt (r(1) + r2(1)) dt = fri(1) d (c any constant) ri(t) dt + [r(t) dt
Prove the Substitution Rule [where g is a differentiable scalar function with an inverse]: So a r(g(t))g' (t)dt = (g-(b) g-(a) r(u) du
Prove that if ΙΙr(t)ΙΙ ≤ K for t ∈ [a, b], then | Srovals r(t) dt K(b-a
What is the domain of r(t) = eti +1/tj + (t + 1)−3k?
Evaluate r(2) and r(−1) for r(t) = (sint, t, (t + 1)-).
What is the domain of r(s) = esi + √sj + cos sk?
Does either of P = (4, 11, 20) or Q = (−1, 6, 16) lie on the path r(t) = (1 + t, 2 + t2, t4)?
Find a direction vector for the line with parametrization r(t) = (4 − t)i + (2 + 5t)j + 1/2tk.
Determine whether the space curve given by r(t) = (sin t, cos t/2, t) intersects the z-axis, and if it does, determine where.
Determine whether the space curve given by r(t) = (t2, t2 − 2t − 3, t − 3) intersects the x-axis, and if it does, determine where.
Determine whether the space curve given by r(t) = (t, t3, t2 + 1) intersects the xy-plane, and if it does, determine where.
Show that the path given by r(t) = (cos t, cos(2t), sin t) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.
Show that the space curve given by r(t) = (1 − cos(2t), t + sin t, t2) intersects the yz-plane in infinitely many points but does not cross through it.
Match the space curves in Figure 7 with their projections onto the xy-plane in Figure 8. * (A) (B) FIGURE 7 (i) W-|- -X (iii) (C) -X -X -X
Match the vector-valued functions (a)–(f) with the space curves (i)–(vi) in Figure 9. (a) r(t) = (t + 15, e0.081 cost, e0.08r sin t) 25t 1 + 1 (c) r(t) = (t,t, (e) r(t) = (t, t, 2t) (b) r(t) = (cost, sint, sin 12t) (d) r(t) = (cos t, sin t, sin 2t) (f) r(t) = (cost, sint, cost sin 12t)
Show that the path given by r(t) = (e−t sin t, e−t cos t, e−t) intersects the sphere x2 + y2 + z2 = 4 once, traveling from outside the sphere to inside as t goes from −∞ to ∞.
Match the space curves in Figure 7 with the following vector-valued functions:(a) r1(t) = (cos 2t, cos t, sin t)(b) r2(t) = (t, cos 2t, sin 2t)(c) r3(t) = (1, t, t)
Which of the following curves have the same projection onto the xy-plane? (a) r(t) = (t, t,e) (b) r(t) = (e, t, t) (c) r3(t) = (t, t, cost)
Match the space curves (A)–(C) in Figure 10 with their projections (i)–(iii) onto the xy-plane. (A) (i) (B) NA (ii) (C) N (iii)
Describe the projections of the circle r(t) = (sin t, 0, 4 + cos t) onto the coordinate planes.
The function r(t) traces a circle. Determine the radius, center, and plane containing the circle.r(t) = 7i + (12 cos t)j + (12 sin t)k
The function r(t) traces a circle. Determine the radius, center, and plane containing the circle.r(t) = (6 + 3 sin t, 9, 4 + 3 cos t)
Consider the curve C given by r(t) = (cos(2t) sint, sin(2t), cos(2t) cos t).(a) Show that C lies on the sphere of radius 1 centered at the origin.(b) Show that C intersects the x-axis, the y-axis, and the z-axis.
Show that the curve C that is parametrized by r(t) = (t2 − 1, t − 2t2, 4 − 6t) lies on a plane as follows:(a) Show that the points on the curve at t = 0, 1, and 2 do not lie on a line, and find an equation of the plane that they determine.(b) Show that for all t, the points on C satisfy the
Let r(t) = (sin t, cos t, sin t cos 2t) be a parametrization of the curve shown in Figure 11.Show that the projection of r(t) onto the xz-plane is the curve X -N
Let C be the curve given by r(t) = (t cos t, t sin t, t).(a) Show that C lies on the cone x2 + y2 = z2.(b) Sketch the cone and make a rough sketch of C on the cone.
Let r(t) = (sin t, cos t, sin t cos 2t) be a parametrization of the curve shown in Figure 11.Find the points where r(t) intersects the xy-plane. X -N
Use a computer algebra system to plot the projections onto the xy- and xz-planes of the curve r(t) = (t cos t, t sin t, t) in Exercise 25.Data From Exercise 25Let C be the curve given by r(t) = (t cos t, t sin t, t).(a) Show that C lies on the cone x2 + y2 = z2.(b) Sketch the cone and make a rough
Parametrize the part of the intersection of the surfaceswhere z ≥ 0 using t = y as the parameter. y-z = x - 2, y +2 = 9
Find a parametrization of the entire intersection of the surfaces in Exercise 29 using trigonometric functions.Data From Exercise 29Parametrize the part of the intersection of the surfaceswhere z ≥ 0 using t = y as the parameter. y-z = x - 2, y +2 = 9
C is the intersection of the surfaces (Figure 12) x + y = z, y = z (a) Separately parametrize each of the two parts of C corresponding to x > 0 and x 0, taking t = z as the parameter. (b) Describe the projection of C onto the xy-plane. (c) Show that C lies on the sphere of radius 1 with its center
Use sine and cosine to parametrize the intersection of the cylinder x2 + y2 = 1 and the plane x + y + z = 1. Then describe the projections of this curve onto the three coordinate planes.
(a) Show that any point on x2 + y2 = z2 can be written in the form (z cos θ, z sin θ, z) for some θ.(b) Use this to find a parametrization of Viviani’s curve (Exercise 31) with θ as the parameter.Data From Exercise 31C is the intersection of the surfaces (Figure 12) x + y = z, y = z (a)
Two paths r1(t) and r2(t) intersect if there is a point P lying on both curves. We say that r1(t) and r2(t) collide if r1(t0) = r2(t0) at some time t0.Determine whether r1(t) and r2(t) collide or intersect, giving the coordinates of the corresponding points if they exist: ri(t) = (t,t,1), r2(t) =
Use sine and cosine to parametrize the intersection of the surfaces x2 + y2 = 1 and z = 4x2 (Figure 13). X
Two paths r1(t) and r2(t) intersect if there is a point P lying on both curves. We say that r1(t) and r2(t) collide if r1(t0) = r2(t0) at some time t0.Determine whether r1(t) and r2(t) collide or intersect, giving the coordinates of the corresponding points if they exist: r(t) = (+ 3,1 + 1,6t-)
Use hyperbolic functions to parametrize the intersection of the surfaces x2 − y2 = 4 and z = xy.
The translated to have center (3, 1, 5) [Figure 14(B)] ellipse (2) + (3) = 1,
Find a parametrization of the curve.
The line passing through (1, 0, 4) and (4, 1, 2)
The circle of radius 1 with center (2, −1, 4) in a plane parallel to the xy-plane
The in the xy-plane, translated to have center (9, −4, 0) ellipse + 2 (5) = 1
The circle of radius 2 with center (1, 2, 5) in a plane parallel to the yz-plane
The in the xz-plane, translated to have center (3, 1, 5) [Figure 14(A)] 2 ellipse ( ) + (3) = 1 NIM
The intersection of the surfaces z = x - y and z = x + xy - 1
The intersection of the plane y = 1/2 with the sphere x2 + y2 + z2 = 1
Let C be the curve obtained by intersecting a cylinder of radius r and a plane. Insert two spheres of radius r into the cylinder above and below the plane, and let F1 and F2 be the points where the plane is tangent to the spheres [Figure 15(A)]. Let K be the vertical distance between the equators
Assume that the cylinder in Figure 15 has equation x2 + y2 = r2 and the plane has equation z = ax + by. Find a vector parametrization r(t) of the curve of intersection using the trigonometric functions y = cos t and y = sin t. F R (A) P F R K P 21 (B) 22
Find the maximum height above the xy-plane of a point on r(t) =(et, sin t, t(4 − t)).
Now reprove the result of Exercise 51 using vector geometry. Assume that the cylinder has equation x2 + y2 = r2 and the plane has equation z = ax + by.Data From Exercise 51Assume that the cylinder in Figure 15 has equation x2 + y2 = r2 and the plane has equation z = ax + by. Find a vector
Let uΙΙv be the projection of u along v. Which of the following is the projection u along the vector 2v and which is the projection of 2u along v? (a) U (b) Ulv (c) 2u|v
Is there any quadric surface whose traces are all parabolas?
The level surface ϕ = ϕ0 in spherical coordinates, usually a cone, reduces to a half-line for two values of ϕ0. Which two values?
Which of the following planes is not parallel to the plane x + y + z = 1?(a) 2x + 2y + 2z = 1 (b) x + y + z = 3(c) x − y + z = 0
Which statement about spherical coordinates is correct?(a) If ϕ = 0, then P lies on the z-axis.(b) If ϕ = 0, then P lies in the xy-plane.
True or false? All traces of a hyperboloid are hyperbolas.
What is the length of −3a if ΙΙaΙΙ = 5?
True or false? All traces of an ellipsoid are ellipses.

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