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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
If ƒ(−θ) = ƒ(θ), then the curve r = ƒ(θ) is symmetric with respect to the (choose the correct answer):(a) x-axis. (b) y-axis. (c) origin.
Find a parametrization c(θ) of the unit circle such that c(0) = (−1, 0).
Find a path c(t) that traces the parabolic arc y = x2 from (0, 0) to (3, 9) for 0 ≤ t ≤ 1.
Find a path c(t) that traces the line y = 2x + 1 from (1, 3) to (3, 7) for 0 ≤ t ≤ 1.
Sketch the graph c(t) = (1 + cos t, sin 2t) for 0 ≤ t ≤ 2π and draw arrows specifying the direction of motion.
Express the parametric curve in the form y = ƒ(x).c(t) = (4t − 3, 10 − t)
Express the parametric curve in the form y = ƒ(x). c(t) = (3 2 1 - ²7,² + ² ) t
Express the parametric curve in the form y = ƒ(x).c(t) = (t3 + 1, t2 − 4)
Express the parametric curve in the form y = ƒ(x).x = tan t, y = sec t
Calculate dy/dx at the point indicated.c(t) = (t3 + t, t2 − 1), t = 3
Calculate dy/dx at the point indicated.c(θ) = (tan2 θ, cos θ), θ = π/4
Calculate dy/dx at the point indicated.c(t) = (et − 1, sin t), t = 20
Find the equation of the given hyperbola.Vertices (±3, 0) and foci (±5, 0)
Find the point on the cycloid c(t) = (t − sin t, 1 − cos t) where the tangent line has slope 1/2.
Find the points on (t + sin t, t − 2 sin t) where the tangent is vertical or horizontal.
Find the area between the two curves in Figure 20(B). r = 2 + sin 20 r = sin 20 (B) ►x
Find the equation of the B´ezier curve with control points P0 = (−1, −1), P1 = (−1, 1), P2 = (1, 1), P3(1, −1)
Find the speed at t = π/4 of a particle whose position at time t seconds is c(t) = (sin 4t, cos 3t).
Convert the points (x, y) = (1, −3), (3, −1) from rectangular to polar coordinates.
Show that r = 4/7 cos θ − sin θ is the polar equation of a line.
Calculate the area of the circle r = 3 sin θ bounded by the rays θ = π/3 and θ = 2π/3 .
Sketch the cardioid curve r = 1 + cos θ.
Calculate the area of one petal of r = sin 4θ (see Figure 1). *. *. n = 4 (8 petals) B n = 2 (4 petals) n = 6 (12 petals)
Calculate the total area enclosed by the curve r2 = cos θesin θ (Figure 2). -X
Find the shaded area in Figure 3. -2 -1 1 r = 1 + cos 20 2 -X
A satellite orbiting at a distance R from the center of the earth follows the circular path x(t) = R cosωt, y(t) = R sinωt.(a) Show that the period T (the time of one revolution) is T = 2π/ω.(b) According to Newton’s Laws of Motion and Gravity,where G is the universal gravitational constant
Find the area enclosed by the cardioid r = a(1 + cos θ), where a > 0.
L is tangent to the circle r = 2 √10 at the point with rectangular coordinates (−2, −6).
L has slope 3 and is tangent to the unit circle in the fourth quadrant.
Prove Theorem 5 in the case 0 −2 − 1). THEOREM 5 Focus-Directrix Relationship Ellipse • If 0 b>0 and c = √a²-b², then the ellipse ()² + ()² = satisfies Eq. (10) with F = (c,0),e=, and vertical directrix x = 2. = 1 Hyperbola • If e > 1, then the set of points satisfying Eq. (10) is a
Prove Theorem 2. THEOREM 2 Hyperbola in Standard Position Let a > 0 and b> 0, and set c = √a² + b². The hyperbola PF - PF2 = ±2a with foci F₁ = (c,0) and F₂ = (-c, 0) has equation ()'-()*²=₁ 1 7
Sketch the vectors v1, v2, v3, v4 with tail P and head Q, and compute their lengths. Are any two of these vectors equivalent? V1 V2 P (2,4) (-1,3) Q V3 V4 (-1,3) (4,1) (4,4) (1,3) (2,4) (6,3)
Answer true or false. Every nonzero vector is:(a) Equivalent to a vector based at the origin.(b) Equivalent to a unit vector based at the origin.(c) Parallel to a vector based at the origin.(d) Parallel to a unit vector based at the origin.
Compute the dot product. (1, 2, 1) (4, 3, 5)
Is the dot product of two vectors a scalar or a vector?
What is the terminal point of the vector v = (3, 2, 1) based at the point P = (1, 1, 1)?
What is the (1, 3) minor of the matrix 3 4 -5 -5 -1 4 0 21 1|? 3
Sketch the vector v = (1, 3, 2) and compute its length.
Sketch the vector b = 3, 4 based at P = (−2, −1).
Compute the dot product. (3,-2, 2) (1, 0, 1)
What can you say about the angle between a and b if a · b < 0?
Let Which of the following vectors (with tail P and head Q) are equivalent to v? V = PoQo, where Po = (1, -2,5) and Qo= (0, 1,-4).
What are the components of the vector v = 3, 2, 1 based at the point P = (1, 1, 1)?
The angle between two unit vectors e and f is π/6. What is the length of e × f?
Suppose that v has components (3, 1). How, if at all, do the components change if you translate v horizontally 2 units to the left?
What is the terminal point of the vector a = (1, 3) based at P = (2, 2)? Sketch a and the vector a0 based at the origin and equivalent to a.
Compute the dot product. (0, 1, 1) (-7, 41, -39)
Which property of dot products allows us to conclude that if v is orthogonal to both u and w, then v is orthogonal to u + w?
If v = −3w, then (choose the correct answer):(a) v and w are parallel.(b) v and w point in the same direction.
Sketch the vector v = (1, 1, 0) abased at P = (0, 1, 1). Describe this vector in the form for some point Q, and sketch the vector v0 based at the origin equivalent to v. PO
What is u × w, assuming that w × u = (2, 2, 1)?
What are the components of the zero vector based at P = (3, 5)?
Let where P = (1, 1) and Q = (2, 2). What is the head of the vector v' equivalent to v based at (2, 4)? What is the head of the vector v0 equivalent to v based at the origin? Sketch v, v0, and v'. v = PÓ,
Compute the dot product. (1,-1, 1) (-2, 4, -6)
Which is the projection of v along v: (a) v or (b) ev?
Which of the following is a direction vector for the line through P = (3, 2, 1) and Q = (1, 1, 1)?(a) (3, 2, 1)(b) (1, 1, 1)(c) (2, 1, 0)
Determine whether the coordinate systems (A)–(C) in Figure 18 satisfy the right-hand rule. (A) (B) Z (C)
Find the cross product without using the formula: (a) (4, 8, 2) × (4,8, 2) (b) (4, 8, 2) × (2, 4, 1)
True or false?(a) The vectors v and −2v are parallel.(b) The vectors v and −2v point in the same direction.
Refer to the unit vectors in Figure 21.Find the components of u. 30° y u 45° 15° q 20⁰ W -x
Compute the dot product. (3, 1). (4,-7)
How many different direction vectors does a line have?
Find the components of the vector P = (1, 0, 1), Q = (2, 1, 0) PO.
What are i × j and i × k?
Explain the commutativity of vector addition in terms of the Parallelogram Law.
Refer to the unit vectors in Figure 21.Find the components of v. 30° y u 45° 15° q 20⁰ W -x
Compute the dot product. (1/1) (3, 1)
Which of the following is equal to cos θ, where θ is the angle between u and v?(a) u · v (b) u · ev (c) eu · ev
True or false? If v is a direction vector for a line then −v is also a direction vector for L,
Find the components of the vector PO.
When is the cross product v × w equal to zero?
Refer to the unit vectors in Figure 21.Find the components of w. 30° y u 45° 15° q 20⁰ W -x
Compute the dot product.k · j
What is the radius of the sphere x2 + y2 + z2 = 5?
Find the components of the vector PO.
Which of the following are meaningful and which are not? Explain. (a) (u. v) x w X (b) (ux v). w (c) ||w||(u - v) (d) ||w||(u × v)
Refer to the unit vectors in Figure 21.Find the components of q. 30° y u 45° 15° q 20⁰ W -x
Which of the following points are on the cylinder (x − 1)2 + y2 = 1? (a) (1,0,0) (d) (0, -1, 1) (b) (0,0,0) (e) (1,-1, 1) (c) (0,0,-1) (f) (1,1,0)
Compute the dot product.k · k
Find the components of the vector PO.
Which of the following vectors are equal to j × i? (a) ixk (b) -k (c) ixj
Compute the dot product.(i + j) · (j + k)
Find the components ofP = (3, 2), Q = (2, 7) Q.
Let R = (1, 4, 3).Calculate the length of OR.
Compute the dot product.(3j + 2k) · (i − 4k)
Find the components ofP = (−3, −5), Q= (4, −6) Q.
Let R = (1, 4, 3).Find the point Q such that has components 4, 1, 1, and sketch v. V=RQ
Compute the dot product.(i + j + k) · (3i + 2j − 5k)
Find the components ofP = (1, −7), Q = (0, 17) Q.
Let R = (1, 4, 3).Find the point P such that has components (3, −2, 3), and sketch w. W = PR
Find the components ofP = (0, 2), Q = (5, 0) Q.
Compute the dot product.(−k) · (i − 2j + 7k)
Let R = (1, 4, 3).Find the components of u = PR, where P = (1,2,2).
Determine whether the two vectors are orthogonal and, if not, whether the angle between them is acute or obtuse. (1, 1,1), (1, -2,-2)
Calculate.(2, 1) + (3, 4)
Let v = 4, 8, 12. Which of the following vectors is parallel to v? Which point in the same direction? (a) (2, 4,6) (c) (-7,-14, -21) (b) (-1, -2,3) (d) (6, 10, 14)
Determine whether the two vectors are orthogonal and, if not, whether the angle between them is acute or obtuse. (0, 2, 4), (-5, 0, 0)
Determine whether is equivalent to AB
Determine whether the two vectors are orthogonal and, if not, whether the angle between them is acute or obtuse. (1, 2, 1), (7,-3,-1)
Determine whether is equivalent to A = (1, 4, 1) B = (−2, 2, 0)P = (2, 5, 7) Q = (−3, 2, 1) AB
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