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mathematics
calculus 4th
Calculus 4th Edition Jon Rogawski, Colin Adams, Robert Franzosa - Solutions
If ƒ is differentiable at (a, b) and ƒx(a, b) = ƒy(a, b) = 0, what can we conclude about the tangent plane at (a, b)?
Assume thatWhich of (a)–(b) is the linearization of ƒ centered at (2, 3)? f(2, 3) = 8, fx (2, 3) = 5, fy(2, 3) = 7
Assume thatEstimate ƒ(2, 3.1). f(2, 3) = 8, fx (2, 3) = 5, fy(2, 3) = 7
Assume thatEstimate Δf at (2, 3) if Δx = −0.3 and Δy = 0.2. f(2, 3) = 8, fx (2, 3) = 5, fy(2, 3) = 7
In the derivation of the equation for the plane determined by ƒx and ƒy, we used w × v for a normal vector to the plane. How would the choice of v × w for a normal vector have affected the resultant equation?
We consider an automobile of mass m traveling along a curved but level road. To avoid skidding, the road must supply a frictional force F = ma, where a is the car’s acceleration vector. The maximum magnitude of the frictional force is μmg, where μ is the coefficient of friction and g = 9.8
We consider an automobile of mass m traveling along a curved but level road. To avoid skidding, the road must supply a frictional force F = ma, where a is the car’s acceleration vector. The maximum magnitude of the frictional force is μmg, where μ is the coefficient of friction and g = 9.8
We consider an automobile of mass m traveling along a curved but level road. To avoid skidding, the road must supply a frictional force F = ma, where a is the car’s acceleration vector. The maximum magnitude of the frictional force is μmg, where μ is the coefficient of friction and g = 9.8
We consider an automobile of mass m traveling along a curved but level road. To avoid skidding, the road must supply a frictional force F = ma, where a is the car’s acceleration vector. The maximum magnitude of the frictional force is μmg, where μ is the coefficient of friction and g = 9.8
We consider an automobile of mass m traveling along a curved but level road. To avoid skidding, the road must supply a frictional force F = ma, where a is the car’s acceleration vector. The maximum magnitude of the frictional force is μmg, where μ is the coefficient of friction and g = 9.8
The orbit of a planet is an ellipse with the sun at one focus. The sun’s gravitational force acts along the radial line from the planet to the sun (the dashed lines in Figure 16), and by Newton’s Second Law, the acceleration vector points in the same direction. Assuming that the orbit has
Suppose that r(t) lies on a sphere of radius R for all t. Let J =|, r × r'. Show that r = (J × r)/ΙΙrΙΙ2. Observe that r and r' are perpendicular.
Prove that aN = ΙΙa × vΙΙ/ΙΙvΙΙ.
Figure 15 shows the acceleration vectors of a particle moving clockwise around a circle. In each case, state whether the particle is speeding up, slowing down, or momentarily at constant speed. Explain. (A) (B) (C)
Explain why the vector w in Figure 14 cannot be the acceleration vector of a particle moving along the circle. Consider the sign of w · N. W N
A particle follows a path r1(t) on the helical curve with parametrization r(θ) = (cos θ, sin θ, θ). When it is at position r(π/2), its speed is 3 m/s and it is accelerating at a rate of 1/2 m/s2. Find its acceleration vector a at this moment. The particle’s acceleration vector does not
A car proceeds along a circular path of radius R = 300 m centered at the origin. Starting at rest, its speed increases at a rate of t m/s2. Find the acceleration vector a at time t = 3 s and determine its decomposition into normal and tangential components.
A satellite orbits the earth at an altitude 400 km above the earth’s surface, with constant speed v = 28,000 km/h. Find the magnitude of the satellite’s acceleration (in kilometers per square hour), assuming that the radius of the earth is 6378 km (Figure 13). 1 1
At time t0, a moving particle has velocity vector v = 2i and acceleration vector a = 3i + 18k. Determine the curvature κ(t0) of the particle’s path at time t0.
Suppose that the Ferris wheel in Example 5 is rotating clockwise and that the point P at angle 45° has acceleration vector a = (0, −50) m/min2 pointing down, as in Figure 12. Determine the speed and tangential component of the acceleration of the Ferris wheel. Ferris wheel 45 -X
In the notation of Example 5, find the acceleration vector for a person seated in a car at (a) The highest point of the Ferris wheel and (b) The two points level with the center of the wheel. EXAMPLE 5 The Giant Ferris Wheel in Vienna has radius R = 30 m (Figure 6). Assume that at time t = to, a
Find the components aT and aN of the acceleration vector of a particle moving along a circular path of radius R = 100 cm with constant velocity v0 = 5 cm/s.
Let r(t) =(t2, 4t − 3). Find T(t) and N(t), and show that the decomposition of a(t) into tangential and normal components is a(t) = 2t 12+4/ 4 T+ () N 1 + 4/
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6. EXAMPLE 6 For r(t) = (,2t, Int), determine the acceleration a(r). At t = 1, decom- pose the acceleration vector into tangential and normal components, and find the curva- ture of the path
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6. EXAMPLE 6 For r(t)=(, 2t, Int), determine the acceleration a(r). At t = 1, decom- pose the acceleration vector into tangential and normal components, and find the curva- ture of the path
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6. EXAMPLE 6 For r(t) = (,2t, Int), determine the acceleration a(r). At t = 1, decom- pose the acceleration vector into tangential and normal components, and find the curva- ture of the path
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6. EXAMPLE 6 For r(t)= (1,2t, Int), determine the acceleration a(r). Atr = , decom- pose the acceleration vector into tangential and normal components, and find the curva- ture of the path
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6. EXAMPLE 6 For r(t) = (1,2t, Int), determine the acceleration a(r). At t = 1, decom- pose the acceleration vector into tangential and normal components, and find the curva- ture of the path
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6. EXAMPLE 6 For r(t) = (,2t, Int), determine the acceleration a(r). At t = 1, decom- pose the acceleration vector into tangential and normal components, and find the curva- ture of the path
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6. EXAMPLE 6 For r(t)= (1,2t, lnt), determine the acceleration a(r). Atr=, decom- pose the acceleration vector into tangential and normal components, and find the curva- ture of the path
Find the decomposition of a(t) into tangential and normal components at the point indicated, as in Example 6.r(t) = (et, 1 − t), t = 0 EXAMPLE 6 For r(t)= (1,2t, Int), determine the acceleration a(r). At t = 1, decom- pose the acceleration vector into tangential and normal components, and find
Use Eq. (6) to find the coefficients aT and aN as a function of t (or at the specified value of t). ar = a. T a-v |v|| an = a N = |la| - Jar|
Use Eq. (6) to find the coefficients aT and aN as a function of t (or at the specified value of t). ar = a. T a-v |v|| ay = a N=|la| - Jar|
At a certain moment, a particle moving along a path has velocity v = (12, 20, 20) and acceleration a = (2, 1, −3). Is the particle speeding up or slowing down?
At a certain moment, a moving particle has velocity v = (2, 2, −1) and acceleration a = (0, 4, 3). Find T, N, and the decomposition of a into tangential and normal components.
A quarterback throws a football while standing at the very center of the field on the 50-yard line. The ball leaves his hand at a height of 5 ft and has initial velocity v0 = 40i + 35j + 32k ft/s. Assume an acceleration of 32 ft/s2 due to gravity and that the i vector points down the field toward
The vector N can be computed using N = B × T [Eq. (16)] with B, as in Eq. (18). Use this method to find N in the following cases:(a) r(t) = (cos t, t, t2) at t = 0(b) r(t) = (t2, t−1, t) at t = 1
Use the formula from the preceding problem to find B for the space curve given by r(t) = sin t, − cos t, sin t. Conclude that the space curve lies in a plane.
Show that r × r" is a multiple of B. Conclude that B = r'xr" ||r'x r'||
Follow steps (a)–(b) to prove dN ds = -KT + TB
Use the identity a × (b × c) = (a · c)b − (a · b)c to prove N × B = T, B × T = N
We investigate the binormal vector further.Torsion means twisting. Is this an appropriate name for τ? Explain by interpreting τ geometrically.
We investigate the binormal vector further.Show that if C is contained in a plane P, then B is a unit vector normal to P. Conclude that τ = 0 for a plane curve.
We investigate the binormal vector further.Follow steps (a)–(c) to prove that there is a number τ called the torsion such that dB ds = -TN dB dN (a) Show that = TX and conclude that dB/ds is orthogonal to T. ds ds (b) Differentiate B. B = 1 with respect to s to show that dB/ds is orthogonal to
We investigate the binormal vector further.Let r(t) = (x(t), y(t), 0). Assuming that T(t) × N(t) is nonzero, there are two possibilities for the vector B(t). What are they? Explain.
Let r(s) be an arc length parametrization of a closed curve C of length L. We call C an oval if dθ/ds > 0 (see Exercise 71). Observe that −N points to the outside of C. For k > 0, the curve C1 defined by r1(s) = r(s) − kN is called the expansion of c(s) in the normal direction.Data From
Show that the curvature of Viviani’s curve, given by r(t) = (1 + cos t, sin t, 2 sin(t/2)), is k(t) = 13 + 3 cost (3 + cost)/2
Let r(t) = (x(t), y(t), z(t)) be a path with curvature κ(t) and define the scaled path r1(t) = (λx(t), λy(t), λz(t)), where λ ≠ 0 is a constant. Prove that curvature varies inversely with the scale factor. That is, prove that the curvature κ1(t) of r1(t) is κ1(t) = λ−1κ(t). This
Two vector-valued functions r1(s) and r2(s) are said to agree to order 2 at s0 if ri (So) = r(so), r (so) = r(so), r(so) = r2(so) Let r(s) be an arc length parametrization of a curve C, and let P be the terminal point of r(0). Let y(s) be the arc length parametrization of the osculating circle
Show thatis an arc length parametrization of the osculating circle at r(t0). 1. 1 y(s) = r(to) + N + ((sin ks)T- (cos KS)N) - K K
Show that both r'(t) and r"(t) lie in the osculating plane for a vector function r(t).
Use Eq. (14) to find the curvature of the general Bernoulli spiral r = aebθ in polar form (a and b are constants).
Use Eq. (14) to find the curvature of the curve given in polar form.ƒ(θ) = eθ K(0) = \(0)+2 f'(0) = f(0)" (0)| ((0) + f'(0))/2
Use Eq. (14) to find the curvature of the curve given in polar form.ƒ(θ) = θ K(0) = \(0)+2 f'(0) = f(0)" (0)| ((0) + f'(0))/2
Use the parametrization r(θ) = (ƒ (θ) cos θ, ƒ (θ) sin θ) to show that a curve r = ƒ(θ) in polar coordinates has curvature K(0) = If (0) +2f'(0) - (0)"(0)| (f(0) + f'(0))/2
Let θ(x) be the angle of inclination at a point on the graph y = ƒ(x) (see Exercise 71). do (a) Use the relation f'(x) = tan to prove that dx (1 + f'(x)) ds (b) Use the arc length integral to show that = 1 + f'(x). dx (c) Now give a proof of Eq. (6) using Eq. (13). =
A particle moves along the path y = x3 with unit speed. How fast is the tangent turning (i.e., how fast is the angle of inclination changing) when the particle passes through the point (2, 8)?
The angle of inclination at a point P on a plane curve is the angle θ between the unit tangent vector T and the x-axis (Figure 21). Assume that r(s) is a arc length parametrization, and let θ = θ(s) be the angle of inclination at r(s). Prove thatObserve that T(s) = (cos θ(s), sin θ(s)). k(s) =
In a recent study of laser eye surgery by Gatinel, Hoang-Xuan, and Azar, a vertical cross section of the cornea is modeled by the half-ellipse of Exercise 69. Show that the half-ellipse can be written in the form x = ƒ(y), where ƒ(y) = p−1 (r − √r2 − py2). During surgery, tissue is
Figure 19 shows the graph of the half-ellipse y = ±√2rx − px2, where r and p are positive constants. Show that the radius of curvature at the origin is equal to r. One way of proceeding is to write the ellipse in the form of Exercise 25 and apply Eq. (11).Exercise 25 k(t) = ab (b cost + a sin
Find an equation of the osculating circle at the point indicated or indicate that none exists.r(t) = (cosh t, sinh t), t = 0
Find an equation of the osculating circle at the point indicated or indicate that none exists.r(t) = (1 − si n t, 1 − 2 cos t), t = π
Find an equation of the osculating circle at the point indicated or indicate that none exists. r(t) = (,1 2t), t = 2
Find an equation of the osculating circle at the point indicated or indicate that none exists.r(t) = cos t, sin t, t = π/4
Find an equation of the osculating circle at the point indicated or indicate that none exists.y = ln x, x = 1
Find an equation of the osculating circle at the point indicated or indicate that none exists.y = ex, x = 0
Find an equation of the osculating circle at the point indicated or indicate that none exists.y = sin x, x = π
Find an equation of the osculating circle at the point indicated or indicate that none exists.y = sin x, x = π/2
Find an equation of the osculating circle at the point indicated or indicate that none exists.y = x2, x = 2
Find an equation of the osculating circle at the point indicated or indicate that none exists.y = x2, x = 1
Use Eq. (10) to find the center of curvature of r(t) = (t2, t3) at t = 1.
Let ƒ(x) = x2. Show that the center of the osculating circle at (xo, x) is given by (-4x, + 3x).
(a) What does it mean for a space curve to have a constant unit tangent vector T?(b) What does it mean for a space curve to have a constant normal vector N?(c) What does it mean for a space curve to have a constant binormal vector B?
Let r(t) =(t, 1 − t, t2).(a) Find the general formulas for T and N as functions of t.(b) Find the general formula for B as a function of t.(c) What can you conclude about the osculating planes of the curve based on your answer to b?
Let r(t) = (cos t, sin t, ln(cos t)).(a) Find T,N, and B at (1, 0, 0).(b) Find the equation of the osculating plane at (1, 0, 0).
Let (a) Find T,N, and B at the point corresponding to t = 1.(b) Find the equation of the osculating plane at the point corresponding to t = 1. r(t) = (1, 1/,1).
Use Eq. (12) to find N at the point indicated. N(t) = v(1)r" (1) - v'(t)r' (t) |v(t)r"(t)- v'(t)r' (1)||
Use Eq. (12) to find N at the point indicated. N(t) = v(t)r"(t)- |v(t)r"(t)- v' (t)r' (t) v'(t)r' (1)||
Use Eq. (12) to find N at the point indicated. N(t) = v(1)r" (t) - v'(t)r' (t) |v(t)r"(t)- v'(t)r'(1)||
Use Eq. (12) to find N at the point indicated. N(t) = v(t)r"(t)- |v(1)r"(t)- v' (t)r' (t) v'(t)r' (1)||
Use Eq. (12) to find N at the point indicated. N(t) = v(t)r"(t)- |v(t)r"(t)- v' (t)r' (t) v'(t)r' (1)||
Use Eq. (12) to find N at the point indicated. N(t) = v(t)r"(t)- |v(t)r"(t)- v' (t)r' (t) v'(t)r' (1)||
Let v(t) = ΙΙr '(t)ΙΙ. Show thatN is the unit vector in the direction T'(t). Differentiate T(t) = r'(t)/v(t) to show that v(t)r"(t) − v'(t)r (t) is a positive multiple of T '(t). N(t) v(t)r'' (t) - v'(t)r' (t) |v(t)r"(t) - v'(t)r' (t)||
Find the normal vector to the clothoid (Exercise 35) at t = π1/3.Data From Eexercie 35Plot the clothoid r(t) = (x(t), y(t)), and compute its curvature κ(t) where = f*sin", du. x(t) = y(t) = f cos COS u 1 - du 3
Find T,N, and B for the curve at the indicated point. After finding T', plug in the specific value for t before computing N and B.r(t) = (t, t, et) at (0, 0, 1)
Find T,N, and B for the curve at the indicated point. After finding T', plug in the specific value for t before computing N and B. r(t) = (t,1, t) at (1, 1, 3)
Find T,N, and B for the curve at the indicated point. After finding T', plug in the specific value for t before computing N and B.r(t) = (cos t, sin t, 2) at (1, 0, 2). In this case, draw the curve and the three resultant vectors in 3-space.
Find T,N, and B for the curve at the indicated point. After finding T', plug in the specific value for t before computing N and B. r(t) = (0,1, 1) at (0, 1, 1). In this case, draw the curve and the three resultant vectors in 3-space.
Find the normal vector to the Cornu spiral (Example 3) at t = √π.
Find the normal vectors to r(t) = (t, cos t) at t = π/4 and t = 3π/4.
Sketch the graph of r(t) = (t, t3). Since r'(t) = (1, 3t2), the unit normal N(t) points in one of the two directions ± (−3t2, 1). Which sign is correct at t = 1? Which is correct at t = −1?
Find the normal vector N(t) to r(t) = (4, sin 2t, cos 2t).
Find the normal vector N(θ) to r(θ) = R (cos θ, sin θ), the circle of radius R. Does N(θ) point inside or outside the circle? Draw N(θ) at θ = π/4 with R = 4.
Plot the clothoid r(t) = (x(t), y(t)), and compute its curvature κ(t) where = f*sin", du. x(t) = y (t) = f cos COS u 1 - du 3
The curve parametrized by x(t) = cos3 t and y(t) = sin3 t, with 0 ≤ t ≤ 2π, is called a hypocycloid (Figure 18). -1
Let for the Bernoulli spiral r(t) = (et cos 4t, et sin 4t) (see Exercise 39 in Section 13.3). Show that the radius of curvature is proportional to s(t).Data From Exercise 39 in Section 13.3Show that the radius of curvature is proportional to s(t). s(t) = J S ||r' (u)|| du
Show that the curvature function of the parametrization r(t) = (a cos t, b sin t) of the ellipse (-). () + = 1 is
Find the curvature of the plane curve at the point indicated.y = t4, t = 2
The involute of a circle (Figure 8), traced by a point at the end of a thread unwinding from a circular spool of radius R, has parametrization (see Exercise 29 in Section 11.2) r(0) = (R(cos0+ 0 sin), R(sin 0 - 0 cos 0))
The unit circle with the point (−1, 0) removed has parametrization (see Exercise 79 in Section 11.1)Use this parametrization to compute the length of the unit circle as an improper integral. Hint: The expression for'(t) simplifies. r(t): = 1-1 2t 1+1 1 + 1, -
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