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mathematics
precalculus

Questions and Answers of Precalculus

Determine the set of points at which the function is continuous.f (x, y, z) = arcsin (x2 + y2 + z2)
Determine the set of points at which the function is continuous.G(x, y) = ln(1 + x - y)
Determine the set of points at which the function is continuous.G(x, y) = √x + √1 - x2 - y2
Determine the set of points at which the function is continuous.H(x, y) = ex + ey/exy - 1
Determine the set of points at which the function is continuous.F(x, y) = 1 + x2 + y2/1 - x2 - y2
Determine the set of points at which the function is continuous.F(x, y) = cos√1 + x - y
Determine the set of points at which the function is continuous.F(x, y) = xy/1+ ex-y
Graph the function and observe where it is discontinuous. Then use the formula to explain what you have observed.f (x, y) = 1/1 - x2 - y2
Graph the function and observe where it is discontinuous. Then use the formula to explain what you have observed.f (x, y) = e1/(x2y)
Find h(x, y) = g( f (x, y)) and the set of points at which h is continuous.g(t) = t + ln t, f (x, y) = 1 - xy/1 + x2y2
Find h(x, y) = t( f (x, y)) and the set of points at which h is continuous.g(t) = t2 + √t , f (x, y) = 2x + 3y - 6
Use a computer graph of the function to explain why the limit does not exist. xy³ lim (x, y)(0, 0) x? + y°
Use a computer graph of the function to explain why the limit does not exist. 2x2 + 3ху + 4у? Зх? + 5у? lim (х, у) — (о, о)
Find the limit, if it exists, or show that the limit does not exist. x²y²z? (x, y, -(0, 0, 0) x2 + y? + z? lim
Find the limit, if it exists, or show that the limit does not exist. ху + уz? + х2? + у? + 2* lim (. у. 3) (0, о, 0) х х? +
Find the limit, if it exists, or show that the limit does not exist. xy + yz (r, y. 2(0,0,0) x? + y? + z? lim
Find the limit, if it exists, or show that the limit does not exist. e tan(xz) lim (x, y, z)- (7, 0, 1/3)
Find the limit, if it exists, or show that the limit does not exist. lim хуч (х. э)-0, 0) х? +у*
Find the limit, if it exists, or show that the limit does not exist. x² + y? (x, y)→ (0, 0) Jx2 + y² + 1 – 1 lim
Find the limit, if it exists, or show that the limit does not exist. ху* (x, y)-(0, 0) x* + y+ lim
Find the limit, if it exists, or show that the limit does not exist. xy? cos y x? + y* lim (x, y)(0, 0)
Find the limit, if it exists, or show that the limit does not exist. х3 — уз х3 lim (. 9) - (0, 0) х2 + ху + у?
Find the limit, if it exists, or show that the limit does not exist. ху x² + y² (x, y)→(0, 0) Jx² + y? lim
Find the limit, if it exists, or show that the limit does not exist. lim (и. 9) (1, 0) (х — 1)? ху — у + у? 1)²
Find the limit, if it exists, or show that the limit does not exist. y sin?x + y* lim (х, ) — (0, 0) х* + у*
Find the limit, if it exists, or show that the limit does not exist. 5y*cos?x lim ços 4 * + y* (x, y)-(0, 0) x
Find the limit, if it exists, or show that the limit does not exist. x* – 4y? lim (x, y)-(0, 0) x? + 2y?
Find the limit, if it exists, or show that the limit does not exist. 2х-у ev2x-y lim (x, y)(3, 2)
Find the limit, if it exists, or show that the limit does not exist. lim (x, y)-(T, 7/2) y sin(x – y)
Find the limit, if it exists, or show that the limit does not exist. x*y + xy² (х, у)— (2, - 1) х2 — у2 lim y²
Find the limit, if it exists, or show that the limit does not exist. lim (x²y³ – 4y²)| (х, у) —(3, 2)
Use a table of numerical values of f (x, y) for (x, y) near the origin to make a conjecture about the value of the limit of f (x, y) as (x, y) → (0, 0). Then explain why your guess is correct.f(x,
Use a table of numerical values of f (x, y) for (x, y) near the origin to make a conjecture about the value of the limit of f (x, y) as (x, y) → (0, 0). Then explain why your guess is correct.f (x,
Explain why each function is continuous or discontinuous.(a) The outdoor temperature as a function of longitude, latitude, and time(b) Elevation (height above sea level) as a function of longitude,
Suppose that lim(x, y)→(3, 1) f(x, y) = 6. What can you say about the value of f (3, 1)? What if f is continuous?
A rocket burning its onboard fuel while moving through space has velocity v(t) and mass m(t) at time t. If the exhaust gases escape with velocity ve relative to the rocket, it can be deduced from
The position function of a spaceship isand the coordinates of a space station are (6, 4, 9). The captain wants the spaceship to coast into the space station. When should the engines be turned off?
If a particle with mass m moves with position vector r(t), then its angular momentum is defined as L(t) = mr(t) x v(t) and its torque as τ(t) = mr(t) x a(t).Show that L'(t) = τ(t). Deduce that if
The magnitude of the acceleration vector a is 10 cm/s2. Use the figure to estimate the tangential and normal components of a. УА х
Find the tangential and normal components of the acceleration vector at the given point.r(t) = 1/t i + 1/t2 j + 1/t3 k, (1, 1, 1)
Find the tangential and normal components of the acceleration vector at the given point.r(t) = ln t i + (t2 + 3t) j + 4√t k, (0, 4, 4)
Find the tangential and normal components of the acceleration vector.r(t) = ti + 2et j + e2t k
Find the tangential and normal components of the acceleration vector.r(t) = cos t i + sin t j + t k
Find the tangential and normal components of the acceleration vector.r(t) = 2t2 i + (2/3t3 - 2t) j
Find the tangential and normal components of the acceleration vector.r(t) = (t2 + 1) i + t3 j, t > 0
(a) If a particle moves along a straight line, what can you say about its acceleration vector?(b) If a particle moves with constant speed along a curve, what can you say about its acceleration vector?
A particle has position function r(t). If r'(t) = c x r(t), where c is a constant vector, describe the path of the particle.
Another reasonable model for the water speed of the river in Exercise 33 is a sine function: f (x) = 3 sins(x/40). If a boater would like to cross the river from A to B with constant heading and a
Water traveling along a straight portion of a river normally flows fastest in the middle, and the speed slows to almost zero at the banks. Consider a long straight stretch of river flowing north,
A ball with mass 0.8 kg is thrown southward into the air with a speed of 30 mys at an angle of 30° to the ground. A west wind applies a steady force of 4 N to the ball in an easterly direction.
A ball is thrown eastward into the air from the origin (in the direction of the positive x-axis). The initial velocity is 50 i + 80 k, with speed measured in feet per second. The spin of the ball
Show that a projectile reaches three-quarters of its maximum height in half the time needed to reach its maximum height.
A medieval city has the shape of a square and is protected by walls with length 500 m and height 15 m. You are the commander of an attacking army and the closest you can get to the wall is 100 m.
A batter hits a baseball 3 ft above the ground toward the center field fence, which is 10 ft high and 400 ft from home plate. The ball leaves the bat with speed 115 ft/s at an angle 508 above the
A rifle is fired with angle of elevation 36°. What is the muzzle speed if the maximum height of the bullet is 1600 ft?
A projectile is fired from a tank with initial speed 400 m/s. Find two angles of elevation that can be used to hit a target 3000 m away.
A ball is thrown at an angle of 45° to the ground. If the ball lands 90 m away, what was the initial speed of the ball?
Rework Exercise 23 if the projectile is fired from a position 100 m above the ground.
A projectile is fired with an initial speed of 200 m/s and angle of elevation 60°. Find (a) The range of the projectile,(b) The maximum height reached, (c) The speed at impact.
A force with magnitude 20 N acts directly upward from the xy-plane on an object with mass 4 kg. The object starts at the origin with initial velocity v(0) = i - j. Find its position function and its
What force is required so that a particle of mass m has the position function r(t) = t3 i + t2 j + t3 k?
The position function of a particle is given by r(t) = k t2, 5t, t2 - 16t) . When is the speed a minimum?
(a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position.(b) Use a computer to graph the path of the particle.a(t) = ti + et j + e-t
(a) Find the position vector of a particle that has the given acceleration and the specified initial velocity and position.(b) Use a computer to graph the path of the particle.a(t) = 2t i + sin t j +
Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.a(t) = sin t i + 2 cos t j + 6t k, v(0) = -k, r(0) = j - 4 k
Find the velocity and position vectors of a particle that has the given acceleration and the given initial velocity and position.a(t) = 2 i + 2t k, v(0) = 3 i - j, r(0) = j + k
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = (t2, sin t - t cos t, cos t + t sin t), t > 0
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = et(cos ti + sin t j + t k)
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = t2 i + 2t j + ln t k
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = √2 ti + etj + e-t k
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = (2 cos t, 3t, 2 sin t)
Find the velocity, acceleration, and speed of a particle with the given position function.r(t) = (t2 + t, t2 - t, t3)
Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of
Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of
Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of
Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of
Find the velocity, acceleration, and speed of a particle with the given position function. Sketch the path of the particle and draw the velocity and acceleration vectors for the specified value of
The figure shows the path of a particle that moves with position vector r(t) at time t.(a) Draw a vector that represents the average velocity of the particle over the time interval 2 < t <
The table gives coordinates of a particle moving through space along a smooth curve.(a) Find the average velocities over the time intervals [0, 1], [0.5, 1], [1, 2], and [1, 1.5].(b) Estimate the
Let’s consider the problem of designing a railroad track to make a smooth transition between sections of straight track.Existing track along the negative x-axis is to be joined smoothly to a track
The DNA molecule has the shape of a double helix (see Figure 3 on page 850). The radius of each helix is about 10 angstroms (1 Å = 10-8 cm). Each helix rises about 34 Å during each complete turn,
Find the curvature and torsion of the curve x = sinh t, y = cosh t, z = t at the point (0, 1, 0).
Use the formula in Exercise 63(d) to find the torsion of the curve r(t) = (t, 1/2 t2, 1/3t3).
Show that the circular helix r(t) = (a cos t, a sin t, bt), where a and b are positive constants, has constant curvature and constant torsion. [Use the result of Exercise 63(d).]
Show that the curvature k is related to the tangent and normal vectors by the equation LP кN ds
The rectifying plane of a curve at a point is the plane that contains the vectors T and B at that point. Find the rectifying plane of the curve r(t) = sin t i + cos t j + tan t k at the point
Show that at every point on the curver(t) = (et cos t, et sin t, et)the angle between the unit tangent vector and the z-axis is the same. Then show that the same result holds true for the unit normal
Show that the osculating plane at every point on the curve r(t) = (t + 2, 1 - t, 1/2 t2) is the same plane. What can you conclude about the curve?
Find equations of the normal and osculating planes of the curve of intersection of the parabolic cylinders x = y2 and z = x2 at the point (1, 1, 1).
Is there a point on the curve in Exercise 53 where the osculating plane is parallel to the plane x + y + z = 1?
At what point on the curve x = t3, y = 3t, z = t4 is the normal plane parallel to the plane 6x + 6y - 8z = 1?
Find equations of the osculating circles of the parabola y = 1/2x2 at the points (0, 0) and (1, 1/2). Graph both osculating circles and the parabola on the same screen.
Find equations of the osculating circles of the ellipse 9x2 + 4y2 = 36 at the points (2, 0) and (0, 3). Use a graphing calculator or computer to graph the ellipse and both osculating circles on the
Find equations of the normal plane and osculating plane of the curve at the given point.x = ln t, y = 2t, z = t2; (0, 2, 1)
Consider the curvature at x = 0 for each member of the family of functions f (x) = ecx. For which members is k(0) largest?
Use the formula in Exercise 42 to find the curvature.x = et cos t, y = et sin t
Use the formula in Exercise 42 to find the curvature.x = t2, y = t3
Use Theorem 10 to show that the curvature of a plane parametric curve x = f (t), y = g(t) iswhere the dots indicate derivatives with respect to t. Tiў — ўх| [i2 + ў2J92 к
The graph of r(t) = (t - 3/2 sin t, 1 - 3/2 cos t, t) is shown in Figure 13.1.12(b). Where do you think the curvature is largest? Use a CAS to find and graph the curvature function.For which values
(a) Graph the curve r(t) = (sin 3t, sin 2t, sin 3t). At how many points on the curve does it appear that the curvature has a local or absolute maximum?(b) Use a CAS to find and graph the curvature

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