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mathematics
calculus 10th edition
Calculus 10th Edition Ron Larson, Bruce H. Edwards - Solutions
The position vector describes the path of an object moving in space.(a) Find 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 r(t) = (cos³ t, sin³ t, 3t) Time t = π
Find the curvature K of the curve.r(t) = 4ti + 3 cos tj + 3 sin tk
A projectile is fired from ground level at an angle of 12° with the horizontal. The projectile is to have a range of 200 feet. Find the minimum initial velocity necessary.
Use a graphing utility to graph the paths of a projectile for the given values of θ and v0. For each case, use the graph to approximate the maximum height and range of the projectile. (Assume that the projectile is launched from ground level.)(a)(b)(c)(d)(e)(f) 0 = 10°, Vo = 66 ft/sec
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = (1, 1², 1³)
Find the open interval(s) on which the curve given by the vector-valued function is smooth.r(t) = eti − e-tj + 3tk
Find T(t), N(t), aT, anda at the given time t for the space curve r(t). Vector-Valued Function r(t) = ti + 2tj - 3tk Time t = 1
Find the curvature K of the curve.r(t) = e²ti + e²t cos tj + e²¹ sin tk
Find the open interval(s) on which the curve given by the vector-valued function is smooth.r(t) = ti -3tj + tan tk
The position vector describes the path of an object moving in space.(a) Find 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 r(t) = (t, -tan t, e¹) Time t = 0
Find the curvature K of the curve at the point P.r(t) = 3ti + 2t²j, P(-3, 2)
Define the unit tangent vector, the principal unit normal vector, and the tangential and normal components of acceleration.
Prove that AN = 2 2 all²-a². aT
Prove that aN |vx all ||A||
Two particles travel along the space curves r(t) and u(t). A collision will occur at the point of intersection P when both particles are at P at the same time. Do the particles collide? Do their paths intersect? r(t) = fi + (9t-20)j + t²k u(t) = (3t+4)i + t²j + (5t − 4)k -
Prove that the vector T'(t) is 0 for an object moving in a straight line.
You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function r. Let r = ||r||, let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of
You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function r. Let r = ||r||, let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of
You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function r. Let r = ||r||, let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.If the curve given by x = ƒ(t), y = g(t), and z = h(t) is a line,then ƒ, g, and h are first-degree polynomial functions of t.
Explain whether the Quotient Rule should be used to find the partial derivative. Do not differentiate. Ә ху əx y² - 3 дх
Use the definition of the limit of a function of two variables to verify the limit. lim (x, y)→(1, 0) x = 1
In Exercises determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.Two particles travel along the space curves r(t) and u(t). The intersection of their paths depends only on the curves traced out by r(t) and u(t), while collision
Find dw/dt usingthe appropriate Chain Rule. Evaluate dw/dt at the given valueof t. Function w = x² + y² x = 2t, y = 3t Value t = 2
Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false.The vector-valued function r(t) = t2i + t sin tj + t cos tklies on the paraboloid x = y² + z².
Use the graph to determine whether z is a function of x and y. Explain. X Z
You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function r. Let r = ||r||, let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of
You are asked to verify Kepler's Laws of Planetary Motion. For these exercises, assume that each planet moves in an orbit given by the vector- valued function r. Let r = ||r||, let G represent the universal gravitational constant, let M represent the mass of the sun, and let m represent the mass of
Use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u = cos θi+ sin θj.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the
Describe the levelsurface F(x, y, z) = 0.F(x, y, z) = 3x - 5y +3z - 15
Explain whether the Quotient Rule should be used to find the partial derivative. Do not differentiate. ху Ә ду 12 3
Find theminimum distance from the point to the plane x - y + z = 3.(0, 0, 0)
Use the definition of the limit of a function of two variables to verify the limit. lim (x, y)→(4, -1) x = 4
Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema.g(x, y) = (x - 1)² + (y - 3)²
Find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t. Function w = √√√x² + y² x = cos t, y = et Value t = 0
Find the total differential.2 = 2x²y³
Use the graph to determine whether z is a function of x and y. Explain. 5 5 y
Use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u = cos θi+ sin θj.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the
Explain whether the Quotient Rule should be used to find the partial derivative. Do not differentiate. ax-y -2 ay x² + 1
Describe the level surface F(x, y, z) = 0.F(x, y, z) = x² + y² + z²-25
Use the definition of the limit of a function of two variables to verify the limit. lim (x, y)→(1, -3) y = -3
Find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t. Function w = x sin y x = e¹, y = π-t Value t = 0
Find the minimum distance from the point to the plane x - y + z = 3.(1, 2, 3)
Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema.g(x, y) = 5 -(x − 3)² – (y + 2)² - -
Find the total differential.z = 2x4y - 8x²y³
Use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u = cos θi+ sin θj.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the
Explain whether the Quotient Rule should be used to find the partial derivative. Do not differentiate. a/x - y axx² + 1,
Describe the level surface F(x, y, z) = 0.F(x, y, z) = 4x² +9y² - 4z²
Determine whether z is a function of x and y.х2z + 3у² - ху = 10
Find the total differential. Z - 1 x² + y² 2
Use the definition of the limit of a function of two variables to verify the limit. lim (x, y)→(a, b) y = b
Findthe minimum distance from the point to the surfacez = √1 - 2x - 2y. (-2, -2, 0)
Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema.ƒ(x, y) =√x² + y² + 1
Find dw/dt using the appropriate Chain Rule. Evaluate dw/dt at the given value of t. Function w = In ² X x = cos t, y = sin t Value TT 4
Use Theorem 13.9 to find the directional derivative of the function at P in the direction of the unit vector u = cos θi+ sin θj.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the
Find the total differential. w= x + y 2 - Зу
Describe the level surface F(x, y, z) = 0.F(x, y, z) = 16x² - 9y² + 36z
Explain whether the Quotient Rule should be used to find the partial derivative. Do not differentiate. ə ху ax x² + 1
Determine whether z is a function of x and y.xz² + 2xy - y² = 4
Find the minimum distance from the point to the surface z = √1 - 2x - 2y. (-4, 1, 0)
Find the indicated limit by using the limits limf(x, y) = 4 and (x, y) b) lim (x, y)→(a, b) g(x, y) = 3.
Find a unit normal vector to the surface at the given point. Surface 3x + 4y + 12z = 0 Point (0, 0, 0)
Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema.ƒ(x, y) = √25- (x - 2)² - y²
UseTheorem 13.9 to find the directional derivative of the functionat P in the direction of v.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the unit vector u = cos i + sin 0j
Determine whether z is a function of x and y. x² 4 9 + z² = 1
Find dw/dt(a) By using the appropriate Chain Rule (b) By convertingw to a function of t before differentiatingw = xy, x = et, y = e-2t
Explain whether the Quotient Rule should be used to find the partial derivative. Do not differentiate. Ә Ә ху дух2 + 1
Find the indicated limit by using the limits limf(x, y) = 4 and (x, y) b) lim (x, y)→(a, b) g(x, y) = 3.
Find a unit normal vector to the surface at the given point. Surface x² + y² + z² = 6 Point (1, 1, 2)
Find threepositive integers x, y, and z that satisfy the given conditions.The product is 27, and the sum is a minimum.
Use Theorem 13.9 to find the directional derivative of the function at P in the direction of v.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the unit vector u = cos i + sin 0j
Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema.ƒ(x, y) = x² + y² + 2x - 6y + 6
Find the total differential.z = x cos y - y cos x
Find the total differential. (( - x- 2 - 2 +x) = 2 ( -
Find dw/dt(a) By using the appropriate Chain Rule (b) By converting w to a function of t before differentiatingW = cos(x - y), x = t², y = 1
Determine whether z is a function of x and y.z + x ln y - 8yz = 0
Find the indicated limit by using the limits limf(x, y) = 4 and (x, y) b) lim (x, y)→(a, b) g(x, y) = 3.
Find three positive integers x, y, and z that satisfy the given conditions.The sum is 32, and P = xy²z is a maximum.
Identify any extrema of the function by recognizing its given form or its form after completing the square. Verify your results by using the partial derivatives to locate any critical points and test for relative extrema.ƒ(x, y) = -x² - y² + 10x + 12y - 64
Use Theorem 13.9 to find the directional derivative of the function at P in the direction of v.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the unit vector u = cos i + sin 0j
Find a unit normal vector to the surface at the given point. Surface x² + 3y + z³ = 9 Point (2,-1,2)
Find dw/dt(a) By using the appropriate Chain Rule (b) By converting w to a function of t before differentiatingw = x² + y² + z², x = cos t, y = sin t, z = et
Find both first partial derivatives.ƒ(x, y) = 2x - 5y + 3
Find and simplify the function values.ƒ(x, y) = xy(a) (3, 2) (b) (-1,4) (c) (30, 5)(d) (5, y)(e) (x, 2) (f) (5, t)
Find the indicated limit by using the limits limf(x, y) = 4 and (x, y) b) lim (x, y)→(a, b) g(x, y) = 3.
Find three positive integers x, y, and z that satisfy the given conditions.The sum is 30, and the sum of the squares is a minimum.
Find a unit normal vector to the surface at the given point. Surface x²y³y²z+ 2xz³ = 4 Point (-1, 1,-1)
Examine the function for relative extrema and saddle points.h(x, y) = 80x + 80y - x² - y²
Use Theorem 13.9 to find the directional derivative of the function at P in the direction of v.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the unit vector u = cos i + sin 0j
Find the total differential.z = ex sin y
Find both first partial derivatives.ƒ(x, y) = x² - 2y² + 4
Find dw/dt(a) By using the appropriate Chain Rule (b) By converting w to a function of t before differentiatingw = xy cos z, x = t, y = t², z = arccos t
Find the limit and discuss the continuity of the function. lim (x, y)→(2, 1) (2x² + y)
Find and simplify the function values.ƒ(x, y) = 4 - x² - 4y²(a) (0, 0)(b) (0, 1)(c) (2, 3)(d) (1, y) (e) (x, 0)(f) (t, 1)
Find three positive integers x, y, and z that satisfy the given conditions.The product is 1, and the sum of the squares is a minimum.
UseTheorem 13.9 to find the directional derivative of the functionat P in the direction of Q.Data from in Theorem 13.9 THEOREM 13.9 Directional Derivative If f is a differentiable function of x and y, then the directional derivative of fin the direction of the unit vector u = cos i + sin 0j
Examine the function for relative extrema and saddle points.g(x, y) = x² - y² - x - y
Find the total differential.w = ey cos x + z²
Find the limit and discuss the continuity of the function. lim (x, y)→(0, 0) (x + 4y + 1).
Find both first partial derivatives.ƒ(x, y) = x²y³
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