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mathematics
calculus 10th edition

Questions and Answers of Calculus 10th Edition

Find both first partial derivatives.z = yey/x
Find an equation of the tangent plane to the surface at the given point.x² + 2z² = y², (1, 3, -2)
Find an equation of the tangent plane to the surface at the given point.x² + 4y² + z² = 36, (2, -2, 4)
A water line is to be built from point P to point S and must pass through regions where construction costs differ (see figure). The cost per kilometer (in dollars) is 3k from P to Q, 2k from Q to R,
Find and simplify the function values.ƒ(x, y) = 2x + y²(a)(b) f(x + Ax, y) = f(x, y)
Examine the function for relative extrema and saddle points. f(x, y) = x² - xy - y² - 3x-y Z 4 3 X 3 -y
Find the limit and discuss the continuity of the function. lim (x, y)→(π/4, 2) y cos xy
Find z = ƒ(x, y) and use the total differential to approximate thequantity.(2.01)²(9.02) - 2² . 9
Find an equation of the tangent plane to the surface at the given point. h(x, y) = cos y, 5, TT 4' √2 2
Find the gradient of the function at the given point.w = 3x² - 5y2 + 2z², (1, 1, -2)
One way to measure species diversity is to use the Shannon diversity index H. If a habitat consists of three species, A, B, and C, then its Shannon diversity index iswhere x is the percent of species
Find aw/asand ∂w/∂t (a) By using the appropriate Chain Rule (b) Byconverting w to a function of s and t before differentiating.W = xyz, x = s + t, y = s - t, z = st²
Find both first partial derivatives.z = x²e²y
Find the gradient of the function at the given point.z = cos(x² + y²), (3, -4)
Common blood types are determined genetically by three alleles A, B, and O. (An allele is any of a group of possible mutational forms of a gene.) A person whose blood type is AA, BB, or OO is
In Exercises 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
Find the curvature of the curve, where s is the arc length parameter. r(t) = (4(sin t - t cos t), 4(cost + t sin t), 2(1²)
Find(a) r'(t)(b) r"(t)(c) r'(t) . r"(t)(d) r'(t) × r”(t). r(t) = (4t + 3)i + t²j + (2t² + 4)k
Define the dot product of vectors u and v.
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = −i + († − 1)j 4
Find T(t), N(t), aT, and an at the given time t for the plane curve r(t). r(t) = (t = t³)i + 2f²j, t = 1 -
Use the properties of the derivative to find the following.(a) r'(t)(b)(c)(d)(e)(f) d dt - [u(t) — 2r(t)]
Find(a) r'(t)(b) r"(t)(c) r'(t) · r"(t). r(t) = 4 cos ti + 4 sin tj
Sketch the curve represented by the vector-valued function and give the orientation of the curve. [¹/^ + !(¹ − S) = (¹).J
Find T(t), N(t), aT, and an at the given time t for the plane curve r(t). r(t) = (t³ − 4t)i + (² − 1)j, t = 0 - -
Use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2.a(t) = -cos ti - sin tj, v(0) = j + k, r(0) = i
Find(a) r'(t)(b) r"(t)(c) r'(t) · r"(t). r(t) = 8 cos ti + 3 sin tj
Use the properties of the derivative to find the following.(a) r'(t)(b)(c)(d)(e)(f) d dt - [u(t) — 2r(t)]
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = t³i + t²j
Find the curvature K of the plane curve at the given value of the parameter.r(t) = 4ti - 2tj, t = 1
Find T(t), N(t), aT, and an at the given time t for the plane curve r(t). r(t) = e¹i + e-2¹j, t = 0
Use the given acceleration vector to find the velocity and position vectors. Then find the position at time t = 2.a(t) = eti - 8k, v(0)=2i + 3j + k, r(0) = 0
Find(a) r'(t)(b) r"(t)(c) r'(t) . r"(t)(d) r'(t) × r"(t) r(t) = t²i- tj + t³k
Find the curvature K of the plane curve at the given value of the parameter. r(t) = ti + ti - + =j, t = 1
Find the indefinite integral. S (i + 3j + 4tk) dt
Find the curvature K of the plane curve at the given value of the parameter.r(t) = t²i + j, t = 2
Find the curvature K of the plane curve at the given value of the parameter. r(t) = ti + ³j, t = 2 1 9°
A baseball is hit from a height of 2.5 feet above the ground with an initial velocity of 140 feet per second and at an angle of 22° above the horizontal. Find the maximum height reached by the
Find the indefinite integral. Sa (t²i + 5tj + 8f³k) dt
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = (t² + t)i + (t²-t)j
Find T(t), N(t), aT, and an at the given time t for the plane curve r(t). r(t) = e¹i + e¹j + tk, t = 0
Find(a) r'(t)(b) r"(t)(c) r'(t) . r"(t)(d) r'(t) × r"(t) r(t) = t³i+ (2t² + 3)j + (3t 5)k
Find T(t), N(t), aT, and an at the given time t for the plane curve r(t). r(t) = et cos ti + e¹ sin tj, t = TT 2
Determine the maximum height and range of a projectile fired at a height of 3 feet above the ground with an initial velocity of 900 feet per second and at an angle of 45° above the horizontal.
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(0) cos i 3 sin ej
Find the indefinite integral. 2 (3√ti + ²7j + k) dt
Find(a) r'(t)(b) r"(t)(c) r'(t) . r"(t)(d) r'(t) × r"(t) r(t) = (cost + t sin t, sin t - t cos t, t)
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = 2 cos ti + 2 sin tj
Find the curvature K of the plane curve at the given value of the parameter. r(t) = (t, sin t), t = 2
Find T(t), N(t), aT, and an at the given time t for the plane curve r(t). r(t) = 4 cos 3ti + 4 sin 3tj, t = π
Find(a) r'(t)(b) r"(t)(c) r'(t) . r"(t)(d) r'(t) × r"(t) r(t) = (e¹, t², tan t)
A baseball, hit 3 feet above the ground, leaves the bat at an angle of 45° and is caught by an outfielder 3 feet above the ground and 300 feet from home plate. What is the initial speed of the ball,
Find the curvature K of the plane curve at the given value of the parameter. r(t) = (5 cos t, 4 sin t), t = E|M 3
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(0) = 3 sec 0i + 2 tan 0j
Find the indefinite integral. S (sin ti + cos tj + e²¹ k) dt
A baseball player at second base throws a ball 90 feet to the player at first base. The ball is released at a point 5 feet above the ground with an initial velocity of 50 miles per hour and at an
Eliminate the parameter t from the position vector for the motion of a projectile to show that the rectangular equation is y = 16 sec² 0 2 0 x² + (tan 0)x+ h.
Find the open interval(s) on which the curve given by the vector-valued function is smooth.r(t) = t²i + t³j
Consider an object moving according to the position vector r(t) = a cos ωt i + a sin ωt j.Find T(t), N(t), aT, and aN.
Find the curvature K of the curve.r(t) = 4 cos 2πti + 4 sin 2πtj
Find an equation of the tangent plane to the surface at the given point.ƒ(x, y) = x² - 2xy + y², (1, 2, 1)
(a) Evaluate ƒ(2, 1) and ƒ(2.1, 1.05) and calculate Δz(b) Use the total differential dz to approximate Δz.ƒ(x, y) = x² + y²
Evaluate the definite integral. -2 (3ti + 2t²j t³ k) dt -
The path of a ball is given by the rectangular equationUse the result of Exercise 29 to find the position vector. Then find the speed and direction of the ball at the point at which it has traveled
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = 2 cos³ ti + 2 sin³ tj
Find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) = 1 t - 1 -i + 3tj
Evaluate the definite integral. [₁ + (ti + √tj + 4tk) dt
Consider an object moving according to the position vector r(t) = a cos ωt i + a sin ωt j.Determine the directions of T and N relative to the position vector r.
Find the curvature K of the curve.r(t) = 2 cos πti+ sin πtj
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = (t + 1)i + (4t + 2)j + (2t + 3) k
Evaluate the definite integral. 2 [²(e1¹/²1 (et/2i 3t²jk) dt
The Rogers Centre in Toronto, Ontario, has a center field fence that is 10 feet high and 400 feet from home plate. A ball is hit 3 feet above the ground and leaves the bat at a speed of 100 miles per
The quarterback of a football team releases a pass at a height of 7 feet above the playing field, and the football is caught by a receiver 30 yards directly downfield at a height of 4 feet. The pass
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = ti + (2t - 5)j + 3tk
Find the open interval(s) on which the curve given by the vector-valued function is smooth.r(θ) = 2 cos³ θi + 3 sin³ θj
Consider an object moving according to the position vector r(t) = a cos ωt i + a sin ωt j.Determine the speed of the object at any time t and explain itsvalue relative to the value of aT.
Find the curvature K of the curve.r(t) = a cos ωti + a sin ωtj
Evaluate the definite integral. TT/3 (2 cos ti+ sin tj + 3k) dt
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = 2 cos ti + 2 sin tj + tk
Find the open interval(s) on which the curve given by the vector-valued function is smooth.r(θ) = (θ + sin θ)i + (1 - cos θ)j
Sketch the graph of the plane curve given by the vector-valued function, and, at the point on the curve determined by r(t0),sketch the vectors T and N. Note that N points toward theconcave side of
Consider an object moving according to the position vector r(t) = a cos ωt i + a sin ωt j.When the angular velocity ω is halved, by what factor is aN changed?
Find the curvature K of the curve.r(t) = a cos ωti + b sin ωtj
A bomber is flying at an altitude of 30,000 feet at a speed of 540 miles per hour (see figure). When should the bomb be released for it to hit the target? (Give your answer in terms of the angle of
Find r(t) that satisfies the initial condition(s).r'(t) = 2ti + etj + e-t k, r(0) = i + 3j - 5k
Find the curvature K of the curve. r(t) = ti + t²j+ 1² 2 k
A bale ejector consists of two variable-speed belts at the end of a baler. Its purpose is to toss bales into a trailing wagon. In loading the back of a wagon, a bale must be thrown to a position 8
Find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) = 2t 8 + 13 i + 21² 8 + 1³ j
Find the open interval(s) on which the curve given by the vector-valued function is smooth.r(θ) = (θ - 2 sin θ)i + (1 - 2 cos θ)j
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = ti + 3 cos tj + 3 sin 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
Sketch the graph of the plane curve given by the vector-valued function, and, at the point on the curve determined by r(t0), sketch the vectors T and N. Note that N points toward the concave side of
Find r(t) that satisfies the initial condition(s).r'(t) = sec ti + tan tj + t²k, r(0) = 3k
Find the curvature K of the curve. r(t) = 2t² i + tj + 1 t²k
Sketch the curve represented by the vector-valued function and give the orientation of the curve. r(t) = 2 sin ti + 2 cos tj + ek
Sketch the graph of the plane curve given by the vector-valued function, and, at the point on the curve determined by r(t0), sketch the vectors T and N. Note that N points toward the concave side of
Find the open interval(s) on which the curve given by the vector-valued function is smooth. r(t) = (t − 1)i + ⁄ j − t²k - -
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
A shot fired from a gun with a muzzle velocity of 1200 feet per second is to hit a target 3000 feet away. Determine the minimum angle of elevation of the gun.
Sketch the graph of the plane curve given by the vector-valued function, and, at the point on the curve determined by r(t0), sketch the vectors T and N. Note that N points toward the concave side of

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