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mathematics
calculus 6th edition

Questions and Answers of Calculus 6th edition

Suppose that f(0) = –3 and f'(x) ≤ 5 for all values of x. How large can f(2) possibly be?
Find the local minimum and maximum values of the function f in Example 1.Data from Example 1Find where the function f(x) = 3x4 – 4x3 – 12x2 + 5 is increasing and where it is decreasing.
Find the absolute maximum and minimum values of the functionf(x) = x3 – 3x2 + 1 –1/2 ≤ x ≤ 4
Calculate ex lim 2 X ∞←x
Prove the identity tan–1 x + cot–1x = π/2.
Find the local maximum and minimum values of the functiong(x) = x + 2 sin x 0 ≤ x ≤ 2π
(a) Use a graphing device to estimate the absolute minimum and maximum values of the function f(x) = x – 2 sin x, 0 ≤ x ≤ 2π. (b) Use calculus to find the exact minimum and maximum values.
Calculate lim ∞04X In x 3√x
Figure 8 shows a population graph for Cyprian honeybees raised in an apiary. How does the rate of population increase change over time? When is this rate highest? Over what intervals is P concave
Find lim X-0 tan x - x 4.3
The Hubble Space Telescope was deployed on April 24, 1990, by the space shuttle Discovery. A model for the velocity of the shuttle during this mission, from liftoff at t = 0 until the solid rocket
Sketch a possible graph of a function f that satisfies the following conditions: 0 on (-, -2) and (2, ), f"(x) 0 on (-∞, -2) and (2, ∞), f"(x)
Find lim X-T 1 sin x COS X
Evaluate lim x ln x. X→0+
Discuss the curve y = x4 – 4x3 with respect to concavity, points of inflection, and local maxima and minima. Use this information to sketch the curve.
Sketch the graph of the function f(x) = x2/3(6 – x)1/3.
Compute lim X-> (π/2)- (sec x (secx - tan x).
Use the first and second derivatives of f(x) = e1/x, together with asymptotes, to sketch its graph.
Calculate lim (1 + sin 4x)cotx x->0+
Find lim x*. x-0+
Evaluate ,1/x lim e¹/x -0-X
Find the volume of the given solid.Under the plane x + 2y – z = 0 and above the region bounded by y = x and y = x4
Use a triple integral to find the volume of the given solid.The solid bounded by the cylinder y = x2 and the planes z = 0, z = 4, and y = 9
Use cylindrical coordinates.Evaluate ∫∫∫E x dV, where E is enclosed by the planes z = 0 and z = x + y + 5 and by the cylinders x2 + y2 = 4 and x2 + y2 = 9.
Find the volume of the given solid.Under the surface z = 2x + y2 and above the region bounded by x = y2 and x = y3
Use polar coordinates to find the volume of the given solid.Below the paraboloid z = 18 – 2x2 – 2y2 and above the xy-plane
Calculate the double integral. fxye¹ dA, R= [0, 1] × [0, 2] R
Use polar coordinates to find the volume of the given solid.Enclosed by the hyperboloid –x2 – y2 + z2 = 1 and the plane z = 2
Use a computer algebra system to find the mass, center of mass, and moments of inertia of the lamina that occupies the region D and has the given density function.D = {(x, y) |0 ≤ y ≤ sin x, 0
Use spherical coordinates.Evaluate ∫∫∫H(9 – x2 – y2) dV, where H is the solid hemisphere x2 + y2 + z2 ≤ 9,z ≥ 0.
Use a triple integral to find the volume of the given solid.The solid enclosed by the paraboloid x = y2 + z2 and the plane x = 16
Calculate the double integral. R x² + y² dA, R= [1,2] × [0, 1]
Find the volume of the given solid.Enclosed by the paraboloid z = x2 + 3y2 and the planes x = 0, y = 1, y = x, z = 0
Use a computer algebra system to find the mass, center of mass, and moments of inertia of the lamina that occupies the region D and has the given density function.D is enclosed by the cardioid r = 1
Use spherical coordinates.Evaluate ∫∫∫E z dV, where E lies between the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 4 in the first octant.
Find the volume of the given solid.Bounded by the coordinate planes and the plane 3x + 2y + z = 6
Use spherical coordinates.Evaluate where E is enclosed by the sphere x2 + y2 + z2 = 9 in the first octant. SSS e√x²+y²+z³ dv. JJE
Use spherical coordinates.Evaluate ∫∫∫E x2dV, where E is bounded by the xz-plane and the hemispheres y = √9 – x2 – z2 and y = √16 – x2 – z2
Explain why the function is differentiable at the given point. Then find the linearization L(x, y) of the function at that point.f(x, y) = sin(2x + 3y), (–3, 2)
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z) = 3x – y – 3z; x + y – z = 0, x2 + 2z2 = 1
Find the first partial derivatives.T(p, q, r) = p ln(q + er)
Find the directional derivative of the function at the given point in the direction of the vector v.g(x, y, z) = (x + 2y + 3z)3/2, (1, 1, 2), v = 2j – k
Verify the linear approximation at (0, 0).2x + 3/4y + 1 ≈ 3 + 2x – 12y
Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint(s).f(x, y, z) = yz + xy; xy = 1, y2 + z2 = 1
Verify the linear approximation at (0, 0).√y + cos2x ≈ 1 + 1/2y
Find the linear approximation of the function f(x, y) = √20x – x2 –7y2 at (2, 1) and use it to approximate f(1.95, 1.08).
Find the extreme values of f on the region described by the inequality.f(x, y) = e–xy, x2 + 4y2 ≤ 1
Find all second partial derivatives of f.f(x, y) = 4x3 – xy2
Use a tree diagram to write out the Chain Rule for the given case. Assume all functions are differentiable.w = f(r, s, t), where r = r(x, y), s = s(x, y), t = t(x, y)
Find the linear approximation of the function f(x, y) = ln(x – 3y) at (7, 2) and use it to approximate f(6.9, 2.06). Illustrate by graphing f and the tangent plane.
Find the linear approximation of the functionand use it to approximate the number f(x, y, z) = √√√x² + y² + z² at (3, 2, 6)
Find the directional derivative of f(x, y, z) = xy + yz + zx at P(1, –1, 3) in the direction of Q(2, 4, 5).
Find the maximum rate of change of f at the given point and the direction in which it occurs.f(x, y) = y2/x, (2, 4)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f(p. q) = qe–p + pe–q, (0. 0)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f(x, y, z) = (x + y)/z, (1, 1, –1)
Find the differential of the function.z = x3 ln(y2)
Find the maximum rate of change of f at the given point and the direction in which it occurs.f(x, y, z) = √x2 + y2 + z2, (3, 6, –2)
Find the differential of the function.v= y cos xy
Use Equation 6 to find dy/dx.√xy = 1 + x2yData from equation 6 dy dx aF ax ƏF ay Fx F
Find the maximum rate of change of f at the given point and the direction in which it occurs.f(x, y, z) = tan(x + 2y + 3z). (–5, 1, 1)
Use Equation 6 to find dy/dx.y5 + x2y3 = 1 + yex2Data from equation 6 dy dx aF ax ƏF ay Fx F
Sketch the graph of the function.f(x, y) = 4x2 + y2 + 1
Find the absolute maximum and minimum values of f on the set D.f(x, y) = 1 + 4x – 5y, D is the closed triangular region with vertices (0, 0), (2, 0), and (0, 3)
Use Equation 6 to find dy/dx.cos(x – y) = xeyData from equation 6 dy dx aF ax ƏF ay Fx F
Find the differential of the function.R = αβ2cos γ
Find the absolute maximum and minimum values of f on the set D.f(x, y) = 3 + xy – x – 2y, D is the closed triangular region with vertices (1, 0), (5, 0), and (1, 4)
Find the differential of the function.W = xyexz
Use Equation 6 to find dy/dx.sin x + cos y = sin x cos yData from equation 6 dy dx aF ax ƏF ay Fx F
Find the first partial derivatives of the function.f(x, y, z)= x sin(y – z)
Use Equations 7 to find ∂z/∂x and ∂z/∂y.x2 + y2 + z2 = 3xyzData from Equation 7 az ax aF ax aF əz az ay aF ay ƏF az
Use Equations 7 to find ∂z/∂x and ∂z/∂y.xyz = cos(x + y + z)Data from Equation 7 az ax aF ax aF əz az ay aF ay ƏF az
Use Equations 7 to find ∂z/∂x and ∂z/∂y.x – z = arctan(yz)Data from Equation 7 az ax aF ax aF əz az ay aF ay ƏF az
Find the first partial derivatives of the function.w = zexyz
Find the first partial derivatives of the function.u = xy sin–1(yz)
Use Equations 7 to find ∂z/∂x and ∂z/∂y.yz = ln(x + z)Data from Equation 7 az ax aF ax aF əz az ay aF ay ƏF az
Describe how the graph of g is obtained from the graph of f. (a) g(x, y) = f(x - 2, y) (c) g(x, y) = f(x + 3, y - 4) (b) g(x, y) = f(x, y + 2)
Find the first partial derivatives of the function.f(x, y) = y5 – 3xy
Find the first partial derivatives of the function.f(x, y, z) = xz – 5x2y3z4
Find the first partial derivatives of the function.u = tew/t
Find the first partial derivatives of the function.f(x, t) = arctan(x√t)
Find the first partial derivatives of the function.f(r, s) = r ln(r2 + s2)
Find the first partial derivatives of the function.w = sin α cos β
Find the first partial derivatives of the function.f(x, y) = x – y/x + y
Find the first partial derivatives of the function.z = tan xy
Find the first partial derivatives of the function.z = (2x + 3y)10
Find the first partial derivatives of the function.f(x, t) = √x In t
Find the first partial derivatives of the function.f(x, t) = e–t cos πx
Find the first partial derivatives of the function.f(x, y) = x4y3 + 8x2y
Find the limit, if it exists, or show that the limit does not exist. lim (5x³x²y²) (x,y) → (1,2)
Find the limit, if it exists, or show that the limit does not exist. 4 xy lim (x,y) → (2.1) x² + 3y²
Find the limit, if it exists, or show that the limit does not exist. lim (x, y) (1,0) In 1 + y² x² + xy 2
Find the limit, if it exists, or show that the limit does not exist. y* lim (x,y) → (0.0) x² + 3y+
Find the limit, if it exists, or show that the limit does not exist. x² + sin²y lim (x,y) → (0,0) 2x² + y²
Find the limit, if it exists, or show that the limit does not exist. xy cos y lim x,y) → (0,0) 3x² + y²
Find the limit, if it exists, or show that the limit does not exist. 6x³y lim (x,y) (0,0) 2x + y + 4 4
Find the limit, if it exists, or show that the limit does not exist. lim (x,y) → (0,0) ху x² + y2 -2
Find the limit, if it exists, or show that the limit does not exist. lim (x,y) → (0,0) x² - y² x² + y²
Find the limit, if it exists, or show that the limit does not exist. x²ye' lim (x,y) → (0,0) x² + 4y² 4
Find the limit, if it exists, or show that the limit does not exist. x² sin²y lim (x,y) → (0,0) x² + 2y²
Find the limit, if it exists, or show that the limit does not exist. xy* 4 lim (x,y) → (0,0) x² + y²

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