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mathematics
precalculus

Questions and Answers of Precalculus

The paraboloid z = 6- x - x2 - 2y2 intersects the plane x = 1 in a parabola. Find parametric equations for the tangent line to this parabola at the point (1, 2, -4). Use a computer to graph the
You are told that there is a function f whose partial derivatives are fx(x, y) = x + 4y and fy(x, y) = 3x - y. Should you believe it?
If a, b, c are the sides of a triangle and A, B, C are the opposite angles, find ∂A/∂a, ∂A/∂b, ∂A/∂c by implicit differentiation of the Law of Cosines.
The kinetic energy of a body with mass m and velocity v isK = 1/2 mv2. Show that әк әк K дт ди?
The average energy E (in kcal) needed for a lizard to walk or run a distance of 1 km has been modeled by the equationE(m, v) = 2.65m0.66 + 3.5m0.75/vwhere m is the body mass of the lizard (in grams)
In the project on page 344 we expressed the power needed by a bird during its flapping modewhere A and B are constants specific to a species of bird, v is the velocity of the bird, m is the mass of
One of Poiseuille’s laws states that the resistance of blood flowing through an artery isR = C L/r4where L and r are the length and radius of the artery and C is a positive constant determined by
A model for the surface area of a human body is given by the functionS = f (w, h) = 0.1091w0.425h0.725where w is the weight (in pounds), h is the height (in inches), and S is measured in square feet.
The wind-chill index is modeled by the functionW = 13.12 + 0.6215T - 11.37v0.16 + 0.3965Tv0.16where T is the temperature (°C) and v is the wind speed (km/h). When T = 215°C and v = 30 km/h, by how
For the ideal gas of Exercise 88, show thatT∂P/∂T ∂V/∂T = mR
The gas law for a fixed mass m of an ideal gas at absolute temperature T, pressure P, and volume V is PV = mRT, where R is the gas constant. Show that∂P/∂V ∂V/∂T ∂T/∂P= -1
The van der Waals equation for n moles of a gas is(P + n2a/V2) (V - nb) = nRTwhere P is the pressure, V is the volume, and T is the temperature of the gas. The constant R is the universal gas
Cobb and Douglas used the equation P(L, K) = .01L0.75K0.25 to model the American economy from 1899 to 1922, where L is the amount of labor and K is the amount of capital. (See Example 14.1.3.)(a)
The total resistance R produced by three conductors with resistances R1, R2, R3 connected in a parallel electrical circuit is given by the formula1/R = 1/R1 + 1/R2 + 1/R3Find ∂R/∂R1.
The temperature at a point (x, y) on a flat metal plate is given by T(x, y) = 60/(1 + x2 + y2), where T is measured in 8C and x, y in meters. Find the rate of change of temperature with respect to
The diffusion equation∂c/∂t = D ∂2c/∂x2where D is a positive constant, describes the diffusion of heat through a solid, or the concentration of a pollutant at time t at a distance x from the
If f and g are twice differentiable functions of a single variable, show that the functionu(x, t) = f (x + at) + g(x - at)is a solution of the wave equation given in Exercise 78.
Show that each of the following functions is a solution of the wave equation ut t = a2uxx.(a) u = sin(kx) sin(akt)(b) u = t/(a2t2 - x2)(c) u = (x - at)6 + (x + at)6(d) u = sin(x - at) + ln(x + at)
Verify that the function u = 1/√x2 + y2 + z2 is a solution of the three-dimensional Laplace equation uxx + uyy + uzz = 0.
Verify that the function u = e-a2k2t sin kx is a solution of the heat conduction equation ut = a2uxx.
Level curves are shown for a function f. Determine whether the following partial derivatives are positive or negative at the point P.(a) fx (b) fy (c) fxx (d) fxy (e) fyy УА 10
Use the table of values of f (x, y) to estimate the values of fx(3, 2), fx(3, 2.2), and fxy(3, 2). 1.8 2.0 2.2 2.5 12.5 10.2 9.3 18.1 3.0 17.5 15.9 3.5 26.1 20.0 22.4
If g(x, y, z) = √1 + xz + √1 - xy , find gxyz.
If f (x, y, z) = xy2z3 + arcsin(x√z), find fxzy.
Find the indicated partial derivative(s).u = xaybzc; ∂6W/∂x∂y2∂z3,
Find the indicated partial derivative(s).w = x/y + 2z; ∂3W/∂z∂y∂x, ∂3W/∂x2∂y
Find the indicated partial derivative(s).V = In (r + s2 + t3); ∂3V/∂r∂s ∂t
Find the indicated partial derivative(s).W = √u + v2 ; ∂3W/∂u2 ∂v
Find the indicated partial derivative(s).g(r, s, t) = er sin(st); grst
Find the indicated partial derivative(s).f (x, y, z) = exyz2; fxyz
Find the indicated partial derivative(s).f (x, y) = sin(2x + 5y); fyxy
Find the indicated partial derivative(s).f (x, y) = x4y2 - x3y; fxxx, fxyx
Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx.u = ln(x + 2y)
Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx.u = cos(x2y)
Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx.u = exy sin y
Verify that the conclusion of Clairaut’s Theorem holds, that is, uxy = uyx.u = x4y3 - y4
Find all the second partial derivatives.w = √1 + uv2
Find all the second partial derivatives.v = sin(s2 - t2)
Find all the second partial derivatives.T = e-2r cos θ
Find all the second partial derivatives.z = y/2x + 3y
Find all the second partial derivatives.f (x, y) = ln(ax + by)
Find all the second partial derivatives.f (x, y) = x4y - 2x3y2
Find ∂z/∂x and ∂z/∂y.(a) z = f (x) g(y) (b) z = f (xy)(c) z = f (x/y)
Find ∂z/∂x and ∂z/∂y.(a) z = f (x) + g(y) (b) z = f (x + y)
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.yz + x ln y = z2
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.ez = xyz
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.x2 - y2 + z2 - 2z = 4
Use implicit differentiation to find ∂z/∂x and ∂z/∂y.x2 + 2y2 + 3z2 = 1
Use the definition of partial derivatives as limits (4) to find fx(x, y) and fy(x, y).f (x, y) = x/x + y2
Use the definition of partial derivatives as limits (4) to find fx(x, y) and fy(x, y).f (x, y) = xy2 - x3y
Find the indicated partial derivative. f (x, y, z) = xyz; fz (e, 1, 0)
Find the indicated partial derivative. f(x, y, z) = ln 1 - √x2 + y2 + z2/1 + √x2 + y2 + z2; fy (1, 2, 2)
Find the indicated partial derivative. f (x, y) = y sin-1(xy); fy (1, 1/2)
Find the indicated partial derivative.R(s, t) = tes/t; Rt (0, 1)
Find the first partial derivatives of the function.f (x, t) = √3x + 4t
Find the first partial derivatives of the function.u = sin(x1 + 2x2 + ∙ ∙ ∙ + nxn)
Find the first partial derivatives of the function. и %3D Vx? + x? +... + x?
Find the first partial derivatives of the function.(x, y, z, t) = ax + By2/yz + δt2
Find the first partial derivatives of the function.h(x, y, z, t) = x2y cos(z/t)
Find the first partial derivatives of the function.u = xy/z
Find the first partial derivatives of the function.p = √t4 + u2 cos v
Find the first partial derivatives of the function.w = y tan(x + 2z)
Find the first partial derivatives of the function.w = ln(x + 2y + 3z)
Find the first partial derivatives of the function.f (x, y, z) = xy2e-xz
Find the first partial derivatives of the function.f (x, y, z) = x3yz2 + 2yz
Find the first partial derivatives of the function.
Find the first partial derivatives of the function. x, cos(e') di F(x, y) Jy
Find the first partial derivatives of the function.f (x, y) = xy
Find the first partial derivatives of the function.R(p, q) = tan-1(pq2)
Find the first partial derivatives of the function.u(r, θ) = sin(r cos θ)
Find the first partial derivatives of the function.g(u, v) = (u2v - v3)5
Find the first partial derivatives of the function.w = ev/u + v2
Find the first partial derivatives of the function.f (x, y) = ax + by/cx + dy
Find the first partial derivatives of the function.f (x, y) = x (x + y)2
Find the first partial derivatives of the function.f (x, y) = x/y
Find the first partial derivatives of the function.z = x sin(xy)
Find the first partial derivatives of the function.z = ln(x + t2)
Find the first partial derivatives of the function.f (x, y) = x2y - 3y4
Find the first partial derivatives of the function.f (x, y) = x4 + 5xy3
Find fx and fy and graph f, fx, and fy with domains and viewpoints that enable you to see the relationships between them.f (x, y) = y/1 + x2y2
Find fx and fy and graph f, fx, and fy with domains and viewpoints that enable you to see the relationships between them.f (x, y) = x-2y3
If f (x, y) = √4 - x2 - 4y2 , find fx(1, 0) and fy(1, 0) and interpret these numbers as slopes. Illustrate with either handdrawn sketches or computer plots.
A contour map is given for a function f. Use it to estimate fx(2, 1) and fy(2, 1). -3 6. -2 10 12 14 16 2 3 18
The following surfaces, labeled a, b, and c, are graphs of a function f and its partial derivatives fx and fy. Identify each surface and give reasons for your choices. 4 -4 a -8 -3 -2 -1 0 1 2 х 3.
Determine the signs of the partial derivatives for the function f whose graph is shown.(a) fxy(1, 2) (b) fxy(-1, 2) ZA х.
Determine the signs of the partial derivatives for the function f whose graph is shown.(a) fxx(-1, 2) (b) fyy(-1, 2) ZA х.
Determine the signs of the partial derivatives for the function f whose graph is shown.(a) fx(-1, 2) (b) fy(-1, 2) ZA х.
Determine the signs of the partial derivatives for the function f whose graph is shown.(a) fx(1, 2) (b) fy(1, 2) ZA х.
The wave heights h in the open sea depend on the speed v of the wind and the length of time t that the wind has been blowing at that speed. Values of the function h = f (v, t) are recorded in feet in
The wind-chill index W is the perceived temperature when the actual temperature is T and the wind speed is v, so we can write W = f(T, v). The following table of values is an excerpt from Table 1 in
At the beginning of this section we discussed the function I = f (T, H), where I is the heat index, T is the temperature, and H is the relative humidity. Use Table 1 to estimate fT(92, 60) and fH
Show that the function f given by f (x) = |x| is continuous on Rn.Consider |x - a|2 = (x - a) • (x - a).]
Graph and discuss the continuity of the function(a) Show that f (x, y) → 0 as (x, y) → (0, 0) along any path through (0, 0) of the form y = mxa with 0 < a < 4.(b) Despite part (a), show
At the beginning of this section we considered the functionf (x, y) = sin(x2 + y2)/x2 + y2and guessed on the basis of numerical evidence that f (x, y) → 1 as (x, y) → (0, 0). Use polar
Use polar coordinates to find the limit. [If (r, θ) are polar coordinates of the point (x, y) with r > 0, note that r → 0+ as (x, y)→ (0, 0).] e-x²-y² – 1 lim (x, y)- (0, 0) x + y?
Use polar coordinates to find the limit. [If (r, θ) are polar coordinates of the point (x, y) with r > 0, note that r → 0+ as (x, y)→ (0, 0).] (x² + y²) In(x² + y²) lim (х, у) — (0, 0)
Use polar coordinates to find the limit. [If (r, θ) are polar coordinates of the point (x, y) with r > 0, note that r → 0+ as (x, y)→ (0, 0).] х + у lim ,3 (х, 9) — (о, о) х? + у?
Determine the set of points at which the function is continuous. ху if (x, y) + (0, 0) f(t. ) — { x* + ху + у? if (x, y) = (0, 0)
Determine the set of points at which the function is continuous. .2. x*y if (x, y) + (0, 0) if (x, y) = (0, 0) 2x + y? f(x, y) =
Determine the set of points at which the function is continuous.f (x, y, z) = √y - x2 ln z

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