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mathematics
calculus 4th

Questions and Answers of Calculus 4th

A cylinder of radius r and height h has volume V = πr2h.(a) Use the Linear Approximation to show that(b) Estimate the percentage increase in V if r and h are each increased by 2%.(c) The volume of a
Use the Linear Approximation to show that if I = xayb, then 1 za- - y + b y
The monthly payment for a home loan is given by a function ƒ(P, r, N), where P is the principal (initial size of the loan), r the interest rate, and N the length of the loan in months. Interest
Automobile traffic passes a point P on a road of width w feet at an average rate of R vehicles per second.Although the arrival of automobiles is irregular, traffic engineers have found that the
The volume V of a right-circular cylinder is computed using the values 3.5 m for diameter and 6.2 m for height. Use the Linear Approximation to estimate the maximum error in V if each of these values
Show that if ƒ(x, y) is differentiable at (a, b), then the function of one variable ƒ(x, b) is differentiable at x = a. Use this to prove that ƒ(x, y) = √x2 + y2 is not differentiable at (0, 0).
This exercise shows directly (without using Theorem 1) that the function ƒ(x, y) = 5x + 4y2 from Example 1 is differentiable at (a, b) = (2, 1). THEOREM 1 Confirming Differentiability If fx(x, y)
Show directly, as in Exercise 43, that ƒ(x, y) = xy2 is differentiable at (0, 2).Data From Exercise 43This exercise shows directly (without using Theorem 1) that the function ƒ(x, y) = 5x + 4y2
Use the definition of differentiability to prove that if is differentiable at (a, b), then ƒ is continuous at (a, b).
Let ƒ(x) be a function of one variable defined near x = a. Given a number M, set e(x) = f(x) - L(x) e(x) L(x) = f(a) + M(x a), Thus, f(x)=L(x) + e(x). We say that f is differentiable at x =a if M
Define g(x, y) = 2xy(x + y)/(x2 + y2) for (x, y) ≠ 0 and g(0, 0) = 0. In this exercise, we show that g(x, y) is continuous at (0, 0) and that gx(0, 0) and gy(0, 0) exist, but g(x, y) is not
Match graphs (A) and (B) in Figure 18 with the functions: (i) f(x, y) = -x + y (ii) g(x, y) = x + y
Match each of graphs (A) and (B) in Figure 19 with one of the following functions: (i) f(x, y) = (cos x) (cos y) (ii) g(x, y) = cos(x + y)
Match the functions (a)–(f) with their graphs (A)–(F) in Figure 20. (a) f(x, y) = (b) f(x, y) = (c) f(x, y) = (d) f(x, y) = (e) f(x, y) = x + [y] cos(x - y) -1 1+9x + y cos(y2)e-0.1(x + y) -1 1 +
Match the functions (a)–(d) with their contour maps (A)–(D) in Figure 21. (a) f(x, y) = 3x + 4y (c) h(x, y) = 4x - 3y (b) g(x,y)=x-y (d) K(x,y)= x - y
Sketch the graph and draw several vertical and horizontal traces. f(x, y) = 12-3x - 4y
Sketch the graph and draw several vertical and horizontal traces. f(x, y) = 12-3x - 4y
Sketch the graph and draw several vertical and horizontal traces. f(x, y) = x + 4y
Sketch the graph and draw several vertical and horizontal traces.ƒ(x, y) = y2
Sketch the graph and draw several vertical and horizontal traces.ƒ(x, y) = sin(x − y)
Sketch the graph and draw several vertical and horizontal traces. f(x, y) = 1 x + y +1
Sketch contour maps of ƒ(x, y) = x + y with contour intervals m = 1 and 2.
Sketch contour maps of ƒ(x, y) = x + y with contour intervals m = 1 and 2.
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves. f(x, y) = x - y
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves. f(x, y) = y x
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves. f(x,y) = X
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves.ƒ(x, y) = xy
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves.ƒ(x, y) = x2 + 4y2
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves.ƒ(x, y) = x + 2y − 1
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves.ƒ(x, y) = x2
Draw a contour map of ƒ(x, y) with an appropriate contour interval, showing at least six level curves.ƒ(x, y) = 3x2 − y2
Find the linear function whose contour map (with contour interval m = 6) is shown in Figure 22. What is the linear function if m = 3 (and the curve labeled c = 6 is relabeled c = 3)? -6 -3 y 3 + x 6
Use the contour map in Figure 23 to calculate the average rate of change:(a) From A to B. (b) From A to C. B -6 -4 -2 y A 2 4 C c=-3 c=0 + x 6
Refer to the map in Figure 24.(a) At which of A–C is pressure increasing in the northern direction?(b) At which of A–C is pressure increasing in the westerly direction? 1024 1012 1020 1006 1024
Refer to the map in Figure 24.For each of A–C indicate in which of the four cardinal directions, N, S, E, or W, pressure is increasing the greatest. 1024 1012 1020 1006 1024 1020 1016 1004 1028
Refer to the map in Figure 24.Rank the following states in order from greatest change in pressure across the state to least: Arkansas, Colorado, North Dakota, Wisconsin. 1024 1012 1020 1006 1024 1020
Let T(x, y, z) denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given. T(x, y, z)= 2x + 3y -z, T = 0, 1, 2
Let T(x, y, z) denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given. T(x, y, z)= x-y + 2z, T = 0,1,2
Let T(x, y, z) denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given. T(x, y, z)= x + y -z, T = 0, 1,2
Let T(x, y, z) denote temperature at each point in space. Draw level surfaces (also called isotherms) corresponding to the fixed temperatures given. T(x,y,z) = x - y + 2, T = 0, 1, 2,-1, -2
ρ(S, T) is seawater density (kilograms per cubic meter) as a function of salinity S (parts per thousand) and temperature T (degrees Celsius). Refer to the contour map in Figure 25.Calculate the
Let Using polar coordinates, prove that and that does not exist. Show that g(x, y) = cos2θ and observe that cos θ can take on any value between −1 and 1 as (x, y) → (0, 0). f(x,y) = x/(x
Use any method to evaluate the limit or show that it does not exist. lim (x,y)(0,0) x - y x + y
Use any method to evaluate the limit or show that it does not exist. lim (x,y) (0,0) x - y x + y
Use any method to evaluate the limit or show that it does not exist. lim (x,y) (0,0) 3x + 2y
Use any method to evaluate the limit or show that it does not exist. x4-y4 lim (x,y) (0,0) x4 + xy + y4
Show that the limit does not exist by approaching the origin along one or more of the coordinate axes. x+y+z lim 2 (x,y,z) (0,0,0) x + y + z
Show that the limit does not exist by approaching the origin along one or more of the coordinate axes. x - y+z lim (xyz) (0,0,0) x2 + y + z
Use the Squeeze Theorem to evaluateSqueeze Theorem lim (x-16) cos (x,y) (4,0) 1 (x-4) + y,
Evaluate tan x sin lim (0'0)+(^x)
Evaluate the limit or determine that it does not exist. lim (z,w)(-2,1) s(W) ez+w
Evaluate the limit or determine that it does not exist. lim (zw - 9z) (z,w) (-1,2)
Evaluate the limit or determine that it does not exist. lim (x,y) (4.2) y-2 x2 - 4 x
Evaluate the limit or determine that it does not exist. x + y lim (x,y)(0,0) 1 + y2
Evaluate the limit or determine that it does not exist. 1 lim (xy)+(3,4) x + y
Evaluate the limit or determine that it does not exist. lim (x,y)>(0,0) x + y2
Evaluate the limit or determine that it does not exist. COS X lim (x,y) (7,0) siny
Evaluate the limit or determine that it does not exist. lim (x,y)(0,0) cos x sin y y
Evaluate the limit or determine that it does not exist. lim (x,y) (1,-3) ex-y In(x - y)
Evaluate the limit or determine that it does not exist. |xx| lim (x,y)(0,0) [x] + [yl
Evaluate the limit or determine that it does not exist. (xy + 4xy) lim (x,y) (-3,-2)
Evaluate the limit or determine that it does not exist. e-t-y lim (x,y) (2,1)
Evaluate the limit or determine that it does not exist. lim tan(x + y) tan- (xy)(0,0) 1 x2 + y2
Evaluate the limit or determine that it does not exist. lim (x+y+2)e=/(x+y) (x,y) (0,0)
Evaluate the limit or determine that it does not exist. x + y lim (x,y) (0,0) x + y + 1 -1
Evaluate the limit or determine that it does not exist.Rewrite the limit in terms of u = x − 1 and v = y − 1. x + y - 2 lim (x,y) (1,1) |x-1| + y - 1|
Let (a) Show that(b) Show that |ƒ(x, y)| ≤ |x| + |y|.(c) Use the Squeeze Theorem to prove that f(x, y) = x + y x + y
Let a, b ≥ 0. Show that and that the limit does not exist if a + b ≤ 2. xyb lim (x,y) (0,0) x + y = 0 if a + b> 2
Figure 7 shows the contour maps of two functions. Explain why the limit in (A) does not exist. Does appear to exist in (B)? If so, what is its limit? lim_ f(x,y) (xy)
Evaluate (1 + x)y/x. lim (x,y) (0,2)
Is the following function continuous? x + y if x + y < 1 if x + y 1 f(x,y) = 3) = { + 3 1
The function ƒ(x, y) = sin(xy)/xy is defined for xy ≠ 0.(a) Is it possible to extend the domain of ƒ to all of R2 so that the result is a continuous function?(b) Use a computer algebra system to
Compute the first-order partial derivatives. - 2 X
Compute the first-order partial derivatives. Z = X x-y
Compute the first-order partial derivatives.z =√9 − x2 − y2
Compute the first-order partial derivatives. = 2 X x + y
Compute the first-order partial derivatives.z = (sin x)(cos y)
Compute the first-order partial derivatives.z = tan(uv3)
Compute the first-order partial derivatives. z = COS 1 - x y
Compute the first-order partial derivatives.θ = tan−1(xy2)
Compute the first-order partial derivatives.w = ln(x2 − y2)
Compute the first-order partial derivatives.P = sin(2s − 3t)
Compute the first-order partial derivatives.W = er+s
Compute the first-order partial derivatives.Q = reθ
Compute the first-order partial derivatives.z = exy
Compute the first-order partial derivatives.R = e−v2/k
Compute the first-order partial derivatives.z = e−x2−y2
Compute the first-order partial derivatives.P = e√y2+z2
Compute the first-order partial derivatives. U = en r
Compute the first-order partial derivatives.z = yx
Compute the first-order partial derivatives.z = sinh(x2y)
Compute the first-order partial derivatives.z = cosh(t − cos x)
Compute the first-order partial derivatives.w = xy2z3
Compute the first-order partial derivatives. W = X y + z
Compute the first-order partial derivatives. Q= L M -Lt/M
Compute the first-order partial derivatives. W = X (x + y +z)/2
Compute the given partial derivatives. f(x,y) = 3xy + 4xy - 7xys, fx(1,2)
Compute the given partial derivatives.g(u, v) = u ln(u + v), gu(1, 2)
Compute the given partial derivatives.h(x, z) = exz−x2z3 , hz(3, 0)
The heat index I is a measure of how hot it feels when the relative humidity is H (as a percentage) and the actual air temperature is T (in degrees Fahrenheit). An approximate formula for the heat

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