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mathematics
calculus

Questions and Answers of Calculus

Establish Green's First IdentityBy applying Gauss's Divergence Theorem to F = f(g?
Establish Green's Second Identity:
In Problems 1-3, use Stokes's Theorem to calculate?1. F = x2 i + y2 j z2 k; S is the hemisphere z = (1 - x2 - y2 and n is the upper normal. 2. F = xy i + yz j + xy k; S is the triangular surface with
Suppose that the surface S is determined by the formula z = g (x, y). Show that the surface integral in Stokes's Theorem can be written as a double integral in the following way:Where n is the upward
Let F = x2 i - 2xy j + yz2 k and ˆ‚S be the boundary of the surface z = xy, 0 ( x ( 1, 0 ( y ( 1, oriented counter clock wise as viewed from above. Use Stokes's Theorem and Problem 13 to
Let F = 2i + xzj + z3 k and ˆ‚S be the boundary of the surface z = xy2, 0 ( x ( 1, 0 ( y 1, oriented counter clock wise as viewed from above. Evaluate
Let F = 2i + xzj + z3 k and ˆ‚S be the boundary of the surface z = x2 y2, x2 + y2 ( a2, oriented counter clock as viewed from above. Evaluate
Let F = 2zi + 2yk, and let ˆ‚S be the intersection of the cylinder x2 + y2 = ay with the hemispherea > 0. Assuming distance in meters and force in newtons, find the work done by the force F
A central force is one of the form F = f(||r||)r, where f has a continuous derivative (expect possibly at ||r|| = 0). Show that the work done by such a force in moving an object around a closed path
Let S be a solid sphere (Or any solid enclosed by a "nice" surface ˆ‚S). Show that(a) By using Stokes's Theorem.(b) By using Gauss's Theorem. Div (Curl F) = 0?
Show that
In Problems 1-3, use Stokes's Theorem to calculate?1. F = 2zi + x j + 3y k; C is the ellipse that is the intersection of the plane z = x and the cylinder x2 + y2 = 4, oriented clock wise as viewed
Sketch a sample of vectors from the vector field F(x, y) = x i + 2y j?
Calculate the flux of F = xi + yj + 3 k across the sphere x2 + y2 + z2 = 1?
EvaluateWhere G is the part of the plane z = x + y above the triangular region with vertices (0, 0, 0), (1, 0, 0), and (0, 2, 0)?
EvaluateWhere And G is the part of the sphere x2 + y2 + z2 = 2 above the plane z = 1 and n is the upward unit normal?
EvaluateWhereF = sin x i + (1 - cos x) y j + 4z kAnd G is the closed surface bounded by z = (9 - x2 - y2, and z = 0 with outward unit normal n?
Let C be the circle that is the intersection of the plane ax + by + z = 0 (a ( 0, b ( 0) and the sphere x2 + y2 + z2 = 9. For F = yi - x j + 3yk, evaluateUse Stokes's Theorem?
Find div F, curl F, grad (div F), and div(curl F) if F(x, y, z) = 2xyz i - 3y2 j + 2y2z k?
Find a function f satisfying (a) (f = (2xy + y)i + (x2 + x + cos y)j; (b) (f = (yz - e-x)i + (xz + ey)j + xy k;
Evaluate:(a)C is the quarter circle from (0, -1) to (1, 0), centered at the origin.(b) C is the curve x = t, y = cos t, z = sin t, 0 ( t ( (/2?
Show thatIs independent of path, and use this to calculate the integral on any path from (0, 0) to (1, 2)?
1. Find the work done by F = y2 i + 2xy j in moving an object from (1, 1) to (3, 4)2. Evaluate
EvaluateIf (a) C is the square path (0, 0) to (1, 0) to (1, 1) to (0, 1) to (0, 0); (b) C is the triangular path (0, 0) to (2, 0) to (2, 1) to (0, 0); (c) C is the circle x2 + y2 = 1 traversed in the
In Problems 1-3, solve each differential equation? 1. d2y / dx2 + 3 dy / dx = ex Suggestion: Let u = dy / dx. 2. y" - y = 0 3. y" - 3y' + 2y = 0, y = 0, y' = 3 when x = 0
Suppose that glucose is infused into the bloodstream of a patient at the rate of 3 grams per minute, but that the patient's body converts and removes glucose from its blood at a rate proportional to
A spring with a spring constant k of 5 pounds per foot is loaded with a 10-pound weight and allowed to reach equilibrium. It is then raised 1 foot and released. What are the equation of motion, the
In Problem 15, what is the absolute value of the velocity of the moving weight as it passes through the equilibrium position?
Suppose the switch of the circuit in Figure 1 is closed at t = 0. Find / as a function of time if C is initially uncharged. (The current at t = 0 will equal zero, since current through an inductance
In Problems 1-3, solve each differential equation? 1. Y" - 5y' + 6y = 0 2. Y" + 5y' - 6y = 0 3. Y" + 6y' - 7y = 0; y = 0, y' = 4 at x = 0
Solve y" - 4y = 0 and express your answer in terms of the hyperbolic functions cosh and sinh?
Show that the solution ofCan be written as Y = ebx (D1 cosh (b2 + c2 x + D2 sinh (b2 + c2 x)
Solve y(4) + 2y(3) + 3y" + 2y' + y = 0. First show that the auxiliary equation is (r2 + r + 1)2 = 0?
Solve y" - 2y' + 2y = 0 and express your answer in the form ceax sin (Î²x + ().Let sin ( = C1 / c and cos ( = C2 / c, where
Solve x2 y" + 5xy' + 4y = 0 by first making the substitution x = ez?
Show that the substitution x = ez transforms the Euler equation ax2 y" + bxy' + cy = 0 to a homogeneous linear equation with constant coefficient?
Show that if r1 and r2 are distinct real roots of the auxiliary equation, then y = C1er1x + C2er2x is a solution of y" + a1 y' + a2 y = 0?
Show that if r1 and r2 are distinct real roots of the auxiliary equation, then y = C1er1x + C2er2x is a solution of y" + a1y' + a2 y = 0?
Recall that complex number have the form a + bi, where a and b are real. These numbers behave much like the real numbers, with the proviso and i2 = -1. Show each of the following: (a) ebi = cos b + i
Let the roots of the auxiliary equation r2 + a1r + a2 = 0 be a ( βi. From Problem 25c, it follows, just as in the real case, that y = c1e(a( βi)x + c2e(a-βi)x satisfies (D2 + a1D + a2)y = 0. Show
Use a CAS to solve each of the following equations: 1. Y" - 4y' - 6y = 0; y(0) = 1, y'(0) = 2 2. Y" + 5y' + 6.25y = 0; y(0) = 2, y'(0) = - 1,5 3. 2y" + y' + 2y = 0; y(0) = 0, y'(0) = 1.25
In Problems 1-3, use the method of undetermined coefficients to solve each differential equation? 1. Y" - 9y = x 2. Y" + y' - 6y = 2x2 3. Y" - 2y' + y = x2 + x
In Problems 1-3, solve each differential equation by variation of parameters? 1. Y" - 3y' + 2y = 5x + 2 2. Y" - 4y = e2x 3. Y" + y = cse x cot x
Let L(y) = y" + by' + cy = 0 have solution u1 and u2, and let yp = v1u1 + v2u2. Show that L(yp) = v1(u"1 + bu'1 + cu1) + v2(u"2 + bu'2 + cu2) + b(v'1u1 + v'2u2) + (v'1 u1 + v'2u2)' + (v'1u'1 +
A spring with a spring constant k of 250 newtons per meter is loaded with a 10-kilogram mass and allowed to reach equilibrium. It is then raised 0.1 meter and released. Find the equation of motion
Use Figure 6.(a) Find Q as a function of time. Assume that the capacitor is initially uncharged.(b) Find I as a function of time.
Using Figure 7, find the current as a function of time if the capacitor is initially uncharged and S is closed at t = 0, the current at t = 0 will equal 0, since the current through an inductance
Using Figure 8, find the steady-state current s a function of time; that is, find a formula for I that is valid when I is very large (t †’ ()?
Suppose that an un-damped spring is subjected to an external periodic force so that its differential equation has the form(a) Show that the equation of motion for A B is' Y = C1 cos Bt + C2 sin Bt +
Show that C1 cos Î²t + C2 sin fit can be written in the form A sin (Î²t + y). Letsin ( = Cil A, and cos y = C2 / A?
Show that the motion of part (a) of Problem 14 is periodic if B / A is rational?
Refer to Figure 9, which shows a pendulum bob of mass m supported by a weightless wire of length L. Derive the equation of motion; that is, derive the differential equation satisfied by 0.
The equation derived in Problem 17 is nonlinear, but for small 0 it is customary to approximate it by the equationHere g = GM / R2, where G is a universal constant, M is the mass of the earth, and R
A spring with a spring constant k of 100 pounds per foot is loaded with a 1-pound weight and brought to equilibrium. It is then stretched an additional 1 inch and released. Find the equation of
In Problem 1, what is the absolute value of the velocity of the moving weight as it passes through the equilibrium position?
A 10-pound weight stretches a spring 4 inches. This weight is removed and replaced with a 20-pound weight, which is then allowed to reach equilibrium. The weight is next raised 1 foot and released
A spring with a spring constant k of 20 pounds per foot is loaded with a 10-pound weight and allowed to reach equilibrium. It is then displaced 1 foot downward and released. If the weight experiences
Determine the motion in Problem 5 if the retarding force equals four times the velocity at every point?
In Problem 5, how long will it take the oscillations to diminish to one-tenth their original amplitude?
Using Figure 5, find the charge Q on the capacitor as a function of time if S is closed at t = 0. Assume that the capacitor is initially uncharged?
In following question use ∈ or ∉ to indicate whether the given object is an element of the given set in the following problems.a. 12 {1, 2, 3, 4, . . . }b. 5 {x: x is a natural number greater
In following question use ( or ( to indicate whether the given object is an element of the given set in the following problems. 1. 12 {1, 2, 3, 4, . . . } 2. 5 {x: x is a natural number greater than
In following question use ( notation to indicate which set is a subset of the other. 1. C = {a, b, 1, 2, 3} and D = {a, b, 1} 2. E = {x, y, a, b}, F = {x, 1, a, y, b, 2} 3. A = {6, 8, 7, 4}, B = {8,
In following question indicate whether the two sets are equal. 1. F = {x: x is a natural number greater than 6}, G = {7, 8, 9, . . . . . } 2. From the following list of sets, indicate which pairs of
In following question find A ( B. 1. A = {2, 3, 4, 5, 6} and B = {4, 6, 8, 10, 12} 2. A = {a, b, c, d, e} and B = {a, d, e, f, g, h} 3. A = ( and B = {x, y, a, b} 4. A = {x: x is a natural number
In following question find A ( B. 1. A = {1, 2, 4, 5} and B = {2, 3, 4, 5} 2. A = {a, e, i, o, u} and B = {a, b, c, d} 3. A = ( and B = {1, 2, 3, 4} 4. A = {x: x is a natural number greater than 5}
In following question letA = {1, 3, 5, 8, 7, 2}B = {4, 3, 8, 10}C = {2, 4, 6, 8, 10}and U be the universal set of natural numbers less than 11. Find the following.1. A'2. B'3. A' ( B'4. (A ( B)'
The difference of two sets, A - B, is defined as the set containing all elements of A except those in B. That is, A - B = A ( B'. Find A - B for each pair of sets in Problems 1-4 if U = {1, 2, 3, 4,
The following table shows information about yearly lows, highs, and percentage changes for the years 2000 to 2012. Let L be the set of years where the low was greater than 8000. Let H be the set of
The number of jobs in 2000, the number projected in 2025, and the projected annual growth rate for jobs in some cities are shown in the following table. Consider the following sets.A = set of cities
Suppose that the following table summarizes the opinions of various groups on the issue of carbon emission controls. Use this table1. Identify the number of individuals in each of the following
In following question write the following sets a second way. 1. {x: x is a natural number less than 8} 2. {x: x is a natural number greater than 6, less than 10} 3. {3, 4, 5, 6, 7} 4. {7, 8, 9, 10,
A survey of 100 aides at the United Nations revealed that 65 could speak English, 60 could speak French, and 40 could speak both English and French. (a) Draw a Venn diagram representing the 100
Suppose that a survey of 100 advertisers in U.S. News, These Times, and World found the following.14 advertised in all three30 advertised in These Times and U.S. News26 advertised in World and U.S.
Records at a small college show the following about the enrollments of 100 first-year students in mathematics, fine arts, and economics. 38 take math 42 take fine arts 20 take economics 4 take
In a survey of the dining preferences of 110 dormitory students at the end of the spring semester, the following facts were discovered about Adam's Lunch (AL), Pizza Tower (PT), and the Dining Hall
Blood types are determined by the presence or absence of three antigens: A antigen, B antigen, and an antigen called the Rh factor. The resulting blood types are classified as follows: type A if the
In following question which of ∅, A, and B are subsets of B? 1. A = 1, 2, 3, 4} and B = {1, 2, 3, 4, 5, 6} 2. A = {a, b, c, d} and B = {c, d, a, b}? 3. Is A ( B if A = {a, b, c, d} and B = {a, b,
In following question indicate whether the given expression is one or more of the following types of numbers: rational, irrational, integer, natural. If the expression is meaningless, so state.1. (a)
In following question evaluate each expression. 1. -322 + 10 ( 2 2. (-3)2 + 10 ( 2 3. 4 + 22 / 2 4. (4 + 2)2 / 2
Which property of real numbers is illustrated in each part of Problems 1 and 2? 1. (a) 8 + 6 = 6 + 8 (b) 5(3 + 7) = 5(3) + 5(7) (c) 6(4 ( 5) = (6 ( 4)(5) (d) -15 + 0 = -15 2. (a) -e ( 1 = -e (b) 4 +
In following questions graph the subset of the real numbers that is represented by each of the following and write your answer in interval notation. 1. (- (, 4) ( (- 3, () 2. [-4, 17] ( [- 20, 10] 3.
A sales clerk's take-home pay is found by subtracting all taxes and retirement contributions from 1gross pay (which consists of salary plus commission). Given the following information, complete
The expenditures E for government public health activities (in billions of dollars) can be approximated by E = 5.03t2 + 100t + 1380 where t is the number of years past 2000 (Source: Centers for
Using data from 2002 projected through 2016, the number of worldwide users of the Internet, in billions, can be approximated quite accurately either by (1) y = 0.207t - 0.000370 or by (2) y =
From data adapted from the National Center for Health Statistics, the height H in inches and age A in years for boys between 4 and 16 years of age are related according to H = 2.31A + 31.26 To
Use the following federal tax table for a single person claiming one personal exemption. Taxable Income I ___________________________ Tax Due T 0-8375
Write all answers without using exponents. 1. (-4)4 2. -53 3. -26 4. (-2)5
Simplify the expressions with all exponents positive. 1. 65 ( 63 2. 84 ( 82 ( 8 3. 108 / 109 4. 78 / 73
Rewrite the expression with positive exponents (x, y, z ( 0). 1. -x-3 2. x-4 3. xy-2z0 4. 4-1x0y-2
Use the rules of exponents to simplify so that only positive exponents remain. 1. x3 ( x4 2. α5 ( α 4. x-5 ( x3 3. y-5 ( y-2
Compute and simplify so that only positive exponents remain. 1. (2x-2y)-4 2. (-32x5)-3 3. (-8α -3b2) (2α5b-4) 4. (-3m2y-1) (2m-3y-1)
If $P is invested for n years at rate I (as a decimal), compounded annually, the future value that accrues is given by S = P(1 + i)n, and the interest earned is I = S - P. Find S and I for the given
If an investment has a goal (future value) of $S after n years, invested at interest rate i (as a decimal), compounded annually, then the present value P that must be invested is given by P = S(1 +
For selected years from 1960 to 2018, total U.S. personal income I (in billions of dollars) can be approximated by the formulaI = 492.4(1.070)twhere t is the number of years past 1960.(a) What
The total number of endangered species y can be approximated by the formulawhere t is the number of years past 1980.(a) The actual numbers of endangered species for selected years were as follows.For
By using Social Security Administration data for selected years from 1950 and projected to 2050, the U.S. population, ages 20-64, P (in millions) can be approximated by the equationwhere t is the
The national health care expenditure H (in billions of dollars) can be modeled (that is, accurately approximated) by the formula H = 738.1(1.065)t where t is the number of years past 1990. (a) What
Use a calculator to evaluate the indicated powers. 1. 1.24 2. (-3.7)3 3. (1.5)-5 4. (-0.8)-9
Find the powers and roots, if they are real numbers. 1. (a) (256/9 (b) (1.44 2. (a) 5(-323 (b) 4(-1655 3. (a) 163/4 (b) (-16)-3/2 4. (a) -27-1/3 (b) 323/5

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