Differential equations Questions and Answers

Use a half angle formula to fill in the blanks in the identity below: (cos(4x))² = H +1 Cos x)

Solve 8 cos(4x) 6 for the smallest three positive solutions. Give your answers accurate to at least two decimal places, as a list separated by commas

Find for the following: dy da 2Ty = y T²+y Ody= da dy da dy dr dy da II -2xy+2y T²-2x+2y -xy+y -x+y | ********* -Ty+y 2x²-x+y De 2xy -x²+2x Jhy

Find the inverse of y = sin 2x. O2 sin¹(x) O sin¹(2x) O sin-1(x)/2 O2 sin¹ (₂)

4x √x² - 4x Determine the derivative of f. Let f(x) df dx = Determine the slope of fat x = 9. f'(9) =

Houston Pumps recently reported $222,500 of sales, $140,500 of operating costs other than depreciation, and $9,250 of depreciation. The company had $35,250 of outstanding bonds that carry a 6.75% interest rate, and its federal-plus-state income tax rate was 35%. In order to sustain its operations and thus generate future sales and cash flows, the firm was required to spend $15,250 to buy new fixed assets and to invest $6,850 in net operating working capital. What was the firm's free cash flow? O a. $26,517 O b. $27,894 O c. $33,060 O d. $34,438 Oe. $39,948

Finance. A person wishes to have $22,600 cash for a new car 6 years from now. How much should be placed in an account now, if the account pays 5.4% annual interest rate, compounded weekly?

2. The data in the table shows the times for Men's 500-m Speed Skating event at the Winter Olympics. Let x be the number of years since 1980. Year 1984 1988 1992 1994 1998 2002 2006 Time (sec) 38.19 36.45 37.14 36.33 35.59 34.42 34.84 SOURCE: www.infoplease.com DDDDD a. Find a quadratic model for the data set. b. Find a cubic model for the data set. c. Find a quartic model for the data set. d. Compare the models and determine which one is more appropriate. Explain your choice.

We continue to guess-check-revise by guessing smaller and smaller widths until we have a total area of 2,880 square inches for the mulched border. 156 in + ? in 60 in + 2(?) in Border width of border Complete the table. Use the given width of the border to determine the width, length, and area of the of the large rectangle. Subtract the area of the small rectangle from the area of the large rectangle to determine the area of the border. 11 in 10 in 9 in 8 in 60 in 7 in 156 in Shed width of large rectangle 60 in + 2(11 in) = 82 in 80 in 78 in in 74 in length of large rectangle 156 in + 11 in = 167 in 166 in in 164 in in area of large rectangle 13,694 in 2 in 2 12,870 in² in ² in ² area of small rectangle 9,360 in ² 9,360 in 2 9,360 in ² 2 9,360 in ² 9,360 in ² area of border 4,334 in² in² in² in² in²

For f(x)=√x and g(x) = 4x + 1, find the following composite functions and state the domain of each. (a) fog (b) gof (c) fof (d) gog OB. The domain of g of is all real numbers. (c) (fof)(x)=√√√√x (Simplify your answer.) Select the correct choice below and fill in any answer boxes within your choice. A. The domain of fofis xx (Type an inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) OB. The domain of fo f is all real numbers. (d) (gog)(x) = (Simplify your answer.) Select the correct choice below and fill in any answer boxes within your choice. O A. The domain of g og is {x}. (Type an inequality. Simplify your answer. Use integers or fractions for any numbers in the expression.) OB. The domain of g og is all real numbers.

Define (a) Show that S(h) and state C1. − f (x+2h) +4f(x + h) − 3f(x) = 2h f'(x) - S(h) = c₁h² + c₂h³ + c3h4+...

Define (a) Show that and state C₁. S(h) -f(x+2h) +4f(x+h)-3f(x) 2h f'(x) - S(h) = c₁h² + c₂h³ + c3h² + ... 4

[5 P] For the following differential equation, determine (without solving) an interval on which the solution of the equation is certain to exist and be unique. Justify your answer. (int)y + y = c²cost, y()=2

Linearize the following ODE and find the corresponding initial conditions: x + 2x + sin x = 0.5, x (0) = -1, x(0) = 0

Choose a suitable method, find the general solution of the following second order differential equations. a) d²y/dx² + 4y = 5sin(3x) b) 2(d²y/dx²) + 18y = 6tan(3x) Justify the reason for the chosen method.

Consider the differential equation dy/dx = 3(2x + 1)sin(x²+x+π/2). (A) Find the equation of the line tangent to the solution curve at the point (0, 3). (B) Find the second derivative at (0, 3) and use it to determine the concavity of the solution curve at point. Explain. Find the particular solution y = f(x) with initial condition f(0) = 3.

What is the area of the region in the first quadrant enclosed by y=e^(-x^2/4) and the line y = 0.5. A) 0.516 sq units B) 0.240 sq units C) 0.480 sq units D) 1.032 sq units

Which of the following integrals correctly gives the area of the region consisting of all points above the x-axis and below the curve y= - x^2 + 2x + 8. 4 a. ∫(8 +2x - x^2)dx -2 4 b. ∫ (x^2 - 2x -8)dx -2 4 c. ∫ (8+ 2x - x^2)dx -2 4 d. ∫ (x^2 - 2x -8)dx -2

Suppose the rate of growth of bacteria in a Petri dish is given by q(t) = 6ᵗ, where t is given in hours and g(t) is given in thousands of bacteria per hour. If a culture starts with 7000 bacteria, find a function Q(t) that gives the number of bacteria in the Petri dish at any time t. How many bacteria are in the dish after 5 hours? You should round your answer to the nearest whole number. Do not include any commas in your final answer (if applicable).

For the following functions, f(x, y) = 4x^2 + 2y^2 – 8xy. Find the minimum using steepest descent method starting from initial point (2,3). Calculate optimum step size at every step.

Find grad f(x) for each of the following functions: (a) f(x) = xo ' x (b) f(x) = |x|, x≠0. (c) f(x) = (xo ' x)^2

Solve the equation in the real number system: 2x³ + 3x² + 2x + 3 = 0.

Determine the domain of the function of two variables f(x ,y) = √y + 9x.

Determine the intervals on which n is continuous. Enter the solution using interval notation. n(x)=13x/8-x n is continuous on=____

Solve the following second order differential equation and initial conditions for the time range shown using the following method. y" + y = u(t-2), y(0) = 0 and y' (0) = 2, 0 ≤ t ≤ ∞ ii) The method of undetermined coefficients, note there will be 2 separate solutions.

Sketch the graph of y²/6² - x²/2² = 1. (Graph by selecting a type of graph, then select the center, then moving the cursor until the graph is correct).

Find the length of the missing side. Leave your answer in simplest radical form. A right angle triangle with 11 cm, 4 cm.

Given x = π/3, what is the exact value of cos(2π - x)? a. 1/2 b. -1/2 c. √3/2 d. -√3/2

Determine the zero(s) of p(x)= x³ + 13x² + 42x. The zeros are x =

Suppose an investment account is opened with an initial deposit of $12,000 earning 7.2% interest compounded continuously. How much will the account be worth after 30 years? How much less would the account from the first exercise be worth after 30 years if it were compounded monthly instead?

Graph and find the solutions for the quadratic function x²- 4x + 4

The total number of solution of the equation (x - 2)² + {x – 2} = 4 is equal to (where {:} is fractional part function)

Sodium chlorate crystals are easy to grow in the shape of cubes by allowing a solution of water and sodium chlorate to evaporate slowly. If V is the volume of such a cube with side length x, find (dV/dx) (in mm³/mm) when x = 4mm. V'(4) = ________ mm³/mm Explain the meaning of V'(4) in the context of this problem. (A) V'(4) represents the rate at which the volume is increasing with respect to the side length as x reaches 4 mm. (B) V'(4) represents the volume as the side length reaches 4 mm. (C) V'(4) represents the rate at which the side length is increasing with respect to the volume as x reaches 4 mm. (D) V'(4) represents the rate at which the volume is increasing as x reaches 12 mm. (E) V'(4) represents the rate at which the volume is increasing with respect to the side length as V reaches 12 mm³.

Find the indicated derivatives of the following functions. No need to simplify. a. Find f'(x) where f(x) = arctan (1 + ∛x) b. Find dy/dx where y is implicit defined by sin(2yx) – sec (y^2) – x = arctan (π) c. Find f'(x) where f(x) = {(1 + x^2)^1/x} + (That's (1 + x^2) raised to the power 1/x) d. Find {d^(47)y} / {dx^(47)} | x=2 (the 47th derivative of y with respect to x at x=2) where y = ln (1 + 2x)

If t = sin 42", express csc 84 in terms of t: (A) 1/2t√1-t² (B) 1/2t (C) 2t (D)√1-t² (E) 2/t√1-t²

(10 pts) Compute the Fourier series for the function f(x) = on (-,). Then find a value for which the resulting sum is conditionally convergent, but not absolutely convergent. (You ust show that this value of r makes the sum converge conditionally. No guesses.)

Find the solution to x' = y y' =12x +4y if (0) = 0 and y(0) = 8.

.Find a general solution to y" - 2y + 1y = 12.5elt t² + 1 Use a and b for the constants of integration associated with the homogeneous solution.

A spherical balloon is being inflated. Find the rate (in ft2/ft) of increase of the surface area (S = 4πr²) with respect to the radius r when r is each of the following. (a) 2 ft = _______ ft²/ft (b) 3 ft = _______ ft²/ft (c) 7 ft = ________ ft²/ft

Consider the transformation of the 4-vector (1,0) under a boost in the direction: (1,0)→ (7₁-√²-1), where r= √v²-1 parameterises the boost. Writing r = cosh∅ show that the boost corresponds to the transformation matrix Q = exp (σ₃/2)(note that there is no i in our matrix representation).

Note the following binomial expression. (3x + 3y)³ The expansion of this binomial has 4 terms. How many of the four terms shown below are correct: 81x³ + 81x²y + 81xy² + 27y^4 A) The number of correct terms is 2. B) The number of correct terms is 1. C) All four terms are correct. D) None of these are correct.

A company makes computer chips from square wafers of silicon. A process engineer wants to keep the side length of a wafer very close to 18 mm and needs to know how the area A(x) of a wafer changes when the side length x changes. Find A'(18) (in mm²/mm). A'(18) =_________________ mm²/mm Explain the meaning of A'(18) in the context of this problem. (A)A'(18) represents the rate at which the area is increasing with respect to the s ide length as x reaches 18 mm. (B)A'(18) represents the rate at which the side length is increasing with respect to the area as x reaches 18 mm. (C)A'(18) represents the area as the side length reaches 18 mm. (D)A'(18) represents the rate at which the area is increasing as x reaches 36 mm. (E) A'(18) represents the rate at which the area is increasing with respect to the side length as A reaches 36 mm².

Without graphing the function y= 7 cos(-9x), determine its amplitude and period. Leave answers in exact form; type pi for π. amplitude = □ period = □

Consider the following problem: maximize (minimize) 3x² + y subject to the constraints: 4x-3y= 9 and x² + z² = 9 (a) Write down the Lagrangian function. (b) What are the first-order conditions? (c) Find the solutions for the given problem.

Solve the seperable differential equation: dy/dx = x²/e²ˣsin²y

Find the complete integral and singular integral of non-linear PDE : u²(1+|Dul²)=1 Partial differential equation

Solve the linear differential equation: y' + 3y = sin 2x

Find the amount of money that results from 1000 dollars invested at 4 percent compounded quarterly after the period of 6 months.

Suppose that p(x, y)-represents the production of a two-product firm. The company produces x units of the first product at a cost of c₁ each and y units of the second product at a cost of c₂ each. The budget constraint, B, is a constant given by the following formula. Use the method of Lagrange multipliers to find the value of λ in terms of Px, Py, c₁, and c₂. The resulting equation holds for any production function p and is called the Law of Equimarginal Productivity. B=c₁x + c₂y A. λ = Px/c₁ = Py/c₂ B. λ = Px = c₁ = Py = c₂ C. λ = Py/c₁ = Px/c₂ D. λ = c₁/Px = c₂/Py

Does the equation specify a function with independent variable x? If so, find the domain of the function. If not, find a value of x to which there corresponds more than one value of y. 3x + 8y = 37 Select the correct choice below and fill in the answer box to complete your choice. (a) The equation specifies a function with independent variable x. The domain of the function is (Type your answer in interval notation.) (b). The equation does not specify a function with independent variable x. One value of x to which there corresponds more than one value of y is x =_____