Differential equations Questions and Answers
Calculus
Differential equationsUse a half angle formula to fill in the blanks in the identity below:
(cos(4x))² =
H
+1
Cos
x)
Calculus
Differential equationsSolve 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
Calculus
Differential equationsFind 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
Calculus
Differential equationsFind the inverse of y = sin 2x.
O2 sin¹(x)
O sin¹(2x)
O sin-1(x)/2
O2 sin¹ (₂)
Calculus
Differential equations4x
√x² - 4x
Determine the derivative of f.
Let f(x)
df
dx
=
Determine the slope of fat x = 9.
f'(9) =
Calculus
Differential equationsHouston 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
Calculus
Differential equationsFinance. 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?
Calculus
Differential equations2. 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.
Calculus
Differential equationsWe 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
Calculus
Differential equationsFor 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.
Calculus
Differential equationsDefine
(a) Show that
S(h)
and state C1.
− f (x+2h) +4f(x + h) − 3f(x)
=
2h
f'(x) - S(h) = c₁h² + c₂h³ + c3h4+...
Calculus
Differential equationsDefine
(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
Calculus
Differential equations[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
Calculus
Differential equationsLinearize the following ODE and find the corresponding initial conditions:
x + 2x + sin x = 0.5,
x (0) = -1, x(0) = 0
Calculus
Differential equationsChoose 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.
Calculus
Differential equationsConsider 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.
Calculus
Differential equationsWhat 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
Calculus
Differential equationsWhich 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
Calculus
Differential equationsSuppose 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).
Calculus
Differential equationsFor 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.
Calculus
Differential equationsFind 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
Calculus
Differential equationsDetermine the domain of the function of two variables f(x ,y) = √y + 9x.
Calculus
Differential equationsDetermine the intervals on which n is continuous. Enter the solution using interval notation. n(x)=13x/8-x
n is continuous on=____
Calculus
Differential equationsSolve 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.
Calculus
Differential equationsSketch 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).
Calculus
Differential equationsFind the length of the missing side. Leave your answer in simplest radical form.
A right angle triangle with 11 cm, 4 cm.
Calculus
Differential equationsGiven x = π/3, what is the exact value of cos(2π - x)?
a. 1/2
b. -1/2
c. √3/2
d. -√3/2
Calculus
Differential equationsSuppose 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?
Calculus
Differential equationsThe total number of solution of the equation (x - 2)² + {x – 2} = 4 is equal to (where {:} is fractional
part function)
Calculus
Differential equationsSodium 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.
Calculus
Differential equationsFind 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)
Calculus
Differential equationsIf 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
Calculus
Differential equations(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.)
Calculus
Differential equations.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.
Calculus
Differential equationsA 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
Calculus
Differential equationsConsider 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).
Calculus
Differential equationsNote 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.
Calculus
Differential equationsA 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.
Calculus
Differential equationsWithout graphing the function y= 7 cos(-9x), determine its amplitude and period. Leave answers in exact form; type pi for π.
amplitude = □
period = □
Calculus
Differential equationsConsider 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.
Calculus
Differential equationsFind the complete integral and singular integral of non-linear PDE :
u²(1+|Dul²)=1
Partial differential equation
Calculus
Differential equationsFind the amount of money that results from 1000 dollars invested at 4 percent compounded quarterly after the period of 6 months.
Calculus
Differential equationsSuppose 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
Calculus
Differential equationsDoes 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 =_____