Differential equations Questions and Answers

y p t y q t y 0

t t 1 y 1 t 1 y 0 t 1 Solution of this ODE on interval 1 infinite is n y1 t e t Find another solution y2 so y1 y

dy dt sin 2y t 1 cos y 2 t 1 y 0 TT O 6

2t y 3y e y 0 a lim t 0 t

2 We want to find the points on y x that are closest to 0 3 0 3 Y D x x x X Figure 2 Distance to 0 3 on y x a For each point x x on the parabola find a formula for its distance to 0 3 Call this distance D x See Figure 2 b Let f x D x 2 which is the squared distance between x z2 and 0 3 Finding where D x is minimal is the same as finding where f x is minimal Determine all x where f x has an absolute minimum The points x z2 for such a are the dlogget points to 03 on

Name Instructions Use algebraic techniques to solve problems unless otherwise instructed Show the calculations leading to your answers except basic arithmetic Simplify and circle your final answers 1 6 pts Calculate the difference quotient of f x e and simplify it as was done in class 2 6 pts If 4 700 is invested at an interest rate of 4 5 per year compounded continuously find the value of the investment at the end of the given number of years Round your answers to the nearest cent a 2 years b 3 years c 6 years 3 8 pts What are the linear y ax b and exponential functions y a b whose graphs contain the points 0 5 and 3 320

Consider a W N weight suspended by two wires as shown in the accompanying figure If the magnitude of vector F2 is 140 N find w and the magnitude of vector F What is the weight w 48 F W 35 F2 W N Type an integer or decimal rounded to three decimal places as needed

Consider the system of linear equation 11 p 11 2 9 a Find a particular solution for the given system of linear equation h 6 5 15 2 1 1 3 9 5 5 14 6 1 4 b Find a non trivial homogeneous solution for the given system of linear equation

Solve the following initial value probl 3 g 7x 47 12 g 1 24 B h A 1 6 sont h 35 3

Solve the following initial value problems 2x fla 4 200d sil f 1 0 x x 4 4 1 0 0 f x xex

Let T3 a be the Taylor Polynomial of degree three for f z centered at 0 If f 0 15 f 0 14 f 0 11 T3 x f 0 5

a List the x values of all critical points of f x B Use interval notation to indicate where f x is increasing A function is increasing if larger x values give larger y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Increasing for a in C Use interval notation to indicate where f x is decreasing A function is decreasing if larger x values give smaller y values Don t include points where the function changes from increasing to decreasing or from decreasing to increasing Decreasing for a in D List the x values of all local maxima of f x values of local maxima E List the x values of all local minima of f x values of local minima F Use interval notation to indicate where f x is concave up Concave up for a in G Use interval notation to indicate where f x is concave down Concave down for in H List the x values of all the inflection points of f values of inflection points BEB B

Water is leaking out of an inverted conical tank at a rate of 12300 cubic centimeters per min at the same time that water is being pumped into the tank at a constant rate The tank has height 12 meters and the diameter at the top is 4 meters If the water level is rising at a rate of 28 centimeters per minute when the height of the water is 1 meters find the rate at which water is being pumped into the tank in cubic centimeters per minute Leave answers in 3 decimal places cm

At noon ship A is 50 nautical miles due west of ship B Ship A is sailing west at 24 knots and ship B is sailing north at 18 knots How fast in knots is the distance between the ships changing at 5 PM Note 1 knot is a speed of 1 nautical mile per hour

Find the particular solution to the differential equation below such that y 0 7 y 5e 9x 7x Do not include y in your answer

4 Determine the equation of the line that is tangent to the function f x 2sinx sin x at the point x 7

2 Jypu un muyor ora B xx Type an integer or a simplified fraction C x x 5 Type an integer or a simplified fraction Use a comma to separate answers as needed D The domain is the set of all real numbers Write R x in lowest terms Select the correct choice below and if necessary fill in the answer box to complete your choice A R x Simplify your answer B R x is already in lowest terms Locate the intercept s of the graph Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The graph has x intercept s and y intercept Simplify your answers Type integers or fractions Use a comma to separate answers as needed Type each answer only once OB The graph has y intercept and no x intercept Simplify your answer Type an integer or a fraction OC The graph has x intercept s and no y intercept Simplify your answer Type an integer or a fraction Use a comma to separate answers as needed Type each answer only once OD The graph has neither x intercepts nor y intercepts

Follow the steps for graphing a rational function to graph the function P x x x 6 36 Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer or once OC The graph will cross the x axis at x Type an integer or a simplified fraction Use a comma to separate answers as needed Type each answer on once D There is no x intercept Determine the vertical asymptote s if one exists Select the correct choice below and if necessary fill in the answe box es to complete your choice OA The function has one vertical asymptote Type an equation Use integers or fractions for any numbers in the equation B The function has two vertical asymptotes The leftmost asymptote is Type equations Use integers or fractions for any numbers in the equations and the rightmost asymptote is the middle asymptote is OC The function has three vertical asymptotes The leftmost asymptote is rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations D The function has no vertical asymptote anc

Find the points on the ellipse 4x2 y2 4 that are farthest away from the point 1 0 1 4 2 3 3 x smaller y value x y x y 1 4 2 3 3 x larger y value

A box with a square base and open top must have a volume of 62 500 cm Find the dimensions of the box that minimize the amount of material us sides of base cm height cm

A poster is to have an area of 210 in2 with 1 inch margins at the bottom and sides and a 2 inch margin at the top Find the exact dimensions that will give the largest printed are width height in in MY NOTES

Question Given the function 3x 9 X 5 use a linear approximation of f x at x 4 to approximate the value of f 4 1 Provide your answer below f x

Estimate the error if P x 1 2 X is used to estimate the value of cos x at x 0 6 2 Error Simplify your answer Round to seven decimal places as needed

Using the trapezium rule with eight ordinates at x 3 2 1 0 1 2 3 and 4 calculate correct to 3 decimal places 6x an approximate value of 2 dx 3 WAEC

r 5 cos 50 O A 10 10 OB 10 10 10 O C 10 10 10 OD 10 10 10

QUESTION 3 2 points Find the derivative of f x In cosa 2 1 x 2 4e

Let a h and k be arbitrary real numbers with a Which of the following statements are true about f Select all that apply A An extreme value occurs at the point h k B The graph of y f x is a parabola C If a 0 then f has a global minimum D The graph of y f x is a line E The function has a critical point at x h F The function has an inflection point at x h G The maximum value of f x is h H If a 0 then f has a global maximum I None of the above 0 and let f be the function defined as f x a x h k

2 Find a polar equation for the curve represented by the Cartesian equation 2x y 1

Find the slopes of the curve y4 y x at the two points shown here 4 1 L 3 4 12 X HE The slope of the curve y y x at the point 3 is 4 2 Type an exact answer using radicals as needed The slope of the curve 3 3 4 2 ve y4 y x at the point 3 is Type an exact answer as needed sing radicals

K shifts Midpoints and quarter points are identified by dots The simplest form of an equation for the graph is y Use integers or fractions for any numbers in the expression 3x 12 3x 4 2 0 12 3t

Use a graphing utility to approximate the solutions of the equation in the interval 0 2 Round your answers to three decimal places Enter your answers as a comma separated list 5 sin x 3 cos x 0

Use a graphing utility to approximate the solutions to three decimal places of the equation in the given interval Enter your answers as a comma separated list 6 sec x tan x 9 0 1 2 1 2 X SPEHE

Approximate the x intercepts of the graph Use a graphing utility to check your solutions Enter your answers as a comma separated list Let be any y 4 sin xx 4 cos x X X

If g x O Og x t x 3t x t x 3e3 g x g x e3x g x find and simplify g x t x 3t x e x t x 3t x e 4

1 point Calculate the following for the matrix A det A 21 and A 6 21 7 21 201 2 30 302 6 21

Given that lim f x 0 x a b lim g x 0 x a d lim p x x a 00 evaluate the limits below where possible If a limit is indeterminate enter INDETERMINATE a f x lim x a g x f x lim x a P x h x lim x a p x lim h x 1 x a lim P x x a 9 x lim g x 0 x a

2 Let f be twice differentiable on R and suppose f x 0 for all x ER a Show that f is monotonic on R i e that f is increasing on all of R or decreasing on all of R b Show that f is increasing if and only if f 1 is increasing Note To show this if and only if you must show two implications if f is increasing then f is increasing and if f 1 is increasing then f is increasing c Show that if f is increasing and concave up everywhere then f 1 is increasing and concave down everywhere

6 Consider the function c f d d 5 a What is the independent variable of f b What is the dependent variable of f c Find f

1 For the function F x 5 a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique asymptotes X OA The domain of the given function is xlx is a real number x Type an integer or a simplified fraction B The domain of the given function is xlx is a real number x 0 Type an integer or a simplified fraction Use a comma to separate answers as needed C The domain of the given function is x x is a real number x Type an integer or a simplified fraction OD The domain of the given function is the set of all real numbers What is the range of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice OA The range of the given function is yly is a real number y Type an integer or a simplified fraction B The range of the given function is yly is a real number y Type an integer or a simplified fraction Use a comma to separate answers as needed OC The range of the given function is yly is a real number y Type an integer or a simplified fraction OD The range of the given function is the set of all real numbers

1 For the function F x 5 asymptotes a Choose the correct graph below OA X a graph the rational function using transformations b use the final graph to find the domain and range and c use the final graph to list any vertical horizontal or oblique X 20 B 201 OB The domain of the given function is xlx is a real number x Type an integer or a simplified fraction Use a comma to separate answers as needed b What is the domain of the given function Select the correct choice below and if necessary fill in the answer box to complete your choice OA The domain of the given function is xlx is a real number x Type an integer or a simplified fraction OC The domain of the given function is xlx is a real number x Type an integer or a simplified fraction domain of the given function is the set of all real numbers O C D Ay Q Check

f x x 10x 9 9 following polynomial function Write f in factored form The complex zeros of fare i i 3i 3i Simplify your answer Type an exact answer using radicals and i as needed Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed Use the complex zeros to factor f f x Factor completely Type an exact answer using radicals and i as needed Use integers or fractions for any numbers in the expression 4

The rate of change of the function fx 9 sec x 9 cos x is given by the expression 9 sec x tan x 9 sin x Show that this expression can also be written as 9 sin x tan x Use the Reciprocal and Pythagore identities and then simplify 9 sec x tan x 9 sin x 9 sin x cos x 9 sin x

Multiply 5 6 4i 5 6 4i Type your answer in the form a bi

A storage tank is being designed to accomodate 1 000 m of contaminated water for a mining operation The storage tank must be cylindrical with a radius R and height H and will be composed of three steel sections Flat disks as the floor and ceiling and the curved portion comprising the side Find the dimensions R and H that minimize the steel used

Given f x x 4x 10 find the x coordinates of all local minima using the seco derivative test If there are multiple values give them separated by commas If there a local minima enter Sorry that s incorrect Try again

The price of a condominium is 188 000 The bank requires a 5 down payment and one point at the time of closing The cost of the condominium is financed with a 30 year fixed rate mortgage at 8 Use the following formula to determine the regular payment amount Complete parts a through e below PH PMT 1 1 nt a Find the required down payment b Find the amount of the mortgage Round c How much must be paid for the one point at closing Round to the nearest dollar as needed d Find the monthly payment excluding escrowed taxes and insurance S Round to the nearest dollar as needed e Find the total cost of interest over 30 years as needed K

An explanation of Rolle s Theorem How is it a special case of the MVT o What can it tell us about the location of critical point s o What continuity differentiability restrictions apply o Be sure to include a graph to help you explain Rolle s Theorem

Use Newton s method with initial approximation x 1 to find x the second approximation to the root of the following equation x4 x 4 0 x x Enhanced Feedback Please try again and draw a diagram Keep in mind that Newton s method is x X f xn

9 Given y f x e lnr find dy dx Given y f x sin tan Inr