Functions Questions and Answers

4 points A cylinder filled with water has a single hole at the bottom The rate at which the water level decreases is directly proportional to the square root of the depth of the water in the cylinder At time t 0 th depth is 20 cm and the rate of leakage is 4cm min Write and solve a differential equation that expresses this situation Find the time at which the cylinder becomes empty Round to the nearest tenth of a minute Work

6 A car company claims that they increased the MPGs for its cars Below are MPGs in 2015 and then in 2020 of some of their models 33 25 30 36 25 32 2015 2020 23 25 27 29 24 25 At 0 05 can we claim that the mean difference is less than zero Does the manufacturer have a valid claim Test using a a hypothesis test b a confidence interval

Calculate the Fourier Transform in Continuous Domain CFDF of the following signs 15 f t e tu t cos 3t 23 2 x t 1 1 1 2 t

180 2 2 45 180 380 6 If the base graph is y sin x describe in words the transformations that occur toy sin x 45 1 3 P 540 720 900 1080 B

Find the equation of the tangent line to y f r at the origin O 0 0 10 points Let g r be a continuous function on the interval a b where a b Suppose that g is differentiable on the interval a b Show that there is c a b such that 3c g b g a g c b a

a Simplify 68 and write your answer without using negative exponents b Simplify 575 2 and write your answer without using negative exponents

Let G be a simple undirected graph with n vertices and m edges where n 3 Prove or disprove the following statement If every vertex in G has degree at least n 2 then G must contain a cycle

123 PL K LLL What functions are represented by the following power series a kzk b k k k 1 Solution a k 0 Multiply by z obtain Differentiate the geometric series 1 k 0 k 1 1 We get zk Multiply by 22 obtain k 1 k 1 k k 1 k 2 b Differentiate the geometric series oz twice obtain k 0 02 obtain k 0 kzk 1 kzk k 1 2 1 2 1 1 z k k 1 1 z k k 1 zk 2 k 2 22 1 z 22 2 1 k 2 2 1 2 k k 1 z k 2 k 2 z 22 1 z 2 1 z

Recall that the degree sequence of a graph is a list consisting of the degree of each vertex Degree sequences are typically arranged so that the values are arranged from largest to smallest Which of the following degree sequences are possible for a simple graph DA 6 5 5 5 4 4 3 B 8 8 8 8 7 7 7 7 6 C 5 4 4 4 1 1 1 1 OD 10 9 9 8 8 4 4 3 3 3

Given x is number of items Demand function d x 532 4 0 5x Supply function s x 0 6x Find the equilibrium quantity 22 Find the consumers surplus at the equilibrium quantity 1839 2

II Using the comparison test and exercise V 1 3 show that the following series converge a jj i b j 1 j 1 2jk

1 2 3 P 111 L LLL K Denna beh ver Nog fixas till Lite Show that if u is a harmonic function on C that is bounded above then u is constant Hint Express u as the real part of an analytic function and expo nentiate Solution Let u be the real part of the analytic function f u iv where we know that u C for some constant C Then set g eutiv where g is an entire function Now we have that g eu iv e ewv e ew eu We can se that becauce u is bounded above so is g bounded above Now we now that g is both analytic and bounded by Liouville s theorem g is constant Thus f is constant

1 2 3 4 5 7 Using the algorithm below construct the natural cubic spline for the data Suppose to and that 1 h Recall that we want to construct the cubic spline S x that interpolates the function f x The spline equations are the following 6 Include a plot of the spline over the interval 0 1 0 4 and use your spline to esti mate f 0 25 0 0 0 0 aj f x Ac z c co C Cn bj d x f x 0 1 0 62049958 0 2 0 28398668 0 3 0 00660095 0 24842440 0 4 where the matrix A E R n 1 x n 1 and the vector z E R 1 are defined as follows 0 1 4 0 1 0 0 1 0 4 1 aj 1aj h C 1 Cj 3h 1 4 0 0 h cj 1 2c 3 0 0 and 2 0 2a ao Ac z you can use in Matlab the expression c A z to solve it an 2an 1 an 2 0 Note The system that we found in class and this one look a bit different because here we assume that we have an equispaced grid Look at the matrix that we derived in class if we set h h for i 0 1 n 1 and divide both sides by h we get the system above Once you have constructed the linear system

Rewrite these expressions without fractions 1 II 64 H S 00 0 O 1 1 1

n is inversely proportional to m When m 25 n 10 m 25 n 10 23 21

y 3y y 5y 0 y t Do not use d D e E i or I as arbitrary constants since these letters already have defined meanings

Solve the given initial value problem w 4w 2w 0 w 0 0 w 0 1 w t

4 points For a science experiment a projectile was launched Its path was given by h d 4 9d 24d 30 where h d is the height of the projectile above the ground and d is the horizontal distance of the projectile from the launch pad both in metres a What was the initial height of the projectile b What was the maximum height reached by the projectile to the nearest tenth of a metre Show all your work

Directions For each of the questions in this section write your answers in the spaces provided Show all your work for full mark 11 4 points The height of an object h in metres thrown vertically upwards at time in seconds is modelled by h t 5 1 2 9 1 0 1 What are domain and range of the function Justify youranswer Domain

8 2 points The point 2 12 is on the graph of y ax What is the value of a Show all your work

2 points a Make an example of a quadratic function in factored form b What is are the x intercept s of your function

Solve the given initial value problem y 4y 0 y 0 9 y 0 16 What is the auxiliary equation associated with the given differential equation Type an equation using r as the variable

Determine whether the functions Y and Y 1 2 sin t y 8 cos 2t Y2 are linearly dependent on the interval 0 1 Select the correct choice below and if necessary fill in the answer box within your choice OA Since y y on 0 1 the functions are linearly dependent on 0 1 Simplify your answer OB Since y y2 on 0 1 the functions are linearly independent on 0 1 Simplify your answer OC Since y is not a constant multiple of y2 on 0 1 the functions are linearly independent on 0 1 O D Since y is not a constant multiple of y2 on 0 1 the functions are linearly dependent on 0 1

Solve the initial value problem 18y x dx 18yx cos y dy 0 x 1 The solution is Type an equation Type an implicit solution Type an exact answer in terms of t

16 The period in radians for y tan 20 T A B 17 12 NIE 2 C 2 D T

Consider the function 1 log c 19 What is the inverse function

Which is the inverse function of y 4 2 O A O B C y y y 9x 4 2 3x 4 2 OD y 2 2 2x 4 3

The graph of the function f x is shown on the coordinate plane Which value is closest to f 4 0 3

The point 7 10 lies on the graph of the function f x Which of the following points lies on the graph of the functions inverse 10 7 10 7

For the function f x 6 x 2 find the f O O 36 2 2 2 2 2 26

For the function f x 5 x 8 find f O H 8 5 8 5 8 8

Which function represents the inverse of f x 7 3x 7 3x 7 3x 21 3x 21

Consider the function f x 4x What is the f 1 4x 5 3 5 22

In order to solve the equation 2 x 2x 1 a student graphs the functions f x and g x x f x x 2 g x x 2x 1 Which is a solution to the equation f x g x 6 2 1

Consider the two functions f x x g x 3x 4 What is the approximate solution to the equation f x g x 0 1 10 6 2 2

Consider the equation 10 P 200 What is the approximate solution to the equation Op 10 Op 0 65 p 1 15 p 2

Consider the two functions f x ex g x ex 4 5 Which sequence of transformations maps f x onto g x a translation right 4 and down 5 a translation left 4 and down 5 a translation left 4 and up 5 a translation right 4 and up 5

What is the approximate solution to the equation 10 2 3t 50 when solved for t 1 292 0 144 0 774 6 966

Solve the initial value problem dy 5x x 3 dx x 1 y 1 x Begin by separating the variables Choose the correct answer below 5x x 3 x x 1 A y 1 dy B C y 1 5 5x x 3 dy dx x x 1 y 1 x x 1 5x x 3 D The equation is already separated The solution is dx 2 1 dy dx y 1 y 3 y 2 Inx 3 In x 1 2 29 3 In 2 X 2 Type an implicit solution Type an equation using x and y as the variables

The population of a city after t years is given by P t 18 272e0 034t where t 0 corresponds to the current year How many years from the current year will it take for the population of the city to reach 75 000

Consider the graph of the quadratic function A B 10 What are the zeros of the function 5 4 and 1 5 and 0 only 4 0 and 1 n 4 and 1 only C NOUVENISE 10 Hills Men enes t 10

4 5 On the graph below Sketch the function h x 2x 6 x 6 6 x 0 0 x 4

Find the surface area of the pyramid 9 yd 7 79 yd 9 yd

Find the surface area of the pyramid The surface area is Type a whole number or a decimal 13 ft 12 ft 12 ft

7 The interior lateral surface of the pipe is coated with sealant to prevent leakage and corrosion What is the surface area of the interior part for one section as shown A B C A 3141 6 cm B 3769 9 cm 2 2 C 3927 0 cm D 6283 2 cm 50 cm

9 Which of the following describes the dimensions of a rectangular prism with a volume of 120 cubic units 00 B A 4x5x8 B 2 x 10 x 10 C 2x6x10 D 3x5x10

10 What is the volume of a right triangular prism whose height is 20 units and whose base is a right triangle with side lengths of 3 4 and 5 hint draw the base A B D 3 A 100 units B 120 units C 240 units D 1200 units

12 A right cylinder has a height of 20 inches and base radius of 5 inches What is the volume of this cylinder leave your answer in exact form do not multiply by pi O O O O ABCO 3 A 100m in B 3 1000m in 4 C 500m in D 2000m in fan area of this

In Exercises 4 and 5 find the surface area 4 L 7 in 3 in 4 in

Solve the equation dx dt 2 Xe 9x Begin by separating the variables Choose the correct answer below OA dx 1 9x xe dt 2 B Xdx dt et 9x 2 C xexdx dt OD The equation is already separated An implicit solution in the form F t x C is C where C is an arbitrary constant Type an expression using t and x as the variables