Calculus Questions

Given g x is a transformation of the graph f x 2x determine which of the below correctly describe g x f x 2 54 3 2 1 N 7 21 3 S Asymptote y 0 123 g x 2 x g x 2 x g x 2 x g x 2x 1 Asymptote y 0 5 4 3 2 1 21 3 g x 1 2 3 4

A exponential function is of the form f x a b c If the graph of f x has a horizontal asymptote at y 4 which of the following is true about the parameters of f x The parameter a must be 4 a 4 The parameter b must be 4 b 4 The parameter c must be 4 c 4 The product of the parameter a and b must be 4 a b 4

Rewrite as an augmented matrix and solve by row reduction x 5y 7 2x 7y 20

Use the inverse matrix equation X A 1 B to find the solution to 4x 7y 2 x 2y 5 4 2 thn 4 7 A 1 2 Hint If A

Perform the indicated operation 13 1 4 5 3 3 8 2 6

Consider the function f x 2x 3x 72x 2 on the interval 6 8 Find the average or mean slope of the function on this interval By the Mean Value Theorem we know there exists a c in the open interval 6 8 such that f c is equal to this mean slope For this problem there are two values of c that work The smaller one is and the larger one is

Given this matrix 10 113 01 5 a Rewrite as a system of equations b How many solutions does it have

Given that A B and A 13 y 22 11 B Solve for X 2x 5 8 28 7 and 4 32 7

Two books and three DVDs costs 54 Three books and one DVD costs 46 Find the cost of a single book and a single DVD SET UP AS AN AUGMENTED MATRIX BUT DO NOT SOLVE

Determine the dimension of the matrix 3 2 7 5 80

Problem 4 Which of the following sequences converge 1 1 n i an n 6 Problem 4 ii an iii an cos 2n 1 2 iv an sin 2n 1 2 D 1 1 n n 8 A all of them B i and iii only C ii and iv only D ii and iii only E 1 only F G none of them H i and iv only V

roblem 2 Evaluate the following integral x 5 x 6x dx

Problem 6 Use the Comparison Theorem to determine which of the following integrals converge S Problem 6 arctanx Spo 2 201 dx Select dx 0 CT 2 2 sinx x512 dx A ii and iii only B i and iii only C i and ii only D i only E none of them F 1 only G iii only H all of them

Problem 1 a 1 an 1 an 8 To show that an is monotonic using mathematical induction which of the following would be the second step A Assume az 1 8 ak 2 8 and show that a az 1 B Assume a 2 8 a 1 8 and show that at 1 az C Assume ak 1 a and show that ax 2 8 ax 1 8 x 18 D Assume ak ak 1 and show that a 8 E Assume az 1 8 a 8 and show that az 1 ak F Assume a 8 a 1 8 and show that a a 1 ax 1 8 ax 2 8 G Assume a ak 1 and show that H Assume ak 1 az and show that a 1 8 8 G

xyl sin xy y 1 find general solution of Differental equation

FIND SOLUTION OF DIFFERENTIAL EQUATIONS x x y y 2x y 0 y 1 2

2 Solve each scenario using a different method graphing substitution elimination

Problem 6 Use the Comparison Theorem to determine which of the following integrals converge S arctanx dx SVS 2 dx 26 0 2 2 sinx x512 dx A ii and iii only B i and iii only C i and ii only D i only E none of them F ii only G iii only H all of them

oblem 4 Which of the following sequences converge 1 1 n 72 6 1 1 2 8 ii an iii an cos 2n 1 2 iv an sin 2n 1 2 A all of them B i and iii only C ii and iv only D ii and iii only E 1 only F G none of them H 1 and iv only

Problem 5 Evaluate the following improper integral el x x2 Problem 5 So dx If the integral does not converge then write divergent Enter your answer symbolically as in these examples If the integral does not then write divergent

Problem 2 Suppose that a sequence is defined by san 2 an Problem 2 a 1 an 1 Assuming that an is convergent find its limit

Problem 1 Suppose that a sequence is defined by a 1 an 1 an 8 To show that a is monotonic using mathematical induction which of the following would be the second step Problem 1 A Assume az 1 8 a 2 8 and show that a ax 1 B Assume ax 2 8 a 1 8 and show that C Assume at 1 a and show that ax 2 8 D Assume a az 1 and show that ax 8 E Assume ax 1 8 a 8 and show that az 1 a F Assume a 8 ax 1 a 1 8 and show that a az 1 8 G Assume ak ak 1 and show that H Assume ak 1 a and show that az 2 8 az 8 ax 1 ax a 1 8 ak 1 8 az 1 8 M

Problem 5 Evaluate the following sum 43 1 2 i 1 i 1 i 2 1

oblem 4 Which of the following sequences converge 1 1 n n 6 i an 1 1 n 8 ii an iii an cos 2n 1 2 iv an sin 2n 1 2 A all of them B i and iii only C ii and iv only D ii and iii only E i only G none of them H i and iv only

Problem 2 Evaluate the following integral 4 1 x 5 x 6x dx

em 1 Suppose that a sequence is defined by a 1 an 1 1 a 8 To show that a is monotonic using mathematical induction which of the following would be the second step A Assume ax 1 8 x 2 8 and show that a ax 1 B Assume a 2 8 ax 1 8 and show that az 1 ak C Assume ax 1 a and show that a 2 8 x 1 8 D Assume a ax 1 and show that a 8 ax 1 8 E Assume ax 1 8 a 8 and show that ax 1 ax F Assume a 8 ax 18 and show that a ax 1 G Assume a ak 1 and show that a 1 8 x 2 8 H Assume a 1 a and show that a 1 8 x 8

roblem 3 Find the limit of the following sequences a a ln 5n 5 In 6n b a n sin 8 n If the sequence does not converge then write divergent Enter your answer symbolically as in these examples Problem 3 a Enter your answer symbolically If the sequenc then write div If the sequenc

a Express each of the following numbers as a product of prime factors in index notation 1 25 ii 150 2x3x5 FEEDEC 2 marks b If the L C M of x and 25 is 150 find two possible values of x by using your answers in a 4 marks SPANNE DATORSVARE

Find the extremals for the following functional f y 2y dx

1 Write 7 3 without exponents Use the equation editor by clicking insert math above or take a photo of what you wrote and upload it

1 a Find the vertical Asymptote b Find the Horizontal Asymptote end behavior 5x 6 4 2x 1 x 5 2 3 x 4x 10x 2 x 5x x 4 7 2 x

A plane is going to fly 300 miles at a planned speed of 530 mph The flight will have an average headwind of w miles per hour the entire time meaning the plane is flying directly against the wind The time T in hours of the flight is a function of the speed of the headwind w in mph and can be modeled by T w 300 530 w a What does T 110 mean in this situation b At what value of w does the graph have a vertical asymptote Explain how you know and what this asymptote means in the situation T travel time 10 A 8 4 2 Here is the graph of y T w 100 400 200 300 w headwind speed 500 600

Use division to justify the expression that describes the end behavior for f x a Approaches y 0 b Approaches y 3x 10 C Approaches y 3x 2 d Approaches y 3 3x 4x 8x x 2x

In a biology lab 500 bacteria reproduce by splitting Every hour on the hour each bacterium splits into two bacteria 30000 20000 10000 What is a reasonable domain for the given function Explain What is a reasonable range for the given function Explain HHH Submit

3 Match the graph of each function in a d with the graph of 3 Match the graph of each function in a d with the graph of its derivative in 1 IV Clive reasons for your choices A VA A c 111 0 VA d

3 Match the graph of each function in a c with the graph of its derivative in I III 3 b 0 X c 0 111 0 0 II

Find the extremals for f y dx subject to the condition Sydx 3 y 0 1 y 1

Reduced the BVP y xy x with y 0 y 1 0 into a variational problem

B G R Match each equation with a graph above y 2 0 84 y 2 0 75 y 3 1 29 y 2 1 54 y 2 1 29 a blue B b green G C orange 0 d black K

Shari has an electronic entry system on her house To enter one must input a six character code where the first four entries are letters of the alphabet A through Z and the last two are single digit numbers 0 through 9 If repeated letters numbers are not allowed how many different codes are possible

often depe only the size of the population but also on how crowded it is When individuals have to compete for resources etc they cannot invest in reproduction leading to a decline in the rate of growth of the population which makes constant growth not sustainable Such models are usually called density dependent growth models One of such models is the logistic growth where the growth rate of the population depends only on the density of the population N and is given by the following 1 2 equation Where EXTRA LIST G N rN TN N K r 0 is a constant called intrinsic growth rate K 0 is a constant called the carrying capacity It represents the popula tion density that a given environment can sustain

Problem 1 Let f x 4x 2 and g x x 2 a Find the solutions to the equation f x 3 b Find the domains of f and g Argue why they are or are not injecti c Write fog d Find the inverse of f x e Calculate the first and second derivative of f x

Problem 4 Let f x x log x 1 a Find the local and global extrema of f x b Find the local and global extrema of f x in the interval 1 2 c Calculate f f x dx and the second derivative of f x

x x 2 Problem 3 Let f x 4 7 3 a Is this function continuous at x 1 b Find lim 1 f x c Can we define f 1 such that f x is continuous at 1 If yes what is the value

Theorem 1 If f is a continuous function on the closed interval a b and differ entiable on the open interval a b then there exists a point c in the interval a b such that f c f b f a b a Try to use it to solve the next problem Problem 2 A car traverses a tunel which is 1km long in 50 seconds If the speed limit inside the tunel is 60 km h can we guarantee that the car went over the speed limit

Determine lim f x for the function f x shown in the figure below 20 4 1 10 18 1 0 The limit as 4 is help limits 18

Given lim 5 f x 7 and lim 5 g x 3 evaluate If the limit does not exist enter DNE Limit lim 2 5 f x g x 7 f x

Use the given graph of the function g to find the following limits 1 lim g z help limits 2 lim g z 3 lim g x 4 lim g z 2 10 103

If an arrow is shot upward on the moon with a velocity of 58 m s its height in meters after t seconds is given by y 58t 83t a Find the average velocity over the given time intervals 1 1 1 5 2 1 1 1 3 1 1 01 4 1 1 001 b Find the instantaneous velocity after one second to the nearest hundredth 1 2 3 4 b m s m s m s m s m s

A cardiac monitor is used to measure the heart rate of a patient after surgery It compiles the number of heartbeats after t minutes When the data in the table are graphed the slope of the tangent line represents the heart rate in beats per minute The monitor estimates this value by calculating the slope of a secant line Use the data to estimate the patient s heart rate after 42 minutes using the secant line between the points with the given values of t Round your answers to the nearest hundredth a t 36 and t 42 b t 38 and t 42 c t 40 and t 42 d t 42 and t 44 What are your conclusions t min 36 38 40 42 44 Heartbeats 2530 2661 2806 2948 3080 a b c d