Application of derivatives Questions and Answers

Find parametric equations and a parameter interval for the motion of a 2 particle that starts at a 0 and traces the circle x y a a once clockwise b once counterclockwise d three times counterclockw c three times clockwise a Which of the following sets of equations could trace the circle x y a once clockwise starting at a 0 OA x a cos t y a sin t 0 t 2 B X a cos t y a sin t 0 t 2n O c x asin t y a cos t 0 t 2 O D x asin t y a cos t 0 t 2 b Which of the following sets of equations could trace the circle x y a once counterclockwise starting at a 0 OA x a cos t y a sin t 0 t 2 OB X a cos t y a sin t 0 t 2 O c x asin t y a cos t 0 t 2 D x a sin t y a cos t 0 t 2

Find a parameterization for the circle x 12 y 36 starting at the point 6 0 and moving clockwise once around the circle Find parametric equations for the circle X y 0 0 2

Find a parametrization for the curve described below the line segment with endpoints 3 3 and 2 2 Complete the pair of parametric equations using linear functions of t One possible parametrization is x t 3 y 0sts 1

2 2 2 particle that starts at a 0 and traces the circle x y a a once clockwise c three times clockwise b once counterclockwise d three times counterclockw 2 a Which of the following sets of equations could trace the circle x y once clockwise starting at a 0 OA x a cos t y a sin t 0 t 2 OB X a cos t yasin t 0 t 2 O c x asin t y a cos t 0 t 2 O D x asin t y a cos t 0 t 2

The parametric equations and parameter interval for the motion of a particle in the xy plane are given below Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion of the graph traced by the particle and the direction of motion x 5 sint y 4 cos t 0 t n The Cartesian equation for the particle is

The parametric equations and parameter intervals for the motion of a particle in the xy plane are given below Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion of the graph traced by the particle and the direction of motion x cos 2t y sin 2t 0stst The Cartesian equation for the particle is

The equation below gives parametric equations and parameter intervals for the motion of a particle in the xy plane Identify the particle s path by finding a Cartesian equation for it Graph the Cartesian equation Indicate the portion of the graph traced by the particle and the direction of motion x 4t 3 y 16t t Find a Cartesian equation for the particle s path y 0

Use the given graph of f over the interval 0 7 to find the following a The open intervals on which f is increasing Enter your answer using interval notation 0 b The open intervals on which f is decreasing Enter your answer using interval notation c The open intervals on which f is concave upward Enter your answer using interval notation d The open intervals on which f is concave downward Enter your answer using interval notation e The coordinates of the points of inflection x y smallest x value x y x y largest x value

Find the limit Use l Hospital s Rule if appropriate If there is a more elementary method consider using it sin 4x lim x 0 tan 7x Submit Answer

Consider the equation below f x 2x 3x 72x a Find the interval on which f is increasing Enter your answer in interval notation Find the interval on which f is decreasing Enter your answer in interval notation b Find the local minimum and maximum values of f local minimum local maximum c Find the inflection point x y Find the interval on which f is concave up Enter your answer in interval notation Find the interval on which fis concave down Enter your answer in interval notation

Find the limit Use l Hospital s Rule where appropriate If there is a more elementary method consider using it x2x lim x 0 2x 1

Find the equation of the tangent line to f x 10x 6 1 when r

3 Below is the graph of f x x 5x 6x 2 On the interval 0 3 determine the maximum and minimum value of the slope of the graph i e the maximum and minimum values of g x f x 3 2 1 Y 0 1 2 X 3

2 Use calculus to find the absolute maximum and minimum values of the following func tions on the given intervals Give your answers exactly and show supporting work a f x x 2x x 1 on 0 1 b x 2 on 1 3 T

2 a Use six rectangles to find estimates of each type for the 12 area under the given graph of f from x i Lo sample points are left endpoints ii Ro sample points are right endpoints iii M6 sample points are midpoints b Is Lo an underestimate or overestimate of the true area c Is R an underestimate or overestimate of the true area d Which of the numbers L6 R6 or M6 gives the best estimate Explain YA 8 4 0 4 y f x 8 0 to x 12 x

1 Assume S is a solid in three space lying entirely between the planes x a and x b each of whose x cross sections S is a circle of radius r x where r is a continuous function Fix positive scaling factors a and 6 and let T be the solid whose points are of the form ax By Bz where P x y z is a point of S In other words T is obtained by stretching S by an x scaling factor of a in the x direction and by a yz scaling factor of in the y and z directions a Using the volume via cross sections definition derive a formula for vol T in terms of vol S a and B Hint describe the x cross sections T in terms of the x cross sections of S What kind of shapes are they How do their dimensions relate to the dimensions of the cross sections of S Suggestion it may help to do the sketch in part b to help you visualize what is going on b Consider the particular example where S is the solid sphere of radius 1 centered at the origin and T is the solid obtained by stretching S as above with the x scaling factor a 4 and the yz scaling factor 2 i Provide a sketch of T that includes x y and z axes with scales Hint start with 6 easy to plot points on the sphere S and determine what happens to them when you stretch by c and d Computo

Use implicit differentiation to find the equation of the tangent line to the curve xy xy 4 at the point 2 1 The slope of this tangent line is m The equation of the tangent line is y

olve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTIO 3 sin 0 7 sin 8 2 0 2 80 k x N

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to three decimal places where appropriate If there is no solution enter NO SOLUTI tan 0 9 2 cos 8 1 0 e

Find the vertical horizontal and oblique asymptotes if any of the following rational function 5x 17x 12 2x 7x 4 Q x Find the vertical asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice 1 The function has one vertical asymptote x 2 Type an equation Use integers or fractions for any numbers in the equation OB The function has two vertical asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no vertical asymptote Find the horizontal asymptotes Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The function has one horizontal asymptote Type an equation Use integers or fractions for any numbers in the equation OB The function has two horizontal asymptotes The leftmost asymptote is and the rightmost asymptote is Type equations Use integers or fractions for any numbers in the equations OC The function has no horizontal asymptote

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 X asymptotes 1ype an integer or a simpiimea traction 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 D The range of the given function is the set of all real numbers c What is are the vertical asymptote s Select the correct choice below and if necessary fill in the answer box es to complete your choice OA The left vertical asymptote is The right vertical asymptote is Type equations Use integers or fractions for any numbers in the equations OB There is one vertical asymptote It is Type an equation Use integers or fractions for any numbers in the equation There is no vertical asymptote

Find all solutions of the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropria cos 0 1 0 0 rad

Solve the given equation Enter your answers as a comma separated list Let k be any integer Round terms to two decimal places where appropriate cos 8 1 4 0 1 32 rad

Consider the initial value problem x 1 y y y ex y 1 1 y 1 1 Does the Existence and Uniqueness Theorem guarantee the existence of a unique solution to this initial v problem Justify your answer DO NOT SOLVE THE DIFFERENTIAL EQUATION

Suppose that Find an equation for the tangent line to the graph of fat x 1 Tangent line y f x 3x 32x

2 Let f x 1 x 11 Write an 3 equation for g x the inverse of f

5 If 9 4 is on the graph of f which of the following ordered pairs is on the graph of f A 9 4 B 9 4 C 4 9 D 4 9

Michelle got a new job through the McCartney Employment Agency The job pays 49 400 per year and the agency fee is equal to 35 of one month s pay How much must Michelle pay the agency Round to the nearest cent

Solve the equation Check for possible extraneous solutions log x 6 loge 3 2 x 6 X 5 no solution x 2

In Exercises 71 82 find the domain of the function 71 f x 5x 2x 1 72 g x 1 2x 4 73 h t 75 77 g x g y y 1 79 f s 81 f x X 10 3 x 2 x 4 x s 1 S 4 3y y 5 76 f t 3 t 4 10 74 s y 78 h x 80 f x 82 f x x 2x x 6 6 x x 2 x 10

a Applying the two step Adam Bashforth method see Lec 8 to approximate y 2 5 where y is the solution to y 1 y 1 t 1 5 y 1 1 with h 0 5 You may use Euler s method to get the starting value at t 1 5 exact solution is y t 2t 1 b Compare your approximation with the exact solution write down your observation and try to explain why very briefly optional

4 Use the graph given for each problem to determine the values If the value is between two integers approximate to one decimal place a f 0 b f 1 c If f x 2 then x d If f x 0 then the possible values of x are x x or x or x or x 1 2 1 0

Find the following limits lim 00 jum b lim 6x5 1 7x4 4x 1 6x5 1 7x4 4x 1 II 8 8

Use the graph of the piecewise function y fx given above to answer questions 15 18 A 15 What is the domain of the function y f x A 4 4 B 2 6 C 16 What is the absolute maximum value of the function y f x A 3 17 Evaluate 0 A I B 4 C 4 0 U 0 4 B2 C 5 C 3 D 1 5 D 6 D nonexistent 18 What is the equation of the line segment of the piecewise function y f x from 0 x 4 A y x 3 B y x C y x y 1 x 1

IF F x x 1 ar x 3 4x then

The function f x 2x 33x 108x 5 has two critical numbers Use the derivative to algebraically find the critical numbers The smaller one equals and The larger one equals Hint Factor carefully

Next question Find the following using the table below X 1 2 3 4 f x 1 3 2 4 f x 4 2 1 3 g x 2 3 1 4 g x 4 2 1 3 h 3 if h x f x g x 3 f x g x h 3 if h x Get a similar question 1 h 3 if h x f g x 4 OFF

0 26 1 04 Points Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f If an answer does not exist enter DNE 16 x2 3x 2 f x absolute maximum absolute minimum local maximum DETAILS local minimum 16 0 DNE PREVIOUS ANSWERS DNE if 4 x 0 if 0 x 4 X SCALCET9 4 XP 1 008 X Enhanced Feedback Please try again Use the first and second derivatives to find the intervals where the function is increasing or decreasing and where the function is concave up or down

Find the critical numbers of the function Enter your answers as a comma separated list If an answer does not exist enter DNE h t t3 4 gt 4 2 9 0 0 t 9 X Your Your answer cannot be understood or graded More Information

coth x sech x csch x Need Help Submit Answer Read It DETAILS 5 0 78 1 04 Points Sketch the graph of f and use your sketch to find the absolute and local maximum and minimum values of f Enter your answers as a comma separated list If an answer does not exist enter f x x 9x absolute maximum value absolute minimum value local maximum value s value s Watch It 8 if 1 x 0 if 0 x 1 1 DNE 0 PREVIOUS ANSWERS SCALCET9 4 1 027 X

Sketch the graph of f by hand and use your sketch to find the absolute and local maximum and minimum values of f If a answer does not exist enter DNE f x absolute maximum local maximum J16 x 13x 2 absolute minimum local minimum if 4 x 0 if 0 x 4

If f 2 5 and f x 2 for 2 x 5 how small can f 5 possibly be f 5 2

o refer to later to find your errors and correct them After the quiz due date you will be able to see hints the ebook link s tutorials and Practice Another Version for the problems on the quiz Be sure to review your answers and udge how well you understood the methods used This is the most mportant part of taking each quiz learning from your mistakes Good luck 1 1 04 Points DETAILS b cosh 0 SCALCET9 3 11 001 Find the numerical value of each expression a sinh 0

Suppose that 4 f x 5 for all values of x What are the minimum and maximum possible values of f 8 f 3 f 8 f 3

A study examined brain size measured as pixels counted in a digitized magnetic resonance image MRI of a cross section of the brain and IQ 4 Performance scales of the Weschler IQ test for college students The scatterplot to the right shows the Performance IQ scores vs the brain size Describe the association between brain size and IQ as seen in the scatterplot Choose the correct answer below OA There is a very weak or possibly no association between the two variables OB There is a fairly strong positive straight association between the two variables OC There is a weak negative curved association between the two variables but there is at least one outlier OD There is a moderately weak negative curved association between the two variables OE There is a moderately weak positive straight association between the two variables Performance IQ 140 120 100 x xxxx 80 x X 825 X Submit test X x X 975 Brain Size pixels

Researchers have linked an increase in the incidence of lung cancer in Austria to a pollutant released by an industrial accident The study identified 995 people who lived near the site of the accident and were under age 40 at the time Seventeen of the people had developed lung cancer at an unusually young average age of 45 Medical records showed that they had heightened concentrations of the pollutant in their blood and that each tenfold increase in the level of the pollutant was associated with a doubling of the risk of lung cancer Complete parts a through e below a Identify whether the research is an observational study or an experiment The research is an b Identify whether i It is c Identify the subje observational study experiment elected Choose the correct answer below OA Austrian peop OB People exposed to the pollutant from the industrial accident OC The medical records of the people exposed to the pollutant mit test

Refer to the graph to the right and follow the directions in parts a through 1 a is a 1 or is 0 a 1 b Give the domain and range of f and the equation of the asymptote c Sketch the graph of g x a d Give the domain and range of g and the equation of the asymptote e Sketch the graph of h x a f Give the domain and range of h and the equation of the asymptote f x ax 1 5 1 a 0 1 a is a 1 or is U a 1 Choose the correct answer below A For the function f x a a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis B For the function f x a a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis OC For the function f x a 0 a 1 because as a gets closer to 0 the graph becomes steeper moving to the left of the y axis b Give the domain and range of f and the equation of the asymptote The domain of fis OD For the function f x a 0 a 1 because as a becomes larger the graph becomes steeper moving to the right of the y axis

3x 6 For the function f x x 5 X 5 that is one to one a write an equation for the inverse function in the form y f x b graph f and on the same axes and c give the domain and the range of f and r If the function is not one to one say so a Write an equation for the inverse function in the form y f x Select the correct choice below and if necessary fill in the answer boxes to complete your choice OA x x B The function is not one to one

7 Your favorite store sends out coupons in the mail Coupon 1 is for 5 off your next purchase Coupon 2 is a 20 off coupon applied to your total purchase Let x the total cost of your purchase at the store a Write a function f x to represent the cost of your purchase using Coupon 1 b Write a function g x to represent the cost of yoru purchase using Coupon 2 c Find f g 75 and explain what this means in context d Find g f 75 and explain what this means in context e In which orer would you like the coupons applied for YOUR order Why

Find the critical numbers of the function Enter your answers as a comma separated list If an answer does not exist enter DNE F x x4 5 x 1 X Need Help Read It Watch It