Application of derivatives Questions and Answers

Level curves for barometric pressure in millibars are shown for 6 00 a m on a day in November A deep low with pressure 972 mb is moving over northeast Iowa The distance along the red line from K Kearney Nebraska to S Sioux City Iowa is 300 km km O mb km km mb 1012 mb K 1008 1000 1008 1004 996 992 988 S 984 980 976 1012 972 What are the units of the directional derivative mb km 016 1020 wh 1024 Z Estimate the value of the directional derivative of the pressure function at Kearney in the direction of Sioux City Round your answer to two decimal places

Consider the following graph of the function f x y given below f x y 4 x y 3xy point corresponding to local minimum 3 7 point corresponding to saddle point 3 2 Use the level curves to predict the location of the critical points of f and determine whether f has a saddle point local maximum or local minimum at each point Then use derivatives test to confirm your predictions If an answer does not exist enter DUE point corresponding to local maximum x y 6 x y x y

Use the graph to identify the formula for a polynomial function of least degree 5 4 3 72 1 f x 3q048P 5 5 y 5 2 1 0 1 2 3 4 57 1 2 3 2 4 X 5 X

2 10 8 6 4 2 0 2 4 2 W 6 10 12 14 h t 6 8 10 Find a trigonometric function sin or cos for the function shown above

The function f x changes value when x changes from xo to xo dx Find the approximation error af df Round your answer if appropriate 21 f x x2 x0 4 dx 0 05 Write a differential formula that estimates the given change in volume or surface area 22 The change in the surface area S 47r of a sphere when the radius changes from ro to ro dx

10 8 6 g t AX 4 2 0 2 4 2 8 6 4 6 8 6 10 8 cod for the function shown a

Suppose y cos t2 4 Answer the following questions What is the period of the function What is the horizontal shift of the function What is the phase shift of the function

Find dy dx by implicit differentiation S 5 O 57 5 10x 18y O gy 5 10x 18 O 05 O S

Suppose f x 0 at x xo but nowhere else Also assume f x is continuous and has the following values where a x b x f x a 2 Check ALL that apply 20 0 This means that b 8 Of increases after x 0 Of has a local minimum at x xo Of has a local maximum at x xo Of decreases after x 0 Of decreases until x 0 If f xo 0 there is a local minimum at x xo If f o 0 there is a local maximum at x xo The given information is incorrect f xo 0 guarantees neither a local maximum nor a local minimum x xo

Suppose f x is continuous for all x and the graph of y f x has an inflection point at x 8 Complete the following table with appropriate possibilities for the values of f x at x 1 8 12 x f x 1 8 12 Based on your table the graph of y f x is Select an answer until x 8 and is Select an answer after The graph of y f x will have a Select an answer at x 8

Suppose f x has f 8 0 but the graph of y f x does NOT have an inflection point x 8 Complete the following table x f x 2 8 0 13 Based on your table the graph of y f x is Select an answer until x 8 and is Select an answer after

Suppose f x is continuous for all x and the only extreme value of f x is at x 8 Complete the following table with appropriate possibilities for the values of f x at x 3 8 12 x f x 3 Question 8 12 Based on your table f x is Select an answer until x 8 and is Select an answer after The function f x will have a Select an answer Vide Fa Writt at x 8

Use the figure to find the exact value of the trigonometric function Find cos 20 21 29 8

r t t 3 t 2 2 2 4 4 X 4 4 2 y y 4 4 4F 2 2 2 4 4 X

6 12 2 Click on graph to enlarge 6

Show all work to write the formula of the quadratic function graphed below 4 2 0 2 2 4

15 Extra credit Draw the graphs of the real part of the function f z e as a function of a real variable x when z re where x 0 is real for several values of the fixed parameter 3 0 0 2 In particular use 0 0 0 4 and 0 2 Also look at sufficiently many

There is a rational function with the following characteristics 2 and X 2 there are vertical asymptotes at X 1 f 0 and f 1 0 and 4 x f x 0 a Write the formula for the rational function and enter it below b Draw a sketch of the function on your written work

2 H 4 is a plane Show that your A gives 1 you a plane by expressing the solution set as a shift of a span of two vectors Hint Use parametric vector form 2 a Create a 3 x 3 matrix A such that the solution set to Ax

The period is Enter NONE if not periodic help numbers M 8

The time required for workers to produce each unit of a product decreases as the workers become more familiar with the production procedure It is determined that the Function for the learning process is T x 2 0 3 where T x is the time in hours required to produce the xth unit Find the time required for a new worker to produce units 10 through 18 OA 16 18 hr B 34 87 hr OC 36 87 hr D 1 82 hr

5 10 5 0 5 1 t 5 P 7 4 S 10 Using the image above suppose t 29 84 Identify the values of angle t in radians r and

Use the properties of summation and the Summation Formulas Theorem to evaluate the sum Use the summation capabilities c graphing utility to verify your result 1 1 i 1

2 Derive the operator A in polar coordinates r 0 and convert the following domain integral to a boundary line integral on I mrd r y

Use the equations to find z x z av z x z x F x F z z y e 4xyz and and z y F y F z

The percents of girls age x or younger who have been sexually active are given in the table a Create a logarithmic functon that models the data using an input equal to the age of the girls b Use the model to estimate the percent of girls age 17 or younger who have been sexually active c Find the quadratic function that is the best fit for the data d Graph each of these functions on the same axes with the data points to determine which function is the better model for the data Age Cumulative Percent Sexually Active Girls 15 16 17 18 19 20 5 6 12 5 27 1 43 8 62 2 73 9

Given the parametric equations x t 2 2 and y t3 4t where 3 t 3 which of the following graphs represents the curve and its orientation O O 8 7 8 7 5 4 3 2 1 A 3 N 1 2 3 7 do d 1 2 3 4 5 50 7 8

uppose that in a memory experiment the rate of memorizing is given by M t 0 003t where M t is the memory rate in words per minute How many words are memorized in the first 10 min from t 0 to t 10 0 4t In the first 10 minutes words are memorized

Given the parametric equations x t2 2 and y t3 4t where 3 t 3 which of the following graphs represents the curve and its orientation O 54 3 1 7 6 5 4 32 1 1 777 7 9 95 2 4 1 2 3 8

What is one solution to the trigonometric equation siny cosy O in the intervallo 212

Let g x y cos x 2y a Evaluate g 2 1 g 2 1 b Find the domain of g OR2 I 2 x 2y 2 1 x 2y 1 2 x 2 1 y 1 1 x 1 1 2 sys 1 20 c Find the range of g Enter your answer using interval nota

z Use implicit differentiation to find and x z x z av 72 e xyz z y

Water traveling along a straight portion of a river normally flows fastest in the middle and the speed slows to almost zero at the banks Consider a long straight stretch of river flowing north with parallel banks 100 m apart If the maximum water speed is 3 m s we can use a quadratic function as a basic model for the rate of water flow x units from the west bank f x 3 x 100 x 2500 a A boat proceeds at a constant speed of 5 m s from a point A on the west bank while maintaining a heading perpendicular to the bank How far down the river on the opposite bank in m will the boat touch shore m

Q36 Let h x x 2 g x sin x 3 then find the value of goh 2 hoal a

A graphing calculator is recommended The graph of the function f x y 1 x a Find an equation of the tangent line to this curve at the point 2 0 40 O 3 xX 25 16 25 is called a serpentine b Illustrate part a by graphing the curve and the tangent line on the same screen y 1 0 2 0 5 0 5 2 X 2 ESPER y 1 0 0 5 0 5 2 4 B DO 2 y 0 5 1 0 2 4 DO 2 y 1 0 Os 2

Use the theorem given below to find the curvature r t 3ti 5 sin t j 5 cos t k Theorem The curvature of the curve given by the vector function r is x t k t Ir t x r t Ir t

Find all local extrema for the function f x y x 9xy y Find the local maxima Select the correct choice below and if necessary fill in the answer box to complete your choice O A OB There are no local maxima Find the values of the local maxima Select the correct choice below and if necessary fill in the answer box to complete your choice The values of the local maxima are Use a comma to separate answers as needed B There are no local maxima OA There are local maxima located at Type an ordered pair Use a comma to separate answers as needed Find the local minima Select the correct choice below and if necessary fill in the answer box to complete your choice O A OA There are local minima located at Type an ordered pair Use a comma to separate answers as needed OB There are no local minima Find the values of the local minima Select the correct choice below and if necessary fill in the answer box to complete your choice The values of the local minima are Use a comma to separate answers as needed

Differentiate the function g x g x 1 to 1 X T9 3 1 x X

Find the equation of the line tangent to the graph of f x In x at x 6 y 0 Type your answer in slope intercept form Do not round until the final answer Then round to two decimal places as needed

Solve for x and simplify 5 2 2 5 12 A 4 or x 2 B x 4 or x 2 C x 4 or x 2 D NO solutions E NO correct choices

Here is 4 on of the function g 3 2 y 4 3 2 1 2 4 5 Use the graph to find the following If there is more than one answer separate them with commas a All local maximum values of g HAKKAR h All values at which g has a local maximum

Question 7 The graph of f is given below O 10 9 No relative maxima exist 10 9 8 7 6 5 4 3 2 1 8 7 6 23 3 5 6 Determine the location of any relative minima of f If multiple such values exist enter the solutions using a comma separated list 7 8 9 10 5 6 7 8 9 10 No relative minima exist Determine the location of any relative maxima of f If multiple such values exist enter the solutions using a comma separated list

Next question Find three numbers whose sum is 21 and whose sum of squares is a minimum The three numbers are Use a comma to separate answers as needed DE Fore

Find the order 3 Taylor polynomial T3 x of the given function given Use exact values f x 3x 238 at a 6 T3 x

Find three numbers whose sum is 6 and whose sum of squares is a minimum The three numbers are Use a comma

Use substitution to find the given indefinite integral Sx5 2x 5 dx 1 56

Answer each of the questions throughout the problem below Question Sarah walked south for one block and then west for two blocks How far was she from her starting point as the crow flies and what angle does she have to turn to her right from facing west to face her starting point Distance Select Angle To find the missing straight line distance we just need to use the Pythagorean Theorem 2 blocks 1 block To find the angle we can use a trig function to find the angle inside the triangle and then subtract that from 180 to find the angle we are looking for

angles below For more information on this topic look in your textbook at section 4 1 You might also want to look at the Angle and Radian Measure presentation 15 3 2 100 17 2 Choose Choose Choose Choose

18 Fig 6 shows a sleeve type part 40 0 16 mm and 10 0 3mm are the design dimensions As the dimension 100 0 3 is inconvenient for measuring in machining process it can only be ensured indirectly by measuring the depth Li of hole 1 Draw the process dimensional chain of the sleeve determine the closed link increasing link and the decreasing link 2 Try to find L and its ES1 deviations L11 Answers 10 0 3 L1 40 0 16

a b c 9 pts Lat g x Rt dt where is the function whose graph is shown Evaluate g 4 Show work Where does g have a maximum value Explain in detail Sketch a rough graph of g x Explain step by step how you got this graph