Application of derivatives Questions and Answers

Let f R R be a function satisfying 2 x f 2 x and 20 x f x x R On the basis of above information answer the following question 15 50 Read Less

Find the zeros of the following polynomial f x x 49x List the zeros separated by commas For example if you found that the zeros were x 1 x 2 and x 3 you enter 1 2 3 Provide your answer below

Given the polynomial f x x 3x what is the smallest positive integer a such that the Intermediate Value Theorem guarantees a zero exists between 1 and a Enter an integer as your answer For example if you found a 8 you would enter 8 Provide your answer below

Based on the degree of the polynomial f x given below what is the maximum number of turning points the graph of f x can have f x x 8 x 3 x 5

Given the following table of values of the polynomial f x what is the minimum number of zeros of f x that are guaranteed to exist by the Intermediate Value Theorem Select the correct answer below 0 1 2 3 X 1 2 3 4 5 6 f x 3 8 5 10 32

Given the graph of y f x shown as a red dashed curve drag the movable blue point to obtain the graph of y f x 4 2 Provide your answer below 10 5 f x 10 5 5 5 10

y f x 4 2 Provide your answer below 10 5 10 5 0 5 f x 5 10

Given the polar equation r 4sin 30 a Sketch the polar graph by first graphing the cartesian graph Show the exact correspondence between sections of the two graphs I e do it exactly the way that was shown in lecture

2 For the function f x y 4y 5x 2 find f 3 2 f 4 5 f 2 5 and f 0 6 1 f 3 2 61 Type an exact answer using radicals as needed Type an exact answer using radicals as needed f 4 5 6 5 f 2 5 Type an exact answer using radicals as needed

Let f x y 3x 9y 7 Find the following values of f x y for the given coordinates a f 4 1 f 4 1 28 Simplify your answer b f 4 1 f 4 1 14 Simplify your answer c f 2 4 f 2 4 37 Simplify your answer d f 0 8 f 0 8 Simplify your answer

Write the rule of a function that gives the perimeter of a rectangle of length L and width w P L W

For the function f x y 4y 5x find f 3 2 f 4 5 f 2 5 and f 0 6 f 3 2 Type an exact answer using radicals as needed

The graph of f on the interval 4 1 is shown in the figure to the right Complete parts a through f below CO a Find the largest possible open interval s on which f is increasing and the largest possible open interval s on which f is decreasing Select the correct choice below and if necessary fill in the answer box es within your choice OA The function is increasing on the open interval s and decreasing on the open interval s notation Use a comma to separate answers as needed The function is never decreasing Simplify your answer Type your answer in interval OB The function is increasing on the open interval s Simplify your answer Type your answer in interval OC The function is decreasing on the open interval s Simplify your answer Type your answer in interval notation Use a comma to separate answers as needed notation Use a comma to separate answers as needed The function is never increasing OD The function is never increasing nor decreasing

A bicycle manufacturer produces two kinds of frames aluminum and carbon fiber The cost function for these products is C x y 460x 1400y 16 000 where x is the number aluminum frames and y is the number of carbon fiber frames produced Complete parts a and b below a What is the marginal cost of producing one additional aluminum frame when the production of carbon fiber models is held constant The marginal cost would be

dT Avalanche forecasters measure the temperature gradient which is the rate at which the temperature in a snowpack T changes with dh respect to its depth h A large temperature gradient may lead to a weak layer in the snowpack When these weak layers collapse avalanches occur Avalanche forecasters use the rule of thumb that if exceeds 10 C m anywhere in the snowpack conditions are dh dT favorable for weak layer formation and the risk of avalanche increases Assume the temperature function is continuous and differentiable Complete parts a through d below a An avalanche forecaster digs a snow pit and takes two temperature measurements At the surface h 0 the temperature is 14 C At a depth of 1 1 m the temperature is 2 C Using the Mean Value Theorem what can he conclude about the temperature gradient The average temperature gradient from h 0 to h 1 1 is C m Simplify your answer Round to the nearest tenth as needed

Solve the differential equation using only methods taught in class y 3xe 0

Pel fie derivative 2 sec 2x dz the Oz 2lnz tan 2x C Dez 2ln z tan 2x C 0 1 2 In 2 tan 22 C 2 Inz tan 22 C

Find the tangent line to the clipse 1 let the point 2 3 16 4 11 3x 4 Y x A wa

i fai if Forl dr 10 and foo dx 7 And fox dx 0 4 23 62 O 17

Evaluate the integrel by interpreting in terms of overs 3 4 x dx 877 20

2 2x At what values of x does the graph of V X of inflection A X 2 and x 0 D X 1 and X 0 c X 2 2 and x 2 2 1 X 1 1 2 32 and x 1 1 2 have a par

Evaluate the indefinite intergral 5 Cos x sin x dx 7 8 C05 x C 8 7 1 1 8 sin x C c1 18 05 x C D 1 8 cos ox C

Given the following function and it s first and Second clerivatives determine each of the following 4 f x x 8x 5 3 f x 4x 24x 4x x 6 f x 12x 48 x 12x x 4 Find all intervals on which f x is decreasing Find all intervals on which f x is concave up Find the x values where the absolute maxinum and absolute minimum of f occur on the interval 1 1

Use the method of Lagrange Multiplers to find the minimum of the function 2 f x y x 3y 6 subject to the Constraint x 2y 4 Write answer as an ordered pair You may assume that the mininum does exist

Find the inverse of f x O g x O g x O g x g x Select the correct answer below 6x 9 6x 4 6x 9 6x 4 4x 9 6x 6 6x 6 4x 9 6x 9 6x 4 from the answer choices below by checking whether f g x x is true

Use a First and Second Derivative test to identify the Critical Numbers of f x sin x cos x on 0 4 then identify them as either Max s Min s or Neither Make sure to identify where they are by x y Then identify where the point or points of inflection are x y And finally give the intervals of Concavity for the function

Find each derivative y x 8e 5 y for each given curve

Sketch one loop of the curve 9y x 3 x Find the area of the surface of revolution generated by one loop of the cum 9y x 3 x about the y axis Leave your answer in the exact form

Use implicit differentiation to find the equation the line which is tangent to the given curve at e point 1 2 xy 3x y 1 2

Use Gaussian elimination to solve the following system of equations 5u 10v 15w 30 2u 9v 14w 103 and 3u8v 4w 13 Select the correct answer below O 3 3 5 O 3 5 3 O 5 3 3 infinitely many solutions Ono solutions

Use Newton s method to estimate the two zeros of the following function Start with xo 2 for the left zero and Xo 3 for the right zero Then for each case find x 2x x 4 If x 2 then x Type an integer or a simplified fraction

Consider the following integral Ico Given the substitution u 8x find du du cos 8x dx dx Rewrite the given integral in terms of u SCC 1 du Evaluate the integral by making the given substitution Use C for the constant of integr

The acceleration function in m s2 and the initial velocity are given for a particle moving along a line a t t 4 v 0 3 0 t 11 a Find the velocity at time t v t m s b Find the distance traveled during the given time interval

a Find a power series representation for f x centered at c 0 below b Determine the interval of convergence for the power series 12 7x 3 Enter the interval of convergence for the power series in the space provided f x

Given that 3 4 2 and 1 2 3 find a u b x Enter the answer to part a in the space provided c the unit vector in the same direction as

Given the following parametric equation x t 3 y 1 6t t for 0 t 4 choose the graph that has the most accurate and complete curve and includes the correct orientation given the limits on t values Be careful here Look closely at the points plotted before making your choice Double check that the orientation is correct Double check endpoints arrows

Integration by substitution is related to what differentiation method What type of integra Integration by substitution is related to what differentiation method OA Integration by substitution is related to the exponential rule and logarithm rule B Integration by substitution is related to the power rule and quotient rule C Integration by substitution is related to the sum or difference rule D Integration by substitution is related to the chain rule

Use substitution to find the given indefinite integral fx ex 5 ax x 8 8x dx x 8x8 5 dx

Find dy 4 for the equation 2x 4y 1 using implicit differentiation dx

Given that 3y 3z 12 and 15y 14z 1 solve the system of equations using Gaussian elimination Select the correct answer below 59 55 O 21 59 55 59 infinitely many solutions no solutions

Straight load At time seconds the position of the car in feet is s t 21 Find the average velocity of the car over the time intervals given below a t 10 to t 10 2 seconds b t 10 to t 10 02 seconds c t 10 to 1 10 002 seconds a The average velocity of the car over the time interval from t 10 to t 10 2 seconds moving along a straight road is Simplify your answer Round to one decimal place as needed b The average velocity of the car over the time interval from t 10 to t 10 02 seconds moving along a straight road is Simplify your answer Round to two decimal places as needed c The average velocity of the car over the time interval from t 10 to t 10 002 seconds moving along a straight road is Simplify your answer Round to three decimal places as needed feet sec feet D

Write down the augmented matrix corresponding to the system of equations shown below 6x 5y 4z 7 7x 9y z 7 9x 3y 6z 7 Note Write an equation in each answer field and do not solve the system Provide your answer below

Admission to a movie theater is 7 for children and 11 for adults On a certain day 57 people bought tickets at the theater and 499 was collected How many children s tickets were sold and how many adult s tickets were sold Provide your answer below

Use Gaussian elimination to solve the given system of equations 2m 6n 6 and 8m 15n 3 State the solution as an ordered pair If there are no solutions write

f 5 shirts and 8 sweaters cost 222 and 6 shirts and 5 sweaters cost 202 what is the cost of one shirt and what is the cost of one sweater Provide your answer below

Problem 32 Budgeting for maximum production A manufacturing firm has budgeted 60 000 per month for labor and materials If a thousand is spent on labor and y thousand is spent on materials and if the monthly output in units is given by N x y 4xy 8x then how should the 60 000 be allocated to labor and materials in order to maximize N What is the maximum N

Problem 30 Use Theorem 2 below to find the local extrema of f x y 2y 6xy x Theorem 2 If 1 z f x y 2 f a b 0 and fy a b 0 3 All second order partial derivatives of exist in some circular region containing a b as center 4 Afrz a b B fay a b C f a b Then Case 1 If AC B 0 and A 0 then f a b is a local maximum Case 2 If AC B 0 and A 0 then f a b is a local minimum Case 3 If AC B 0 then f has a saddle point at a b Case 4 If AC B 0 the test fails

Use the linear equation y x 32 where x is the degrees Fahrenheit and y is the degrees Celsius to convert each terapera a 83 C to Fahrenheit b 93 F to Celsius a 83 C F Round to one decimal place as needed b 93 F C Round to one decimal n

Cantaloupe is given by V The radius is growing at the rate of 0 8 cm week at a time when the radius is 7 5 cm How fast is the volume changing at that moment The volume is changing at a rate of cm week Round to one decimal place as needed

Evaluate the integral 2 2 x X 1 2x 8 dx X 1 x 2x 8 Type an integer or decimal rounded to three decimal places as ner dx