Differentiation Questions and Answers
Calculus
DifferentiationSuppose a company s revenue function is given by R q q 390g and its cost function is given by C q 350 12g where g is hundreds of units sold produced while R q and C q are in total dollars of revenue and cost respectively A Find a simplified expression for the marginal profit function Be sure to use the proper variable in your answer MP q B How many items in hundreds need to be sold to maximize profits Answer sold Bound to two decimal hundred units must be
Calculus
DifferentiationAssume the function f is defined as f x 5cosx 4sinx Use the corresponding derivative rules to find the second derivative f x and evaluate that second derivative at x 1 7 Be sure your calculator is in radians mode retain at least eight decimal places in your calculations Use the central difference formula for the second derivative to estimate f x at x 1 7 for four different values of the mesh sizes h 0 2 h 0 1 h 0 05 h 0 025 Answer the following two questions Does the error in the central difference approximation drop as the mesh size h is reduced As the mesh size h is reduced by a factor of 2 by what factor is the error in the central difference approximation reduced
Calculus
DifferentiationAssume the function f is defined as f x 5 cos x 4 sin x Use the corresponding derivative rules to find the second derivative f x and evaluate that second derivative at x 1 7 Be sure your calculator is in radians mode Note To avoid round off error retain at least eight decimal places in your calculations Use the central difference formula for the second derivative to estimate f x at x 1 7 for four different values of the mesh sizes oh 0 2 h 0 1 o h 0 05 oh 0 025 O
Calculus
DifferentiationFind the derivative of the given function 3 y 5 log x dy 11 Type an exact answer
Calculus
DifferentiationFind the derivative of the given function 2 y 3 sin x sec x Choose the correct answer below O A O B O C O D dy 6 sec x 2x cos x tan x sin x dx dy 2 2 6 sec x x sin x tan x cos x dx dy 2 2 3 sec x 2x sin x tan x cos x dx dy dx 2 3 sec x sinx 2x tan x 1
Calculus
DifferentiationYou may not use your calculator for this question In which of the following is ya function of x I y 9 x II y x III y x A I only B II only C III only OD I and III only E I II and III
Calculus
Differentiationnd the mass and center of mass of the lamina that occupies the region D and has the given density function D is the triangular region enclosed by the lines x 0 y x and 2x y 6 p x y 7x m x y
Calculus
DifferentiationFind the points on the given curve where the tangent line is horizontal or vertical Write answer in correct order using order pairs to get the credit r 1 cos 0
Calculus
DifferentiationGiven that cos 0 and 0 is in Quadrant IV what is sin 0 4 17 17 Simplify your answer including rationalizing the denominator Provide your answer below
Calculus
DifferentiationGiven the triangle below find csc 0 Enter an exact answer in fraction form 12 13 5 0
Calculus
DifferentiationWrite a possible polynomial equation based on the graph below y 5 4 32 2 1 X 5 4 3 2 10 1 2 3 4 5 6 1 2 3 4 5
Calculus
DifferentiationConsider the polynomial x 4x 7x x 12 Using Decartes Rule of Signs create a table like the one below and fill in the possible combination of roots Positive 3 Negative Imaginary
Calculus
DifferentiationGiven the polynomial x3 7x2 16x 12 answer each of the follwing Be sure to label your answer by labeling the parts a b c etc To get full points show your work for this problem on your scratch paper a State the y intercept b State the degree and what it tells you about the graph of the function c Using Decartes rule of signs state the number of possible real roots zeros d Use the rational root theorem to state all possible rational roots e Using either long division or synthetic division break down the polynomial into factored form Clearly state your result
Calculus
Differentiationxpress the function as the sum of a power series by first using partial fractions Give your power series representation centered at x 0 2x6 6x 5 x DO f x C n 0
Calculus
DifferentiationDetermine if the geometric series converges or diverges If it converges find the sum of the series 04 O converges sum is 4 O converges sum is 2 O converges sum is 6 Odiverges
Calculus
DifferentiationTo find the derivative of the function f x sqrt 3 x 2 we can apply the power rule for differentiation
Calculus
DifferentiationLet part of a two part problem P 8 6 cos 61 sin 6t Enter your answers in terms of the variable t x 0 13 a Show that x 1 is a solution to the system Px by evaluating derivatives and the matrix product 06 86 6 Enter your answers in terms of the variablet 6 sin 6t 6 cos 6t x 1 6 b Show that x 1 is a solution to the system Px by evaluating derivatives and the matrix product 06 31 1 6 X 1
Calculus
DifferentiationO IN Sketch the graph of a function that is continuous on 1 51 and has the given properties absolute maximum at 2 absolute minimum at 5 4 is a critical number but there is no local maximum or minimum there 6 Kinh 1 4 2 6 6 1 2 3 4 5 2 3 ASK YOUR TEA
Calculus
DifferentiationQuestion 1 A car moved distance S meters in a time t seconds according to the formula S 3t 2t 4t 1 Determine 1 1 velocity at i t 0 ii t 1 5s 1 2 Acceleration at i t 0 ii t 1 5s Question 2 Let f x x 2x 1 2 1 Find the equation of the tangent at x 1 2 2 Determine the equation of the normal to the tangent line at x 1 Question 3 If f x 3 1 Determine the equation of the tangent at x 1 3 2 Determine the equation of the normal to the tangent line at x 1 Question 4 Determine the derivative of the following functions with respect to the independent variables 4 1 yxsinx cos 2x sin 4x 4 2 y Question 5 Use logarithm differentiation techniques to determine the y of the following functions 5 1 y 5 2 Inx cos3x x sin3x e t sin 2t 5t Question 6 Determine the derivatives of the following inverse trigonometry functions tan x 6 1 f x 6 2 h t tcos 1 et 5 3 3 3 4 4 5 6
Calculus
Differentiation8 Let E be the solid region above the cone z x y and inside the sphere x y z 1 If the density is given by 8 x y z a Set up an expression in Rectangular Coordinates of the mass b Using Spherical Coordinates find the mass
Calculus
DifferentiationDetermine whether the following series is convergent or divergent If it is convergent find its sum Do not use the Ratio Test or the Root Test Use only methods taught in this class iM8 87 1 7 5
Calculus
DifferentiationFor the following parametric equation x 3 cos t y 2 sint 1 for t TC Give the value of x and y if t 4 Round the values to the the nearest tenth 1 decimal place Give your answer as an ordered pair x y with no spaces in between symbols For example 17 56
Calculus
DifferentiationFor the following parametric equation x t 1 y 3 2t t Give the value of x and y if t 2 Give your answer as an ordered pair x y with no spaces in between symbols For example 17 56 for 4 t 0
Calculus
DifferentiationFind the equation of the tangent line to the curve at the given poin 20x xy y 40 at 2 2 The equation of the tangent line is Type an equation Type an exact answer
Calculus
DifferentiationFind f x for the following function Then find f 3 f 0 and f 5 6 f x X f x 0 Select the correct choice below and fill in the answer box within your choice if needed OA f 3 Simplify your answer OB The answer is not a real number OC The answer is undefined Select the correct choice below and fill in the answer box within your choice if needed OA f 0 Simplify your answer OB The answer is not a real number OC The answer is undefined Select the correct choice below and fill in the answer box within your choice if needed OA f 5 Simplify your answer OB The answer is not a real number OC The answer is undefined
Calculus
DifferentiationThe scrap value of a machine at the end of its useful life is given by Sin C 1 1 where C is the original cost nis the useful life of the machine in years and r is the constant annual percentage of value lost Find the scrap value of the following machine Onginal cost 68 000 Mife 8 years annual rate of value lost 11 S Round to the nearest cent
Calculus
DifferentiationUpload solutions to Problem 1 here Find the following for the function a fx 1 2 b fyy x y c fxy x y f x y 3x y 5x 26 9
Calculus
DifferentiationLet F xyz xy x yz eff cut Use Stokes Theorem to evaluate curlFdS where S consists of the top and the four sides but not the bottom of the cube with o corner at 5 5 5 and the diagonal corner at 1 1 1 Hint Use the fact that if S and S share the same boundary curve C that curl F dS curl F d S S F F dr
Calculus
Differentiation3pts Consider f 1 1 x 3 mly The Definition of the DERIVATIVE to find f 1 e equation of the tangent line to the graph of f at the
Calculus
Differentiation6 6 Suppose a is expanded in descending powers of a a Write down the r 1 th term of the expansion where 0 r 6 b Write down the first and the middle term of the expansion
Calculus
DifferentiationOut of pocket spending in a country for health care increased between 2003 and 2008 The function f x 2573e0 0359x models average annual expenditures per household in dollars In this model x represents the year where x 0 corresponds to 2003 a Estimate out of pocket household spending on health care in 2008 b Determine the year when spending reached 2778 per household a The total expenditures per household in the year 2008 were approximately Round to the nearest dollar as needed b During the year spending reached 2778 per household Round down to the nearest year
Calculus
DifferentiationUse the definition to find an expression for the instantaneous velocity of an object moving with rectilinear motion according to the given function relating s in ft and t in s Then calculate the instantaneous velocity for the given value of t s 3t 4t t 6 The expression for the instantaneous velocity of the object is
Calculus
DifferentiationDetermine the values of x for which the function f x Where is the function continuous or not continuous 4 2 4X is continuous If the function is not continuous determine the reason OA The function is not continuous at x 4 OB The function is continuous for all values of x OC The function is continuous for all values of x greater than 0 OD The function is not continuous at x 0 OE The function is not continuous at x 4 OF The function is continuous for all values of x less than 4 OG The function is continuous for all values of x between 0 and 4 H The function is not continuous at x 0 and x 4
Calculus
DifferentiationFind values of the trigonometric functions of the angle in standard position whose terminal side passes through the given point 4 3 sin 0 cos tan 0 cot 0 3 5 4 5 3 Type an exact answer in simplified form Type an exact answer in simplified form Type an exact answer in simplified form Type an exact answer in simplified form
Calculus
DifferentiationThe electric current i in mA in the circuit i 8 7 1 e 0 37t where t is the time in s Evaluate i for t 7 0 ms i mA Round to four decimal places as needed
Calculus
DifferentiationBy means of the definition of a radian change the given angles in radians to an equal angle expressed in degrees 4 498 rad O 4 498 rad Round to two decimal places as needed
Calculus
DifferentiationEvaluate the given function f x 3x 1 find f x 3 f x 3 f x 3 f x 3 Simplify your answer