Differentiation Questions and Answers

Suppose 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

Assume 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

Assume 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

Find the derivative of the given function 3 y 5 log x dy 11 Type an exact answer

Find 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

lim X 5x x7 et O A OB 5 3 OC 0 O D 1 OE 1

You 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

nd 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

Find 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

Given 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

Given the triangle below find csc 0 Enter an exact answer in fraction form 12 13 5 0

Write 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

Consider 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

Given 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

xpress 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

Find the interval of convergence for the power series 1 Q 1 1 1 1 1 1 1 1

Determine 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

Find the area of the region bounded by the graphs of x y x y O 1 0 1 1 01

To find the derivative of the function f x sqrt 3 x 2 we can apply the power rule for differentiation

Let 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

O 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

Question 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

8 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

Determine 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

Find the derivative y for each curve functo e sx Y x 3x 2

Find each derivative y for each given curve functier 2 y x In x

We know that if u f x then du f x dx Therefore if u x 30 then du dx

For 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

For 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

Find 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

Find 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

The 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

Upload 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

Let 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

cos x 2 1 QUESTION 3 2 points Find the derivative of f x In x 2 4

3pts 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

6 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

Out 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

7 Write the equation of the following function 10 tay x 1

Find the derivative of the function by using the definition y 6x 7x 11

Use 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

Find the derivative dy dx 9 5 y 10x 0 5x

Differentiate dy dx 3 y 4x 16x 16x 64x 7

7 Find g t for the function g t

Determine 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

Find 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

The 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

By 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

Evaluate the given function f x 3x 1 find f x 3 f x 3 f x 3 f x 3 Simplify your answer

Find the y derivative of the function by using the definiti y 2x 5x