Calculus Questions

Use the Shell Method to find the volume V of the solid obtained by rotating the region bounded by the following curves about the x axis Draw the 2D region the resulting 3D volume of revolution a typical shell state dV get everything in terms of x or y queen variable and finally integrate to find V Use only methods taught in this class x 3 y 6 x 19

The rate at which revenue was generated in billions of dollars per year for one large company can be approximated by the equation below where t 5 corresponds to the year 2005 What was the total revenue generated to the nearest billion between the start of 2005 and 2017 f t 1 381 2 82t 112 5sts 17 Write a definite integral to find the total revenue generated to the nearest billion between the start of 2005 and 2017 dt

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

A car leasing firm must decide how much to charge for maintenance on the cars it leases After careful study the firm determines that the rate of maintenance M x on a new car will be approximately M x 53 1 x dollars per year where x is the number of years the car has been in use What total maintenance cost can the company expect for a 2 year lease What minimum amount should be added to the monthly lease payments to pay for maintenance on a 2 year lease Write a definite integral to find the total maintenance cost for a 2 year lease JOdx dx 0

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

If labor x costs 260 per unit materials y cost 123 per unit and capital z costs 82 per unit write a function for total cost G XYZ www

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

Evaluate the integral using the Trigonometric Substitution method Do this problem in the exact way it was shown in lecture x 81 x 3 dx

Solve the initial value problem explicitly du dx 2 8x 3x 6 and u 3 when x 1

Evaluate the integral using only methods taught in class 2 Tsi sin 0 cos 0d0

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

Consider a die with only four sides marked one two three and four a Write out a sample space for rolling a pair of these dice Die One c P three d P four e P five f P six 2 9 P seven 3 4 1 Assuming equally likely outcomes find the probability that the sum of the dice is the given number Enter the probabilities as fracti b P two 2 3 2 Die Two 4

Estimate the value of the sum of series S n 0 n methods taught in this class Do not refer to the function ex anywhere in your solution Yo 1 n to within an error of 0 0002 Use only

a Determine whether class 4n 5n 1 is convergent or divergent Use only methods taught in this

Given the polar equation r 4sin 30 0 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

Find the Maclaurin series for the following function Find its radius of convergence R Use only methods taught in this class f x ln 1 4x

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

Find a power series representation for the function and find I the interval of convergence Use only methods taught in this class f x 8 9 x

Show all work No Calculator is allowed 1 Write the equation in standard form identify the shape and center a and b ar graph it 7x 5y 42x 10y 23 0

Rewrite the integral O O O FIT 0 0 O 2 8 8 p2 0 4 1 8 x 12 x x r dzdrde r dzdrde p sin pdpd qdo 0 0 0 0 p sin und podypde 8 x y dzdxdy in spherical coordinates You do not have to evaluate the integral

scale from 0 to 50 in intervals of 5 Six plotted points are generally falling The approximate coordinates of the points are as follows 0 5 28 1 5 3 1 3 11

the function f is defined as f x 5x4 9x3 2 Use the power rule to find the derivative f x and evaluate that derivative at x 1 7 retain at least six decimal Use the forward difference and centered difference formulas to estimate f x at x 1 7 for three different values of the mesh sizes h 0 1 h 0 01 h 0 001 Calculate derivatives using two finite difference formulas h forward difference approximation centered difference approximation exact

Find Kyo for which the dea of the region x 0 and y 0 dave the x axis and below the cure y xx 9 15 9 19 3 2 4

The following graph describes a matrix A as usual that is a colored vector in the domain on left is mapped to same color in the codomain on the right Find two linearly independent eigenvectors and each one s associated eigenvalue 3 y 3 Y 3 2 2 2 3 1 2 X 3 3 2 1 2 1 2 3 1 2 Xx 3

2 Find the antiderivative 0 4 0 17 S 2 2 3 dx

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

2 In 2 1 dz 0 1 0 0 5 0 C 2 1 32 422 02

4z 32 1 Find S 0 0 24 6 32 1 dz C C C 9 32 11 C

Find Sin z 01 1 sin z sin z 1 sin z COS 2 1 sin z points 0006 z 1 sin

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

Find the vector in the subspace W spanned by Answer which is closest to 23 3 4 2

Given that graph of f passes through the Point 4 69 that the slope of HS fargent line at x f x is 10x 4 find f 1 8 b

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

gchool Find the general indennite intergal Land Sin 20tdt Sin 10 t A shot C 5 6 cas 10t C Sin lot C

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

Find the most general antiderivative of the function for 18x 4x 9 F x 6x 14x C BI x 3bx 14 C F x 6x 7 9x c F x 18x 14x 9X C

Evaluate the integral by making an u Substitution x 2 dxy 312 13 x 2 C B Ya x 2 2 C xXx 7 2 1 2 C 11 11 1 C 312 3 X

Eval Ssin x cos cos x dx Bin casx 2 C 3 2 cos sinx C 3 2 sin los x C wants

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 integral 3 b by y dy 0 A 18 b 36 045

Evaluate the indefinite intera S 4 10X 7 4x 5x 2 2 A 2 7 4x 5x C B 7 4x 5x C C 2 7 4X 5x C

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 matrices A and B below determine if they are inverses of each other 1 Select the correct answer below Yes A and B are inverses No A and B are not inverses A B 3 2

Determine each of the following for the giv function f x y x y 3y n x a f 1 2 5 fx x y fxy x y

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