Mathematics Questions and Answers July 07, 2020

27x y 0 between y 0 and y 2 Let C be the curve 27x y 0 Find the area of the surface generated by revolving C about the y axis Please set up do not evaluate a O 2 54m x 1 2 dx 54 81 5 2 9 0 2 94 b d fy 2 2 v 27 dy 81 dy 81

Find the antiderivative of the given derivativ ex 5e 5x ex e 5x 4 dp dx 11 SEN DE BRANDS HONESTON INTE

A chemist has a 100 gram sample of a radioactive material He records the amount of radioactive material every week for 6 weeks and obtains the following data Complete parts a f Week x 0 1 2 3 4 5 6 Weight y in grams 100 0 86 0 72 0 64 1 53 1 45 4 38 8 a Treating week x as the independent variat use a graphing utility to fit an exponential funct to

Find an equation of the normal line to the curve of y = ??x that is parallel to the line 4x + y = 1.y =

Determine the volume of the solid generated by rotating function f(z) =[4,10].100-z about the x-axis on-

Determine the volume of the solid generated by rotating function f(z) about the xaxis on 4,101.

Is the set of functions {1, sin x, sin 2x, sin 3x, ...) orthogonal on the interval [-?, ?]? Justify your answer.

Find the vertical asymptote x A f x x 3 5x 4

Find the vertical asymptote x 5x 4

*4. Use the comparison theorem for improper integrals to prove thatSverges.sin r1Hint: Use your answer from question 3(a) combined with the fact that sin r ? 1.de con-

23 Given logb2 4 logb3 5 logb5 6 Use properties of logarithms to find the following value log 75 A 86 B 1 7 C 41

Find E(x), E(x), the mean, the standard deviation and variance, over the given interval, of the random variable whose probability density function is given below.f(x) = 1 4, [3,7]

O x is the price in dollars per unit that consumers are willing to pay for x units of an item and S x is the price in dollars per unit that producers are willing to accept for x units Find a the equilibrium point b the consumer surplus at the equilibrium point and c the producer surplus at the equilibrium point D x x 9 S x x 4x 15 a What are the

The battery charge of a cell phone is tracked with a function over the course of one day Graph a possible function below that correctly represents the domain and range of the situation Co Battery charge 150 1 25 100 75 50 25 0 25 4 8 12 16 20 24 28 Time hours

Calculate the following integral using integration by parts We let u X So du 1 and du sec 2 4x da and v and then use the integration by parts formula to find that x sec 4x dx and finally we have a sec 4x dx da da C r sec 42 dr

b It is given that g x is a periodic function with period 1 i e g x 1 g x for any x Using fundamental theorem of calculus show that i 4 S 0 g x dx 4 1 S g x dx 1 3 ii 9 3x dx 1 2 1 g x dx 0

1 2 SSG x y 2 do 2 1 42 S S dzdx S 00 Evaluate the integral found in the previous step SJ G x y z do S

Use a change of variables or the table to evaluate the following definite integra KIN KIN 2 S 2 cotx csc x dx 4 Click to view the table of general integration formulas 2 S 2 cotx csc x dx Type an exact answer

1 Find the partial fraction decomposition of these rational functions a b c 3x 5 x 3 x 9 d 9 x 1 x 8 7x 9 2x x 2x 3x 1

Bonus 1 10th deri f x x 10 10 x

13 onuse1z1Qtt)lcn

Sketch the following polar rectangle R 1 0 15r54 50s 1 rs4 Choose the correct graph below QA I 3 FO 61 Q B 61 6 X ww O G OC D Q Q OD 6 3 134

x 4x 1 5 88 Points DETAILS LARCALCET7 8 5 009 Use partial fractions to find the indefinite integral Remember to use absolute values where appropriate Use C for the constant of integration 2x 16x 39x 32 dr

Antarctica is roughly semicircular with a radius of 2 00 x 103 km see the figure The average thickness of its ice cover is 3 00 x 10 m How many cubic centimeters of ice does Antarctica contain Ignore the curvature of Earth 8000 m 2000 km

From the Table 5 2 we deduce that the value of f x may be made smaller than any given number by choosing a negative real number very close to 0 In symbols we write lim f x 0 x 0 to be read as the left hand limit of f x at 0 is minus infinity Again we wish to emphasise that is NOT a real number and hence the left hand limit of fat 0 does not exist

The region R in the first quadrant is bounded by y 4 sin z y 4 z 2 and y 3x 1 and contains the point 2 4 The area of R is area The volume of the solid generated by rotating R about the x axis is volume The volume of the solid generated by rotating R about the y axis is volume The volume of the solid with R as its base and square cross sections

mLet x be a continuous random variable over [a,b] with probability density function f. Then the median of the x-values is that number m such that?? 1/72Sf(x) dx =athe median.f(x) = -1128x, [0,16]The median is m =Find

Find the area of the region inside one leaf of the three leaved rose r 3 cos 30 The area of the region is Type an exact answer using as needed

6 Graph of f The derivative of a function fis defined by f x X g x for 4 x 0 5e 2 3 3 for 0 x 4 The graph of the continuous function f shown in the figure above has x intercepts at x 2 and 2 3 ln The graph of g on 4 x 0 is a semicircle and f 0 5 a For 4 x 4 find all values of x at which the graph of fhas a point of inflection Justify your answer b

Use a calculator to make a table to approximate the following limit Confirm your result with l H pital s Rule lim 1 3h h 0 Complete the table below Round to four decimal places as needed h 10 1 10 2 1 h 10 10 4 1 3h h h 10 1 2 10 10 10 4 3 1 1 3h

In Problems 51 56 use modal decomposition to solve 23 8 8 0 51 A X 52 A 1 1 2 2 orn 0 Ax f x 0 xo 0 xo 1 X x1 2 Xo

K Watch the video that describes finding inverse cosine values Click here to watch the video Find the exact value of the expression arccos arccos Type an exact answer using a

Evaluate the integral fz 2 4 dz by making the substitution u C NOTE Your answer should be in terms of I and Question Help Video Message instructor

y tan x y sin x y cos x y csc x y sec x y cot x Submit A V 16 5 4 3 2 6 5 4 3 2 1 3 Q

Evaluate the definite integral 1 4e in cos 1 dr

Evaluate thie det

Find the average value fave of the function f on the given interval.f(x)=?x,[0, 16]

Find the average value f of the function f on the given interval. f(X) vG, 0, 16

D 3 x 2 ay Transform D to cylindrical and spherical coordinates

D 3(x z)ay. Transform D to cylindrical and spherical coordinates.

Evaluate SSD x2 y2 dA D x y a2 x2 y2 b2 one where Select

Evaluate , where D(x,y) I a2Q2Y2

Find and simplify the expression if f(x) = x - 8.f(3x)= ________ (Simplify your answer.)

Find and simplify the expression if f(x)x 8. f(3x) f(3x) (Simplify your answer.)

The goal of this problem is to evaluate Part 1 The Partial Fraction Setup 82 16 x 8x Perform the partial fraction decomposition of 8x 16 x28x dx using partial fractions 8x16 x 8x and find the values of the coefficients

Use Stokes Theorem to evaluate VXF dS where M is the hemisphere x y 9 x 0 with the normal in the direction of the positive x direction and F x9 0 M Begin by writing down the standard parametrization of OM as a function of the angle 8 denoted by t in your answer X z y SOM F ds f 0 do where f 0 The value of the integral is use t for theta

What is the degree of the polynomial f x defined below f x 6x 3x 6

What is the degree of the polynomial f (x) defined below (ja2 3.r 6

R be the region bounded by the following curves Use the method of your choice to find the volume of the solid erated when R is revolved about the y axis x y 2x 2 x 3 and x 6 volume of the solid is 108 e an exact answer Save

Set up but do not evaluate an integral for the length of the curve osys112 X 3 sin y 2 3 cos y x dy TO

10The mouse is a particularly useful model for studying human disease because mice and humans are 100 genetically a identical all mice are in essence clones mice do not feel pain and discomfort d mice and humans are the only species whose genomes have been sequenced b C e approximately 90 of human disease associated genes are present in the mouse

Suppose a student finds the following values of x and f x x If f 4 9 3 4 99 877 C 4 999 2393 these values would SUPPORT the claim that lim f x x 5

Solve the following logarithmic equa log x log x 3 1

Solve the following logarithmic equz log x log (x

A baseball player's batting average is 0.337, which can be interpreted as the probability that he got a hit each time at bat. Thus, the probability that he did not get a hit is 1-0.337=0.663. Assume that the occurrence of a hit in any given at-bat has no effect on the probability of a hit in other at-bats. In one game, the player had 4 at-bats.

approximately what values of x between 5 and 5 does the original function y have inflection points W M 3 OA x 2 03 x 0 x 2 03 OB x 2 03 x 2 03 OC x 3 83 x 3 83 OD x 4 91 x 2 03 x 0 x 2 03 x 4 91 O E There are no points of inflection

3 Let F x 3r cos 1 2 dt A Using the First Fundamental Theorem of Calculus find F x Yes this is as easy as it seems B Starting with F x f 31 cos 1 2 dr use the substitution u 1 to rewrite the definite integral You should get a new equivalent expression for F x which consists of this new definite integral

Find the area of the region bounded by the graph of the equation.25.) y = 1 + 3x x = -2, x = 1, y = 0

1 Evaluate the following 2 a tan cos b sin cos Crat sin

I.Evaluate the following a) tan (co (3) b)sin (cos( i)

Problem A random variable X has CDF 860 ft 70 Fx x e x x 0 4 0 x 0 Find P X 2 P X 0 P X 10 What kind of random variable is X

Suppose we want to approximate the integral b x dx by using a left hand sum with 4 rectangles of equal widths Write out this sum using function notation for each term Answer Now approximate the integral b a da by using a right hand sum with 3 rectangles of equal widths Write out this sum using function notation for each term Answer

A company s marginal cost function is MC x x 1 2 5 where x is the number of units If fixed costs are 23 250 find t company s cost function C x

The function defined by years and x 0 corresponds to 1960 325 b When do the seasonal maximum and minimum carbon dioxide levels occur 3 7 11 2 2 2 OA Maximums x 0 OB Maximums X 15 2 2 3 7 11 2 22 159 OC Maximums X 51515 2 2 2 OD Maximums X Minimums x 222 Minimums x 0 Minimums x 5 www Minimums x 2 2 2 2 2 159 325

Solve sin(x)There are two solutions, A and B, with A

Solve sin) 0.28 on O 2m There are two solutions, A and B, with A B and A and B in radians to 4 decimal places.

Question 5 Find the equation of the slant asymptote of the rational function 2x 6x2 7x 10 f x 2x 5 The equation of the slant asymptote is y

If F x f g x where f 2 3 f 2 5 f 4 1 g 4 2 and g 4 9 find F 4 F 4

4 Knowing that 1 1 x x x 1 20 approximate with a tolerance level of 0 0001 the following definite integra 0 2 X 1 z dx

A fence must be built to enclose a rectangular area of 5000 ft2 Fencing material costs 1 per foot for the two sides facing north and south and 2 per foot for the other two sides Fin the cost of the least expensive fence The cost of the least expensive fence is

A solid is obtained by rotating the shaded region about the specified line about the x axis y V 4 3 2 1 y x 2 4 2TY y 6 x 6 8 i a Set up an integral using the method of cylindrical shells for the volume of the solid dy X b Evaluate the integral to find the volume of the solid 10

Determine the area enclosed by the functions g x 4x 3 and f x 6 x 2x 2 between x 4 and x 2

Use the washer method to find the volume of revolution generated by revolving the region bounded by the graphs of y = 2x,y = 8, and the y-axis about the x-axis.(Express numbers in exact form. Use symbolic notation and fractions where needed.)

6 r x sin x cos x 2 0 7

sin(r) cos(r) F 2

Verify the Divergence Theorem for the vector field and region F 6x 92 8y and the region x y2 1 0 z 9 F ds R div F dV

v 2 dr Then find the length of the indicated portion of the curve r 1 4cos 11 4 Find the arc length parameter along the given curve from the point where t 0 by evaluating the integral s t j 5t k where Ostsz

Let F e5y sin x cos7 x Calculate curl F curl F

Let F (e5y, sin x, cos7 x Calculate curl(F). curl(F)

Please Simplify as instructed. sec(t) cos(t) = (A) _________________ ( using sec(t) = 1 / cos(t) )(B) _________________( adding using common denominator )(C) __________________( Using sin(t) + cos(t) = 1 )

Determine whether the following vector field is conservative on R F 2y 2x 2y Is F 2y 2x 2y conservative on R O Yes O No

Use any method to evaluate the integral 30 dx S 225x 1 30 dx 225x 1 S 0

Use any method to evaluate the integral. 30 dx (225k 1) 2 30 dx (2258 1)

a At Use the form 12x dx f12 dx faxd a dx 1 In a C to solve the following indefinite integral

Area of a triangle Given a triangle that has vertices x y x y2 and x3 y3 the area enclosed by the 1 1 1 triangle is given by A IDI where D Determine the area of the triangle X1 X2 X3 V V2 V3 with vertices given by 1 6 6 2 and 6 8 where units of length are measured in inches 8 pts

Find the integral e7a 1 c 7x 4 e7x 4 dx 7x 7x 4 e7 4 c 7e7x 4 c O e 7x c

What is the total change of f(x) if f'(x)=sin(x) from [0, ?]?

Question 2 (3 points) What is the total change of f(x) if f(x) sin(x) from O, rr J Question 3 (3 points) If you are given the rate of change of a function f, and an interval, what is th change over the interval

y sec x y tan x y cot x y csc x y cos x y sin x 4 HU B 2 1 1 3 2 1 1 2 A 3 4 5 6

Suppose that f is the function given in the graph below and that f is a piecewise function whose parts are either portions of lines or portions of circles as pictured 6 3 4 3 A 6 2 A 1 1 6 5 4 3 2 1 2 3 5 6 Let A be the function defined by A 2x f t dt b Compute the followings 5 q a What does the Second FTC tell us about the relationship between A

5 24 Find the general indefinite integral 5 f 3x 4x 1 dx 7 f x cos x dx 9 f x 7x 5 dx 11 5 3x x dx 13 u 4 2u 1 du 6 f 5 2 x dx x 10 x dx 12 u 2u u du 14 1 31 2 dt 8 x dx I

K Find the a amplitude b period c phase shift if any d vertical translation if any and e range of the following function Then graph the function over at least one period y cos 1 2 T a The amplitude is

For the vectors a = (1, 4) and b = (2, 3), find orth ab.orth ab =

(1, 4) and b (2, 3), find orthab. For the vectors a orthab

Use the Trapezoidal Rule to approximate fe da for n 4 Round the result to four decim places Please note the equation is e raised to 3 x O 3 2122 3 3423 O None of the answer choices

Find the most general antiderivative of the function Check your answer by differentiation Use C for the constant of the antiderivative f x x 3 5x 3 F x 5 3 6x 9x cx

Now if u 8 x4 then This evaluates as x s dx us du 1 us du 8 x4 6 dx

4 k 1 Graph the function f x over the given interval Partition the interval into four subintervals of equal length Then add to your sketch the rectangles associated with the Riemann sum 1 ck Axx given that c is the a left endpoint b right endpoint and c midpoint of the kth subinterval Make a separate sketch for each set of rectangles f x sinx OA E

Consider a function h that is continuous over 10 10 and let 10 foto h 0 Use the above information to evaluate each of the following definite integrals 17 forth he 7 h x 7 dx 10 h x 7 dx Assuming h is odd Assuming h is even 10 10 h x dx h x dx 5 10 h x dx

Determine which values of p make the following integrals converge Give your answer in each case by entering an appropriate inequality sign e g or in the box immediately after p and a numerical value in the second box to define a range of p values for which the integral converges If the integral never converges enter none for the numerical value A

11:41"He should do his homeworka little better, because he'llfind out that we've had a 10decline of 5 million in the farmpopulation under thesegovernment programs. He'llalso find that the Democraticadministration has sought toget from Congress [an]extension of the farmprogram to include thatthree-fourths that is nowfree."A. Reagan bases his

Evaluate the integral 6e6t S 6 e6t 6e6t dt C

Evaluate the integral. dt 6e

Find suike dx Find sin e dx Hinto satu e G N 2 pts

(2 pts.)

Solve the following equation by the quadratic formula below.36x + 7x-6=0Give the answers in ascending order. Round your answers to three significant digits.x1 -0.51688x2= 0.32244

Tanisha Vizquez owns a wallpaper hanging busine the number of square feet of wallpaper she hangs

Tanisha Vizquez owns a wallpaper hanging busin

Find the area of the finite part of the paraboloid y = x + z cut off by the plane y = 81. [Hint: Project the surface onto the xz-plane.]

Points 0 2 of 1 A heavy duty shock absorber is compressed 4 cm from its equilibrium position by a mass of 700 kg How much work is required to compress the shock absorber 8 om from its equilibrium position A mass of 700 kg exerts a force in newtons of 700g where g 9 8 m s 0 08 S 171 500y dy 0 Type exact answers of 24 points Set up the integral that

3 x is y x C 15 Find the area shown below Hint The indefinite integral of y 2 87 3 x 1 PL y 4 x 4

Differentiate the function h x h x n x x 2

Differentiate the function. h(x) In x x2 2)

Find the horizontal asymptotes of f f t 711 3 6412 8 1 6 Let y f t and give your answer as a comma separated list of equations in terms of the given variables Express numbers in exact form Use symbolic notation and fractions where needed horizontal asymptote s PRETREATE

5 Consider the boundary value problem as in Q3 b y 4x 2x2y 2r2ln x ze 1 2 y 1 1 2 y 2 ln 2 1 a Write the system of first order IVP regarding y and y2 that we need to compute if we would like to apply the linear shooting method to solve the above BVP b Computational exercise Write a script to solve this problem Let h 0 1 using Euler s method to

Find the area of the region described The region in the first quadrant bounded by y 5 and y 5 sin x on the interval The area of the region is Type an exact answer using x as needed T

Evaluate the integral 16 22 dx by interpreting it in terms of area In other words draw a picture of the region the integral represents and find the area using geometry 16x dx 16 Holn Written Example

Integrate by parts Assome x O x In xdx 11 x

3 Bacteria population A bacteria culture initially contains 100 cells and grows at a rate proportional to its size After an hour the population has increased to 420 a Find an expression for the number of bacteria after t hours b Find the number of bacteria after 3 hours c Find the rate of growth after 3 hours d When will the population reach 10 000

37 ng Question 6 9465 Find an equation for the graph sketched bel 8 7 4 341 5 4 3 2 1 7 2 2 3 4 5 3 5 6 7 8 X 2 3 4 5

Find the volume of the solid enclosed by the paraboloid z = 2 + x + (y - 2)2 and the planes z = 1, x = -3, x = 3, y = 0, and y = 2.