Calculus Questions

If 3x 5 f x x 3x 4 for x 0 find lim f x x 3

Find the limit If an answer does not exist enter DNE t 36 2t 13t 6 lim t 6 Need Help Read It Watch It

An oil tanker breaks apart and starts leaking As time goes on the rate at which the oil is leaking out will diminish Suppo that t hours after the tanker breaks apart the oil is leaking out at a rate of R t million gallons per minute where R t 0 8 1 1 Then the amount of oil that will leak out in the first 9 hours after the shipwreck in millions of gallons is

Values of f x y are shown in the table below x 32 3 32 3 6 y 4 8 y 4 5 10 13 10 12 16 V 5 12 14 7 Let R be the rectangle 3 3 6 4 y 5 Find the values of Riemann sums which are reasonable over and under estimates for f f x y dA with Ar 0 3 and Ay 0 over estimate 5 85 under estimate 6

1 22 Differentiate 1 f x 3 sin x 2 cos x 3 y x cotx 5 h 0 0 sin 0 7 y sec 0 tan 0

Evaluate the integral Answer 484x 5y Answers A Answer S e42 5y dA R 0 6 x 0 8 5y d

Find f f x y dA where f x y x and R 4 8 3 6 X SSR f x y dA

1 f x sin y y cos x dxdy 24 1 2 24 2 3 16 2 3 1 2 12 3 16 8

Find ff f x y dA where f x y 2x 1 and R 2 8 4 2 SR f x y dA 124

Evaluate the double integral of the function over the rectangle Y 1dA R 0 28 x 2 1

Evaluate the integral 1 e42 59 dA

Using Riemann sums with four subdivisions in each direction find upper and lower bounds for the volume under the graph of f x y 1 2zy above the rectangle R with 0 z 1 0 y 3 upper bound lower bound

Values of f x y are shown in the table below x 3x 3 3r 3 6 13 16 y 4 8 10 y 4 5 10 12 y 5 12 14 7 Let R be the rectangle 3 3 6 4 y 5 Find the values of Riemann sums which are reasonable over and under estimates for f f x y dA with Ar 0 3 and Ay 0 over estimate 1 under estimate

Suppose f z y zy and R is the rectangle with vertices 0 0 6 0 6 4 0 4 In each part estimate effs f x y dA using Riemann sums For underestimates or overestimates consistently use either the lower left hand corner or the upper right hand corner of each rectangle in a subdivision as appropriate a Without subdividing R Underestimate Overestimate 1 A b By partitioning R into four equal sized rectangles Underestimate Overestimate

1 point Find the average value of f x y x y over the rectangle R with vertices 3 0 3 7 2 7 2 0 Answer

Calculate the double integral f SR 2x 10y 20 dA where R is the region 0 x 5 0 y 1

Values of f x y are given in the table below Let R be the rectangle 1551 6 25ys 3 2 Find a Riemann sum which is a reasonable estimate for ff z y da with Az 0 2 and Ay 0 4 Note that the values given in the table correspond to midpoints yz 1 1 1 3 1 5 2 2 3 0 5 26 7 5 3 0 7 5 1 4

Find ff f x y dA where f x y x and R 4 8 x 3 6 SSR f x y dA

Calculate a Riemann sum S3 3 on the square R 0 3 x 0 3 for the function g x y f x y 6 The contour plot of f x y is shown in Figure 4 Choose sample points and use the plot to find the values of f x y at these points Use the values of f x y to evaluate g x y accordingly S3 3 PP P P21 P31 0 1 3 FIGURE 4 Contour plot of f x y P32 2

Evaluate the following integral T 6a y da dy

The figure below shows contours of g x y on the region R with 9 x 15 and 4 y 10 Using Az Ay 2 find an overestimate and an underestimate for SR 9 x y dA Overestimate 4 Underestimate 1 10 5 11 15

Compute the Riemann sum S4 3 to estimate the double integral of f x y 6ry over R 1 3 x 1 2 5 Use the regular partition and upper right vertices of the subrectangles as sample points

Find the maximum value of f x y 2 y for z y 20 on the unit circle

roblem 2 Find all solutions to the equation cos x 3 in the interval 2 5T 4 and A and B only C and 5 D and 7 E only F only G only H I ST

Evaluate the iterated integral 2x y dydx

Find the minimum and maximum values of the function f x y x y subject to the given constraint a y 32 fmin

Evaluate the iterated integral f f 4x y dady 2048

lim f x lim f x lim f x x a L 0 M I

0 y 2 lim 1 1 z 0 x lim 848 y 1

8 Use the graph of y f x to answer the following questions show away to lis work sfy 5 4 3 2 506 a Determine the domain stumboro b Determine the x intercept s vollot svo S 21 11231 mot avio2 8

1 x 21 models the percentage of households with an interfaith marriage I x years after 1989 The formula N x 9 models the percentage of households in which 1 The formula 1 x 2 a person of faith is married to someone with no religion N x years after 1989 Answer parts a d below a In which years will more than 34 of households have an interfaith marriage After the year b In which years will more than 19 of households have a person of faith married to someone with no religion After the year c Based on your answers to parts a and b in which years will more than 34 of households have an interfaith marriage and more than 19 have a faith no religion marriage After the year d Based on your answers to parts a and b in which years will more than 34 of households have an interfaith marriage or more than 19 have a faith no religion marriage After the year S 16 2025

Use formulas to find the area of the figure 3 mi b 7 mi 2 OA 12 mi 2 OB 27 mi OC 24 mi OD 10 5 mi 18 mi 8 mi

If construction costs are 160 000 per kilometer find the cost of building the new road in the figure shown to the right The cost of building the new road is Round to the nearest dollar as needed new road 15 000 m 8 000 m

Let f x x x fx 0 S x if x 0 1 I f 0 exists A only I only II C I and II only All of them None of them Which of the following statements are true II lim f x exists III is continuous at x 0 x 0 lim x X 70 x lim x x X 70 o CON SINGE I is when X 0

This problem asks you to evaluate U X 2 using integration by parts Fill in the table below with the appropriate terms for setting up the integration by parts du 1 dx sin 10x dx a sin 10 da Crain 1 V Now substitute all of this into the Integration by Parts formula giving fas x cos 10x 10 S Complete the last integration to get 10 6 6 dv cos 10x dx cos 10x 10 2 da

8 122 12r 2 f x dx 7 6 5 3 2 1 6 1 The graph of the differentiable function f is shown above The area in the first quadrant bounded by the graph of y f x the x axis and the y axis is equal to 9 Use integration by parts to help evaluate the following definite integral using information about the function f

Simplify the expression to a bi form 4 175 100 112

x 3 e 8 da using integration by parts Fill in the table below with the appropriate terms for setting up the integration by parts Complete the last integration to get u u 3 du 1 dx dv e 8x dx Now substitute all of this into the Integration by Parts formula giving x 3 e 8 dx x 3 6 V 0 10 dr 3 e 8x da C

21 8 xf x dx 1 7 7 2 7 8 3 4 6 4 10 5 he function f has a continuous second derivative The table above gives values of f and its first erivative at selected values of x Use integration by parts to help evaluate the following definite integral sing data from the table 2

sin 9x dx x cos 9x 9 sin 9x 81 C

f x x 2 e 9 dx 9x 19 9x 81 e C

Find the average rate of change on the interval specified for real numbers b where b 3 g x 3 2 4 on 3 6 Basic Funcs Trig X rum

This problem asks you to evaluate using integration by parts Fill in the table below with the appropriate terms for setting up the integration by parts u Complete the last integration to get 12 du x ln x dx dx V dv Now substitute all of this into the Integration by Parts formula giving x 1 x dx In o do dx 0 10 C da

e sin 5x dx lick on Part 1 to open the section Enter your answers for the steps in Part 1 then click on Sul answers Once your answers are correct you will be able to open Part 2 Once your answers a or Part 2 you wil be able to open Part 3 Step 1 The first integration by parts In this particular example one can let u be either e or sin 5x We will choose e Fill in t of the table below with the appropriate terms for setting up the integration by parts u e4 du V dx dv 4 sin 5r dr dx Now substitute all of this into the Integration by Parts formula giving

Consider the integral methods 4x 4x4 5 1 dx In the following we will evaluate the integral using two A First rewrite the integral by multiplying out the integrand 9 4 4x4 x 1 dx 4x 4x da Then evaluate the resulting integral term by term 4x4 4x4 xr5 1 dr 10 4x4 x5 1 dx 2x 5 4x4 x5 1 dx x 10dw 5 4x 5 B Next rewrite the integral using the substitution w x5 1 424 25 c Evaluate this integral and back substitute for w to find the value of the original integral 43 C How are your expressions from parts A and B different What is the difference between the two Ignore the constant of integration answer from B answer from A Are both of the answers correct Be sure you can explain why they are

s 4x4 5 1 dx In the following we will evaluate the integral using two Consider the integral methods A First rewrite the integral by multiplying out the integrand 40 4x x 1 dx Then evaluate the resulting integral term by term 4x 4 x x5 1 dx da B Next rewrite the integral using the substitution w x5 1 42 10dw 4x x5 1 dx Evaluate this integral and back substitute for w to find the value of the original integral 14 4x4 x5 1 dx C How are your expressions from parts A and B different What is the difference between the two Ignore the constant of integration answer from B answer from A

Using the method of u substitution 4 S 6x 8 dx 2 where U du a b f u S f f u du enter a function of x dx enter a function of x enter a number enter a number enter a function of u The value of the original integral is

Evaluate the definite integral 2e 0 4 p2x dx

Suppose that A So f 2t dt B 0 125 C 0 625 S05 0 5 f t dt 3 Calculate each of the following f 1 8t dt f 5 8t dt

Evaluate the integral using an appropriate substitution 9e9e T dx c C