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mathematics
february 2020
19

Mathematics Questions and Answers February 19, 2020

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Find fxx(x,y), fxy(x,y), fyx(x,y), and fyy(x,y) for the function f. f(x,y) 9xe7xy fu (X,Y)

what does the following integral equal 41 59 (6f(x) 39(x) h(x)dx 41 f(x)dx 10 59 and 41 59 9(x)dx 13 and 41 h(x)dx 20 59

Flu). S, 34 de 3 a) (amputz Fix) b) Show F is c) Compte (F.) (0)

Suppose F(t) has the derivative f(t) shown below, and F(0) 2. Find values for F(3) and F(T 1 2 3 F(3) F(7) 2 1 2 3 5

In general, research shows that social media is more persuasive when it O a. duplicates the advantages of oneway communication. Ob. matches the format of print journalism. Oc. duplicates the advantages of facetoface communication. Od. matches the format of TV journalism.

T x sin(xy)dxdy Show Transcribed Text Evaluate the following integrals

Find the centroid of the region bounded by the graphs of the functions y 3 sin(x), y 1 X 6 and x 2 , and touching the origin. The centroid is at (x, y) where beginarray barx bary endarrayI

e 8 dx with n 4 a. Find the trapezoid rule approximations to the integral using n and 2n subintervals. b. Find the Simpsons rule approximation to the integral using 2 n subintervals. c. Compute the absolute errors in the trapezoid rule and Simpsons rule with 2n

Use a substitution of the form u ax b to evaluate the indefinite integral below. 4x3 dx S4x3 4x3 dx

Evaluate the following line integral. Sxy ds C is the portion of the unit circle r(s) (cos s, sin s), for 0ss The value of the line integral is (Simplify your answer.)

A table of values of an increasing function f is shown. Use the table to find lower and upper estimates for lower estimate upper estimate X 10 14 18 22 26 30 f(x) 15 8 2 2 5 7 30 f(x)

Find the exact length of the curve. y In(cos(x)), 0 X 26

Answer the following questions about F(x)3x180. (A) Calculate the change in F(x) from x 8 to x 16. (B) Graph F(x) and use geometric formulas to calculate the area between the graph of F(x) and the xaxis from x 8 to x 16. (C) Verify that your answers from (A) and (B) are equal, as guaranteed by the fundamental theorem of calculus. (A) Calculate

Let 2 g(x) (2) dz13, f(x) dz15, 9(z) dz11, 9(2) dr 13, f S Use these values to evaluate the given definite integrals. a. a. S (f(x) 9(x)) dx 6. S (f(x) 9(x)) dx b. C. (3f(x) 29(x)) dr 3 Find the value a such that (af(x) g(x)) dx 0. a

Use a Riemann sum with 4 rectangles of equal width to approximate the area between y x2 and the xaxis on the interval (1,1). Use the righthand endpoint of each subinterval. O 3.5 units O9 units O 8.0625 units O 4.5 units

9 f(x) dx 33 and S 9 g(x) dx 18, find 2f(x) 3g(x) dx.

6) Determine f(x) given that f(x) 60x 60x, f( 1) 1 and f(1) 6.

Oy2 D Question 18 A third degree polynomial f(x) H What would be one of

Find the consumers surplus for the following demand function at the given point. D(x) (x 3) x 3 4.33 7.28 3.25 4.50

Ef is continuous and t f(t) dt 16 f(t) dt 14, find the following integral.

The length of the curve (x e cos(t) y et sin(t) for 0 t 5 is

A volume is described as follows 1. the base is the region bounded by y x14x5 and y x 24x 163 2. every cross section perpendicular to the xaxis is a semicircle. Find the volume of this object. volume

Suppose (s)3253 2 1 25 and f(1) What is the constant term C in f(s) equal to Enter a number only. S S 2

Find the surface area of revolution about the xaxis of y5x6 over the interval 0

(1 point) Solve the separable differential equation Subject to the initial condition y(0) 9. y 11x 6yx1 dy 0. dx

Evaluate the ine integral F.ds where F(x, y) (xy, x) and c is the part of the circle x y 1 that starts at (1,0) and ends at (0, oriented counterclockwise.

36. Suppose that f(x, y) 15,000xexy is the population den sity of a species of small animals. Estimate the population in the region bounded by y x, y 0 and x 1.

For the following exercises, graph the equations and shade the area of the region between the curves. Determine its area by integrating over the xaxis. 7. y x and y x 18x

Find the arclength of y 3x32 on 1 x 3

(1 point) Solve the separable differential equation Subject to the initial condition y(0) 9. y 11x 6yx1 dy 0. dx

Evaluate the integral. 5 3 (2x6) 6)dx 5 (2x fildy

Consider the parametric equations below. x 1 et, y t2, 3 st 3 Set up an integral that represents the length of the curve. 3 LO ) dt Use your calculator to find the length correct to four decimal place

9. Find the integral a. Scot(x)dx 1 x(Ln2x) de

Summarize the pertinent information obtained by apply Find the domain of f(x). Select the correct choice below

Use the graph of f(x) shown below to find the following integrals. 2 2 a. S L 2 f(x)dx b. If the red shaded area in the graph has area A, estimate Your answer may be written in terms of A f(x)dx

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. 391. 32 dx 1x2 dx 12 2.VIX2 393. 394. 12 dx 3 1 x2 13 dx 1 x2 dx 395.

The length of the curve defined by the equation xy f(x) 16, from x a to x b is given by L f f(x)dx where

1 and Let the region R be the area enclosed by the function f(x) x g(x)2x1. Find the volume of the solid generated when the region R is revolved about the line y4. You may use a calculator and round to the nearest thousandth. 2 15 1 0.5 05 1 15 2 10 6

gion in the first quadrant bounded by y 2x2, 2xy4, and the yaxis is rotate about the lin The volume of the resulting solid is Preview My Answers Sub

19g 18 A graph of f(x) 81(x9)2 is shown. Using the geometry of the graph, evaluate the definite integrals. 17 16 15 12 11 18 9 8 6 1 21 1 23 45678 9 18 11 12 13 14 15 16 17 18 19 Graph of y f(x) a. f(x) dx 18 b. f(x) dx 18 C. f(x) dx 18 d. 4f(x) dx help (numbers)

13. Sketch the region between y 5x and y 3x. Find the area of the region.

Evaluate the integral using the following values. L x dx 8 8 x dx x960, xdx24, dx4 L(21 5x x) dx

Convert the numeral to a numeral in base ten. 1111112 The numeral in base 10 is

Test the series for convergence or divergence. n 2 In(n) 2 Extra Credit 3 points if you verify convergencedivergence using two different methods.

3. Let h0 be a fixed real number. Consider the elliptic paraboloid given by the equation y2 9 h h 0

6 5 3 6 5 4 3 1 0 1 2 3 4 5 6 2 3 4 5 Use the graph to estimate the slope of the line tangent to the given graph at (3, 2)). O O O 32 23 22

Verify that the Fundamental Theorem for line integrals can be used to evaluate the following line integral, and then evaluate the line integral using this theorem. Sve cos y) dr, where C is the line from (0,0) to (In 3,2x) Select the correct choice below and fill in the answer box to complete your choice as needed. A. The Fundamental Theorem for

(1 point) The instantaneous rate of change of the value of a certain investment (P) is proportional to its value. That is to say TP. If r 6 and P(0) 3500 BW

Find the area. 22 ft. B C 138 38.09 ft. 109 A D 37 ft. square feet Round to the nearest hundred

Convert to the indicated metric u Convert 50 in to square centime in2

Find the mean of 12, 15, 19, 23 28, Round your answer to the nearest he tenth.

33. Change each number to algebraic form 3(cos 90 i sin 90) A. 3i B. 331 C. 13i D.3i

INT y x 32 3 x1 x4 16. Find the area shown below. (Hint The indefinite integral of y 3r is yxC.) (87) 20 10 y3x x2

f 2. (5 pts) (6.3 148) Sketch the region bounded by x y and y x. Determine the volume when the region is rotated about the line y 2. Sinhly 2 24 2 217

Here is the graph of 5 cos 3x interval , . A 3 over the 4 B C Give the coordinates of point A, the yintercept Give the coordinates of point B, the first xintercept along the positive x axis Give the coordinates of point C, the first peak of the graph with positive x value

7 f(x)dx S Suppose f(x)dx6, f(x)dx8, and g(x)dx4. Evaluate the following integrals. 2 2 2 2 Sg(x)dx 7 g(x)dx (Simplify your answer.) 7 2 4g(x)dx (Simplify your answer.) 7 g(x)f(x)dx (g(x) f(x)dx S19(x) 2 (Simplify your answer.) 7 (19)10x 179(x) f(x)dx 2 (Simplify your answer.)

Use the GaussJordan Elimination method to solve Change the system of equations to a matrix Get a 1 in row 1, column 1 Get a 0 in row 2, column 1 Get a 1 in row 2, column 2 2x 3y 44 6 Show your step

For the following exercises, find the volume generated when the region between the curves is rotated around the given axis. 140. y 3 x, y 0, x 0, and x 2 rotated around the yaxis. 141. yx3, x 0, and y 8 rotated around the yaxis. 142. y x, yx, rotated around the yaxis. 143. y x, y 0, and x 1 rotated around the line x 2. 144. y4, x 1, x

Find the area of the region between y x 12 and y 15 for 0 x 1. area

5. (x1)(2x 1)dx 2 1x2x2x2 24 (2x 1x1) dx 3 2 2 1x 2 29 b 31 2 (2)) Z

2. (10 points) Construct a polynomial q P2(R) such that 1 dx p(x)q(x) dr p(x) cos x dx Tx

2) Find the area of the region bounded by the graph of f and the xaxis on the given interval. f(x)9x2 on 2,3

29. Solve for the unknown part of the triangle, if it exists. If b 125, c 178, and C113, then what does a A. 150 B. 122 C.61

1.0 (a) If F(x)e, find F(x). (b) Find 2xex dr in two ways. i. Using the Fundamental Theorem of Calculus. i. Numerically using a right Riemann sum with n 5. You can use the graph of 2xe 0.8 0.6 0.4 0.2 0.2 0.4 0.6 0.8 1.0. below.

The curve C is the graph of y f t t23t8 dt. Determine xinterval(s) on which the curve is concave upward. C is concave upward on

Shown below is the graph of the flow rate f(t) of a pipe, in gallons per minute. Here, t the number of minutes. Using this information, find S f(t) dt, which gives the total number of gallons of water that flowed through this pipe in 7 minutes. Velocity (GallonsMinute) 1 2 3 4 5 6 Time (In Minutes) SF(1) dt dt 24 gallons f(t) dt25 f(t) dt 25

Find the following indefinite integral (ocscxcotx 25 dx X 6cscx C 25 3x3 25 O6cotx C 3 3x3 6cotx 25 3x3 C

For the regions bounded by y 6cosx and 27 7. Part A Find the limits of integration. (10 points) Part B Write an expression representing the total area. (10 point Part C Solve for the total area. (10 points)

Complete the table to show the approximate area of the region using n number of rectangles of equal width. f(x) 1 X2 3 y 6 4 X 2 4 6

Determine if da converges or diverges. 3 converges to 19 converges to 0 O None of the answer choices O converges to 118

6 Given so f(x)dx 5 f(x)dx 15, find the following 6 5 f(x)8dx O7 103 23

2) Find the surface area and volume if y is rotated around the x axis from 1 to 4. 3) Find the following integrals (a)

In 23, without solving, state whether the triangle has one, two, or no solutions. Justify your answer. 4 2) AFGH, GH 22, FH 39, FG19 3) ADEF, EF 16, DF 5, m2B 42

Use a series to estimate the following integrals value with an error of magnitude less than 108. 0.3 0 e dx 0.3 dx ...

Use the graph of H shown to the right to find the limit. If necessary, state that the limit does not exist. lim H(x) 0x Select the correct choice below and, if necessary, fill in the answer box to complete your choice. OA. lim H(x) X0 (Type an integer or a simplified fraction.) OB. The limit does not exist. 8 8 64 2 OL 24 6

5 points Sketch the region of integration R and evaluate the integral over R in polar coordinates. If you do not transform to polar coordinates to evaluate th will receive no credit. Show all steps. R (r, 0) 2r 4,00

Use a graphing calculator to graph the region R bounded by the graphs of the indicated equations. Use approximation echniques to find intersection points correct to two decimal places. Describe R in set notation with double inequalities, and evaluate the indicated integral correct to two decimal places. yexy53x4y d dA R D. Ay Graph the region R

The region R of the xyplane is bounded by the graphs of y x1 and y 12, and the lines 1 and x 4. A solid S is generated by revolving S about the line x 4. The volume of S is given by the integral dr

Ophion A. find the integral. F CSX3) dx 25x9xC B. 3x9xC C. x15x9xC D. 2515x79xC

Use Stokes Theorem to evaluate Jc F(x, y, z) yzi 3xzj exYk, F.dr where C is oriented counterclockwise as viewed C is the circle x y 25, z 4.

Convert to liters and add. 79 L863 mL 30.6 L 79 L863 mL 30.6 L L (Simplify your answer. Type a whole number or a decimal.)

1. A cardboard box with a square base is to have a volume of 8000cm. a) Find the dimensions of the box that will minimize the amount of cardboard to be used. b) The cardboard for box costs 0.1 per cm, but the cardboard for the bottom is thicker, so it costs three times as much. Find the dimensions that will minimize the cost of the cardboard.

Without graphing, describe the shape of the graph of the function. Find the second coordinates of the points with first coordinates 0 and 1. f(x) 1.3x The graph exponentially decays. Find the second coordinates of the given first coordinates. x 0 f(x)1.3x f(0) f(1) (Round to one decimal place as needed.)

The temperature at a point (x,y,z) of a solid E bounded by the coordinate planes and the plane xyz1 is T(x, y, z) (9xy 8z 20) degrees Celsius. Find the average temperature over the solid. (Answer to 3 decimal places). Average Value of a function using 3 variables N 1 y 0 x

10 8 6 4 2 1 2 3 4 6 8 10 12 The net signed area in the above graph can be written as where c m d and b (mxb)e C

ab numbers, equations, computations, or algorithms it is about understanding. William Paul Thurston, American mathematici 5 5. (5 pts) (6.6 275) Sketch the region bounded by y x and y 5. Compute the center of mass of region. 50 4 50 2 2 4 2 4 6 642 K

1 2 3 P LK 111 LLL KKK K 246 Evaluate the following integrals, for a path y that travels from Ti to Ti in the right halfplane, and also for a path y from i to i in the left halfplane. (a) zdz (b) Syezdz (c) co cos z dz (d) J, sinh zdz Solution The integrals are all independent of path and can be evaluated by finding a primitive. (a) (b) z dz 2 (i)

(6) a) Sketch the function f(x) 2 from x 2 to 2 2. b) Approximate the signed area of f(x) on 2, 2 by using right hand sums with n 4. Is this an overestimate or and underestimate of the actual signed area c) Approximate the signed are of f(x) on 2, 2 by using left hand sums with n timate of the actual signed area 4. Is this an overestimate or

Evaluate the definite integral. 4 4 (x3 x8 tan(x)) dx

Find the most general antiderivative of the function f(z) z7(12)12, where 1 1. Use an uppercase C for any arbitrary constants. F(2) 4.

(a) Estimate the area under the graph of f(x) 3 2x from x 1 to x 2 using three rectangles and right endpoints. R3 Then improve your estimate by using six rectangles. R6

Calculate the integral of zexy over the surface of the box with 0 x 20 y 3,0 z6. (Express numbers in exact form. Use symbolic notation and fractions where needed.)

2) Use cylindrical Coordinates to evaluate fff xyzdv E Where E is the solid in the first octant that lies under the Paraboloid 242242. y

me of the solid A) x 2y, x 0, y 9 about the y axis by rotating the region bo by 829 wing (disk, washer, or shell) x215

Solve each inequality. login (2x1)log12 (5x16) x 5 Ono solution Question 5

4. Partition the given interval into the indicated number of subintervals. 1, 4 6

Tumepage Zoom Meetings Question 3 of 10 (1 point) Attempt 1 of Unlimited Solve the right triangle for the unknown sides and A29.1, b19.1 A b C a B

A function y f (x) passes through P(3, 15) and Q (3, 30). Let the function g(x) f (3x). The transformation that maps the graph of f (x) to the graph of g (x) maps point P to P and point Q to Q where P (a, b) and Q (c, d) are given. Write down the values of a, b, c, d Z in the order given here. 184x2.SS

g(t) 8 7 6 5 4 3 9 8 765432 2 3 234 5 7 1 2 3 4 56789 t Over which of the given intervals does g have a positive average rate of change

Find the area of the shaded region below. Area YA y5r 12 yx (4, 4) x

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