Vector Calculus Questions and Answers

3 Find the formula for the Maclaurin series for f x 1 4 without taking derivatives What is the radius of convergence Now integrate term by term and get the series representation for a particular special function Extra Credit What function is this 1 1 x

Show the derivation of and give the Taylor series for cos x around x 2 Find the formula for the Maclaurin series for

Find the sum of the series 8 n 0 1 n 320 2n

The rectangles in the graph below illustrate a right endpoint Riemann sum for f x interval 3 7 42 7 6 5 4 3 2 1 2 3 4 56 7 The value of this right endpoint Riemann sum is This Riemann sum is Select an answer the area beneath the cup on 2x on the

The rectangles in the graph below illustrate a left endpoint Riemann sum for f x on the interval 4 8 9 8 S96 O 7 5 4 3 2 1 3 4 5 6 4 5 6 7 8 10 The value of this left endpoint Riemann sum is This Riemann sum is Select an answer the area beneath the x 3 3x

Write the equation of the circle in standard form x y2 6x 8y 9 0 Identify its center and radius center x y radius

Use the following for questions 18 20 The sequence is arithmetic Find the common difference the tenth term a recursive rule an an explicit rule for the nth term for the given sequence 8 2 4 10

Let f x Find 1 1 3x 2 4x 3 1 Domain in interval notation 0 1 U 1 3 U 3 00 2 Y intercept s at the point s 0 3 3 X intercept s at the point s 2 0 4 X value of any holes 1 5 Vertical asymptotes at x 3 6 Horizontal asymptote at y 1 7 Oblique asymptote at y DNE x Invalid use of a comma syntax incomplete 4

3 Find all real zeros of the function f x 6x T Question Help Message instructor x H 1 9x 4 Given z 1 is a factor of f x Enter the real zeros separated by commas Give as exact values hs

Find the parametric equation for the cyloid given by a wheel radius 10 meters if it is moving foward at 5 meters per second D Find the area inside r 1 sin 0

6 10Pts Consider the force field F 3x y 2z x 2x a Demonstrate that F is conservative b Determine the scalar potential function p x y z of F

Suppose that an accelerating car goes from 0 mph to 54 5 mph in five seconds Its velocity is given in the following table converted from miles per hour to feet per second so that all time measurements are in seconds Note 1mph is 22 15 feet per sec 22 15 ft s Find the average acceleration of the car including units t 0 v t 1 2 0 27 272 47 272 3 4 5 61 816 72 728 80 Average acceleration over the first second Average acceleration over the second second

Below is a graph of f x the first derivative of f x Use the graph to answer the questions below the 0 5 1 1 5 2 0 5 2 f x a Where is f x increasing Where is f x decreasing b Where are the critical points of f x Classify each critical point as a local maximum local minimum or neither Where is f x concave down Where are the inflection points of f r 16

A food lab conducted a blind taste test to determine if organic fruits taste better than non organic fruits Each participant ate one type of fruit Apples Peaches Blueberries Organic Non organic 4 5 4 6 6 5 What is the probability that a randomly selected participant tasted peaches given that the participant preferred non organic Simplify any fractions

The integral represents the volume of a solid Describe the solid 63 3y 3 y2 dy 2x The solid is obtained by rotating the region 0 SXS 0 sys about the Selecte using cyl

The graph of an exponential function is shown What is the asymptote of the graph 9 8 7 6 5 y 4 Ox 2 Oy 3 7375pngor 3456789 X

The following density function describes a random variable X A Find the probability that X is greater than 1 Probability B Find the probability that X is less than 5 Probability C Find the probability that X is equal to 1 5 Probability f z 1 2 if 0 x 2

a Determine the quadrant the point P lies in b cost Identity tant c State an identity needed to find the exact value of tan t Find tan t Exact value of tant Identity sin 2t 3 5 d State an identity needed to find the exact value of sin 2t Find sin 2t Exact value of sin 2t sint Identity cos 2t e State an identity needed to find the exact value of cos 2t Find cos 2t Exact value of cos 2t

6 Start from the given graph and sketch the graph of the function f State the domain range and asymptote if any Your graph must include at least one period and the y intercept 277 0 1 f x 3 cos x 7 TT 2T X Domain Range Asymptote

Consider the following function Step 2 of 4 Identify the shape of the more basic function found in step 1 Answer 2 Points Ay n x x 3 x

The probability density function p d f of a continuous random variable X is defined to be k for 0 x 2 5 0 otherwise for some constant k For these problems please ensure your answers are accurate to within 3 decimals f x Part a Find the value of k that makes the above function a proper p d f Part b Hence find P 0 5 X 1

J each row containing two more seats than the row preceding it How many seats 8 Pat designs a patio with a trapezoid shaped deck consisting of 16 rows of congruent slate tiles The numbers of tiles in the rows form an arithmetic sequence The first row contains 15 tiles and the last row contains 30 tiles How many tiles are used in the deck Submission status

4 The sequence is arithmetic Find a the common difference b the tenth term c a recursive rule for the nth term and d an explicit rule for the nth term 4 1 6 11

The density function of X is given by If the expectation of X is E X 0 5 find a and b a b f x Ja bx if 0 x 1 otherwise

38 The horizontal and vertical components of a vector are 10 meters and 24 meters respectively Find the magnitudes of the resultant vector and the angle that it forms with the horizontal What is the angle that it forms with the horizontal A 43 B 47 C 67 D 23

43 Given that A 2 31 and B 5T 7j find B A 31 B 11 C 1 D 29

34 Two vectors 5 and 8 centimeters in length form an angle of 65 with each other What is the magnitude of the resultant vector A 9 4 cm B 7 6 cm C 11 1 cm D 10 cm

40 Simplify the following 123 A 1 B 1 C i D i

39 Find x such that i 3 j and 2 i xj are perpendicular to each other 2 3 A B C 32 312 3 2 D 3

30 Find the dot product of the given vectors A 3i 2j B 4i 5j A 2 B 2 C 12 D 7

I 8 A vector 20 inches in length forms angles of 58 and 22 with two component vectors Find the magnitudes of the two vectors The magnitude of the other vector A 49 5 inches B 7 6 inches C 52 6 inches D 10 6 inches

2 Use logarithms to evaluate 6 13 15 03 What is the answer A 0 4379 B 15 63 C 100 37 D 92 13

20 Solve for the unknown part of the triangle if it exists If a 50 b 42 and C 124 then what does c A 44 B 88 C 81 D 28

24 What is one of the three pairs of polar coordinates for the point 5 60 with 360 0 360 A 5 60 B 5 120 C 5 300 D 5 240

6 Given that A 31 4j and B 5i 12j find A B A 8i 16 j B 8i 8j C 2 i 8j D 2 i 16 j I

25 The horizontal and vertical components of a vector are 8 inches and 2 inches respectively What is the magnitude of the resultant vector A 4 1 inches B 12 inches C 8 2 inches D 20 inches

1 21 logx 9 3 X A 3 3 B 81 C 27 D 729

22 Perform the indicated operation leave the answers in the rectangular form a bi 3 i 5 4i A 15 7i B 11 17i C 11 7i D 12 5i

17 Transform from polar coordinates to rectangular coordinates 2 45 A 1 1 B 2 2 C 1 1 D 1 1

I 12 Change 2 i 2 to polar form A 2 cos 45 i sin 45 B 2 cos 45 i sin 45 C 2 cos 45 i sin 45 D 2 cos 45 i sin 45

Vincent was a spectator at his town s air guitar competition Contestants were allowed to play either the acoustic or electric air guitar but not both Vincent recorded which type of guitar each contestant played He also counted the number of contestants wearing different kinds of pants as there were some interesting stylistic choices Leather Velvet Acoustic guitar 1 3 Electric guitar 2 6 What is the probability that a randomly selected contestant was dressed in velvet given that the contestant played an electric guitar Simplify any fractions

You roll a 6 sided die What is P 3 or divisor of 28 5 4 3 4 2 3

Find the Banzhaf power distribution of the weighted voting system 34 19 15 12 6 Give each player s power as a fraction or decimal value P P P3 P on

In a proof by mathematical induction that 8 16 24 8n 4n n 1 for all positive integers n what is the inductive hypothesis elect one 8 b C Dd De Assume 8 16 24 Assume 8 4 1 1 Assume k 1 Assumen 1 Assume 8 16 24 8n 4n n 1 for all positive integers n 8k 4k k 1 for some positive integer k

Which angles are supplementary angles W U V T R ZVUW and ZQRP ZQRU and ZORP S P ZQRP and TUR TUW

Which angles are consecutive interior angles M P N Q S ZRQN and PQS O R ZRQN and ZMNQ

Determine the mean median mode and midrange of the set of data 54 47 34 45 55 42 58 The mean is 47 9 Round to the nearest tenth as needed The median is 47 Select the correct choice below and fill in any answer boxes within your choice OA The mode is Use a comma to separate answers as needed OB There is no mode

Which angles are corresponding angles C F G D B ZHGD and ZCDB FGD and EGI E H ZHGI and FGI FDG

What values of u and v make AABC AGFE 7v U 11 B J V 245 A C F 8u v 43 U 14 E

Look at this diagram 1 M K L G H F