Vector Calculus Questions and Answers

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Determine whether the following statements are true and give an explanation or counterexample Assume x 0 and y 0 a In xy Inx Iny b In 0 1 c In x y Inx Iny d 2 2Inx 1 e The area under the curve y and above the x axis on the interval 1 e is 1 a Choose the correct answer below OA The statement is false because In 35 In 5 In 7 OB The statement is true For x 0 and y 0 In xy Inx In y by the Product Rule of Logarithms OC The statement is false because In 5 In 7 In 5 7 OD The statement is true For x 0 and y 0 In xy Inx In y by the Quotient Rule of Logarithms

Problem 1 Use the method of cylindrical shells to find the volume when the region bounded by y 1 x x 0 x 1 and the x axis and rotated around the y axis

Water has a density of 1 0 g mL Which of these objects will float in water Object I mass 50 0 g volume 53 1 mL Object II mass 71 7 g volume 51 4 mL Object III mass 100 0 g volume 50 0 mL OI only O I III II only O II III

Evaluate the indicated function for f x x2 1 and g x x 3 algebraically If possible use a graphing utility to verify yo answer f g t 5

Use a graphing utility to graph the three functions in the same viewing window f x x 3x 2 g x f x h x f 3x Describe the graph of g relative to the graph of f O g x is a reflection in the y axis O g x is a horizontal shrink O g x is a reflection in the x axis O g x is a horizontal stretch Describe the graph of h relative to the graph of f Oh x is a reflection in the x axis Oh x is a reflection in the y axis O h x is a horizontal stretch O h x is a horizontal shrink

For each problem below the magnitude of the horizontal and vertical components of vector V are given In each case find the magnitude of V 27 Vx 30 V 40 29 Vx 35 0 V 26 0 31 Vx 4 5 V 3 8 28 V8 Vyl 6 30 Vx 45 0 V 15 0 32 Vx 2 2 V 5 8 33 Velocity A bullet is fired into the air with an initial velocity of 1 200 feet per second at an angle of 45 from the horizontal Find the magnitude of the horizontal and vertical components of the velocity vector 34 Velocity A bullet is fired into the air with an initial velocity of 1 800 feet per second at an angle of 60 from the horizontal Find the horizontal and vertical components of the velocity 35 Velocity Use the results of Problem 33 to find the horizontal distance traveled by the bullet in 3 seconds Neglect the resistance of air on the bullet 36 Velocity Use the results of Problem 34 to find the horizontal distance traveled by the bullet in 2 seconds 37 Velocity An arrow is shot into the air so that its horizontal velocity is 35 0 feet per second and its vertical velocity is 15 0 feet per second Find the velocity of the arrow

For each problem below the magnitude of the horizontal and vertical components of vector V are given In each case find the magnitude of V 27 Vx 30 V 40 29 Vx 35 0 V 26 0 28 Vx 8 V 6 30 Vx 45 0 V 15 0

Each problem below refers to a vector V with magnitude V that forms an angle with the positive x axis In each case give the magnitude of the horizontal and vertical components of V 19 V 40 0 30 21 V 13 8 0 24 2 23 V 420 0 36 10 20 V 40 0 60 22 V 17 6 0 67 2 24 V 380 0 16 40

Draw vectors representing the following velocities 1 30 miles per hour due north 2 30 miles per hour due south 3 30 miles per hour due east 4 30 miles per hour due west 5 50 centimeters per second N 30 W 6 50 centimeters per second N 30 E

17 If the distance between A and T on the map in Figure 19 is 0 5 inch find each of the following a The horizontal distance between Amy and Travis b The difference in elevation between Amy and Travis c The angle of elevation from Travis to Amy 0 1000 SO 1546 1 in 1600 ft Bishop Peak 1200 1600 Contour interval FIGURE 17 000 40 ft

The point P 4 1 lies on the curve y x 3 a If Q is the point x x 3 use your calculator to find the slope of the secant line PQ correct to six decimal places for the following values of x 1 3 5 0 585786 ii 3 9 iii 3 99 iv 3 999 v 4 5 vi 4 1 vii 4 01 viii 4 001 b Using the results of part a guess the value of the slope of the tangent line to the curve at P 4 1 0 5 c Using the slope from part b find an equation of the tangent line to the curve at P 4 1 y d Sketch the curve two of the secant lines and the tangent line

ecologist wishes to find the height of a redwood tree that is on the other side of a creek as shown in Figure 14 From point A he finds the angle of elevation to the top of the tree is 10 7 He then walks 24 8 feet at a right angle from point A and finds angle B is 86 6 What is the height of the tree FIGURE 14 24 8 B

1 If tan 0 2 4 find cot 0

Consider the position function s t 7 sin 3t that describes a block bouncing vertically on a spring Complete the following table with the appropriate average velocities Then make a conjecture about the value of the instantaneous velocity at t 2 Time Interval Average Velocity T 2 Time Interval Complete the following table 2 T 2 2 T T 2 2 0 1 0 1 Average Velocity Round to five decimal places as needed T T 2 2 EIN EN 2 2 T The value of the instantaneous velocity at t Round to the nearest integer as needed 2 is 0 01 0 01 T B EN 2 2 T 2 2 0 001 0 001 T 2 2 KIN T T 2 2 0 0001 0 0001

Use the appropriate reciprocal identity to find the value of sin 0 given that csc 0 13 Rationalize denominators when applicable sin 0 Simplify your answer including any radicals Use integers or fractions for any numbers in the expression Rationalize all denominators

an 0 3 ot 0 Simplify your answer including any radicals Use integers or fractions for any numbers in the

Question A solid is formed by rotating the region bounded by the graphs of y x y 24 and x 13 in the first quadrant about the line x 5 Which of the following gives the volume of the solid Select the correct answer below 24 17 2 169 10 J J JU 24 169 10 2 10y 8 169 dy 2 10y 5 64 dy 24 10y 18 169 dy 24 10y 13 64 dy

0 1 pt 399 Details The graph shows the water level in a reservoir as a function of the number of months since the start of the year 100 90 80 70 60 50 40 30 20 10 1 10 1 2 3 4 5 6 7 8 9 10 The highest water level was reached in month the lowest water level was reached in month and

Question A solid is formed by revolving the region bounded on the left by y 2x on the right by y In x above by y 6 and below by x axis about the y axis Which of the following gives the volume of the solid Select the correct answer below O y 25e4y dy Ox 25e y dy OT 4e10 1 dy 4e10y dy 1 2

Find the a period b phase shift if any and c range of the following function y 3 sec X R 2 T a The period is Type an exact answer in terms of b Select the correct choice below and fill in the answer box within your choice OA 3 3 OC 0 00 OA The phase shift is OB The given function has no phase shift c Find the range of the function Choose the correct answer below units Type an exact answer in terms of t OB 3 U 3 OD 3 U 3

t the following radian measure to degree measure

Let x y be any two vectors in space Which of the following are properties of the cross product x x y Select all that apply x x y is orthogonal to the plane determined by x and y xxy y xx x xx y 0 and y x x y 0 Oxxy y x x x x y is the area of the paralellogram determined by x and y Oxxx yx y 0

The velocity of an airplane flying into a headwind is given by v t 30 25 12 mi hr for Osts3 hr Assume that s 0 0 a Determine the position function for 0 t 3 b How far does the airplane travel in the first 2 hr c How far has the airplane traveled at the instant its velocity reaches 500 mi hr

sin 2x Which of the following is equal to sin 3x 2x 3x 2x sin 2x 2x sin 3x 3x sin 2x 3x sin 3x 2 sin x 3 sin x assume0 Select all that apply

XI is moys Su 4 Z 2 2 6 M 223 1 75 a 24 CDS C Z 2 S E 2 get puls of 26 C x 5 5 x UNO NG S

A hemispherical bowl of radius 7 inches is filled to a depth of h inches where 0 sh 7 Find the volume of water in the bowl as a function of h Check the special cases h 0 and h 7 The volume of the water in terms of his in Type an exact answer using as needed What is the volume of water when h 0 V 0 in 3 Simplify your answer Type an exact answer using as needed Use integers or fractions for any numbers in the expression What is the volume of water when h 7 V 7 in Type an exact answer using as needed

The tangent line to a smooth curve r t f t i g t j h t k at t to is the line that passes through the point f to g to h to parallel to v to the curve s velocity vector at to User to and r to to find parametric equations for the line that is tangent to the given curve at the given parameter value t to r t in 7 7 tim 7 x In i What is the standard parametrization for the tangent line X y to 7 Z Type expressions using t as the variable

Factor x 13x 30

The equation r t 3t 1 i 81 6 j 4t k is the position of a particle in space at time t Find the particle s velocity and acceleration vectors Then write the particle s velocity at t 0 as a product of its speed and direction What is the velocity vector v t 3 i 16t j 4 k What is the acceleration vector a t i j k

The tangent line to a smooth curve r t f t i g t j h t k at t to is the line that passes through the point f to g to h to parallel to v to the curve s velocity vector at to Use r to and r to to find parametric equations for the line that is tangent to the given curve at the given parameter value t to r t In X 1 3 1 m K j tln t 5 What is the standard parametrization for the tangent line y Z k to 3

Use a graphing utility to graph the function and estimate its domain and range Then find the domain and range algebraically Enter your answers using interval notation h t 25 2 5 5 5 0 domain range X

The position of a particle in the xy plane at time t is r t t 5 i 1 2 4 j Find an equation in x and y whose graph is the path of the particle Then find the particle s velocity and acceleration vectors at t 4 The equation for the path of the particle is y

The tangent line to a smooth curve r t f t i g t j h t k at t to is the line that passes through the point f to 9 to h to parallel to v to the curve s velocity vector at to User to and r to to parametric equations for the line that is tangent to the given curve at the given parameter value t to r t cos t i What is the standard parametrization for the tangent line X y 12 5 sin t j k to 0 Z

The tangent line to a smooth curve r t f t i g t j h t k at t t is the line that passes through the point f to g to h to parallel to v to the curve s velocity vector at to Use r to and r to to find parametric equations for the line that is tangent to the given curve at the given parameter value t to r t 1 in 6 tim x k What is the standard parametrization for the tangent line X y Z to 3 F Tuno exprossions using t as the variable CIDE

The path r t t i 31 4 j describes motion on the parabola y 3x 4 Find the particle s velocity and acceleration vectors at t 1 and sketch them as vectors on the curve The velocity vector at t 1 is v 1 i j Simplify your answer including any radicals Use integers or fractions for any numbers in the expression

The equation r t 2t i 2t 16t j is the position of a particle in space at time t Find the angle between the velocity and acceleration vectors at time t 0 The angle between the velocity and acceleration vectors at time t 0 is radians Type an exact answer using as needed

1 Particle A moves along a horizontal line with a velocity where is a positive continuous function of t The time t is measured in seconds and the velocity is measured in cm sec The velocity of the particle at selected times is given in the table below t sec v cm sec 0 1 7 2 6 8 5 7 7 4 15 6 Use data from the table to approximate the distance traveled by particle A over the interval 0 ts10 seconds by using a right Riemann sum with four subintervals Show the computations that lead to your answer and indicate units of measure 2 1 3 4 5 2 417 3 f x 141 7cm 2 Let y t represent the population of a town over a 20 year period where y is a differentiable function of t The table below shows the population recorded at selected times t yrs 4 10 13 20 y t people 2500 2724 3108 3697 4283 10 24 9 Use data from the table to find an approximation for y 12 and explain the meaning of y 12 in terms of the population of the town Show the computations that lead to your answer y 12 y 13 y 10 13 10 3697 3108 13 10 Use data from the table and a trapezoidal approximation with four subintervals to approximate the average population of the town over the 20 year period Show the antations that lead to your answer F

Compute the following dx d dx 2x 5 2 1x 5 2x 57 x05 2x 5 2 Simplify your answ

Use the Law of Sines to solve for all possible triangles that satisfy the given conditions If an answer does not exist enter D Round your answers to one decimal place Below enter your answers so that A is smaller than A a 22 c 17 C 22 A 38 3 48 38 3 38 3 b x x 0 0 B b

How much work is required to move an object from x 5 to x 9 measured in meters with a constant force of 6 N acting along the x axis If a constant force F displaces an object a distance d in the direction of the force what is the correct relationship to use to calculate the work done OA workforce distance distance force OB work c work force distance force distance OD work The work required is

The tangent line to a smooth curve r t f t i g t j h t k at t t is the line that passes through the point f to 9 to h to parallel to v to the curve s velocity vector at to Use r to and r to to find parametric equations for the line that is tangent to the given curve at the given paramete value t to r t int i 1 5 tint What is the standard parametrization for the tangent line X y t Int k to 1 Z Type expressions using t as the variable

If each of the following angles were an angle in a triangle how many significant digits would a side in the triangle contain 11 24 13 76 2 4 12 4 3 14 23 45

Give the number of significant digits in each of the following numbers 1 42 3 2 2 05 3 0 76 4 0 84 5 3 5723 6 4 189

Let S x represent the total sales in thousands of dollars for the month x in the year 2005 at a certain department store Represent each statement to the right by an equation involving S or S a Choose the option with equations that best represent a above OA S 7 136 020 S 7 1 7 OC S 7 1 7 S 7 136 020 a The sales at the end of July reached 136 020 and were rising at the rate of 1700 per month b At the end of December the sales for this month dropped to 60 000 and were falling by about 6 day Use 1 month 30 days OB S 7 1 7 S 7 136 020 OD S 7 136 020 S 7 1 7

Set up the problem below and calculate the length of the dime in centime 0 0185 m x 100 m 1 85 Drag and drop your selection from the following list to complete the ans 100 cm 1 cm cm 1 m

erform each of the following calculations 1 72 x 10 1 00 10 8 X 3 62 x 10 4 3 72 x 10 4 5 30 10 5 8 72 10 x 1 00 10 6 53 104 1

many hours are there in exactly 26 d hours answer has bmit Answer significant figures Retry Entire Group

by of your chemistry textbook is found to have a mass of 2360 grams gunit analysis show what the mass of this copy of your chemistry textbook is in gx 2 20 lb and drop your selection from the following list to complete the answer 454 g 237 g OZ 12 oz 28 3 g

The boiling point of a liquid is calculated to be 334 K What is this temperature on the Celsius and Fahre C F