Vector Calculus Questions and Answers

Find the volume of a parallelepiped with one of its eight vertices at A 0 0 0 and three adjacent vertices at B 2 3 0 C 0 4 3 and D 3 5 4 The volume of the parallelepiped is Simplify your answer

Find parametric equations for the line through 2 5 3 parallel to the x axis Choose the correct answer below OA x 2 y 5 t z 3 t t B x 2 t y 5 z 3 t OC x 2 t y 5 z 3 t OD x 2 y 5 t z 3 t t

Classify the graph of the equation as a circle a parabola an ellipse or a hyperbola 9x 25y 15x 250y 543 0 O parabola O ellipse O circle Ohyperbola Submit Answer

Classify the graph of the equation as a circle a parabola an ellipse or a hyperbola x y2 12x 8y 12 0 O parabola O hyperbola O circle O ellipse

Question 10 Find the local minimum and maximum values of the function below f x 3x 12x 108x 6 f x 12x 36x 216x 12x x 6 x 3 From the chart Interval x 3 3 x 0 0 x 6 x 6 12x And f 0 1 T 1 6 x 3 f x T we see that f x changes from negative to positive at 3 so f 3 value by the First Derivative Test Similarly f 2 changes from negative to positive at 6 so f 6 f decreasing on 3 increasing on 3 0 decreasing on 0 6 increasing on 6 is a local minimum is a local minimum value is a local maximum value because f x changes from positive to negative at 0

Use the graph of f to estimate the values of c that satisfy the conclusion of the Mean Value Theorem fo the interval 0 8 Round your answers to the nearest tenth Enter your answers as a comma separated list C 0 y f x X

The graph of the derivative f of a continuous function f is shown Assume the function f is defined only for 0 x 0 y f x 6 a On what interval s is f increasing Enter your answer in interval notation x 8 x On what interval s is f decreasing Enter your answer in interval notation b At what value s of a does f have a local maximum Enter your answers as a comma separated list I At what value s of a does f have a local minimum Enter your answers as a comma separated list c On what interval s is f concave upward Enter your answer in interval notation On what interval s is f concave downward Enter your answer

Find the average value over the given interval f x x x 8 0 16 The average value of f x x x 8 on 0 16 is Type an integer or a simplified fraction

Explain what the shaded area represents Choose the correct answer below OA The average number of kilowatts used in t hours OB The total number of hours it takes to use t kilowatts OC The total number of kilowatts used in t hours O D The average kilowatts per hour in t hours Number of kilowatts

This exercise uses the population growth model The population of the world was 7 1 billion in 2013 and the observed relative growth rate was 1 1 per year a Estimate how long it takes the population to double Round your answer to two decimal places yr b Estimate how long it takes the population to triple Round your answer to two decimal places yr

Let u 5 4 and v 4 6 Find the a component form and b magnitude length of the vector 5 12 13 13 a 5 13 12 13 V

O The matrix is in RRE form O There is a leading entry of a row that is NOT one 1 0 3 0 0 0 0 1 4 O Leading ones DO NOT appear further to the right from row to row O O There is a row of zeros that is NOT at the bottom of the matrix Matrix 4 4 0 1 NOT all other entries in a column with a leading one are zero The matrix is in RRE form 1 3 0 4 004 4 000 0 There is a leading entry of a row that is NOT one

Solve the system below by using Gauss Jordan Elimination This means convert the system to an augmented matrix put that matrix in reduced row echelon RRE form using row operations and determine the solution x y 5z 2 y 4z 1 2x7y 22z 11 a Enter the RRE form of the augmented matrix RRE form of augmented matrix Give the solution to the system If there is a unique solution enter it as an ordered triplet in the form x y z If there is no solution enter Inconsistent If there are free variables enter your answer as an ordered triplet in terms of the free variables

Solve the triangle rounding each nswer to the nearest tenth as needed B a 5 7 b 4 3 59 Y

Write the first five terms of the sequence Assume n begins with 1 an 110 9n a 101 32 92 a3 83 34 74 35 65 Determine whether or not the sequence is arithmetic If it is find the common difference d If the sequence is not arithmetic enter NOR Yes the sequence is arithmetic O No the sequence is not arithmetic X

Find the standard form of the equation of the parabola with the given characteristic s and vertex at the origin Focus 2 0

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin 10 4 0 5 y 10 0 7 2 4 0 5 0 7 2 10 X

Write the first five terms of the sequence Assume n begins with 1 an 110 9n a 32 a3 a4 a5 Determine whether or not the sequence is arithmetic If it is find the common difference d If the sequence is not arithmetic ente O Yes the sequence is arithmetic No the sequence is not arithmetic d

8 Consider the vector whose initial point is P 5 2 and terminal point is Q 1 3 Find the position vector and graph both vectors in the space provided 5

Solve the logarithmic equation for x as in Example 7 Enter your answers as a comma separated list 2 log x log 2 log 5x 8

7 To find the distance between two cities a satellite calculates the distance and the angle shown below not to scale Find the distance between the two cities Round your answer to the nearest tenth 250 km 1 8 210 km

Solve the logarithmic equation for x as in Example 7 Enter your answers as a comma separated list log x log x 3 log 3x X

The graphs below are traces for a function at x If f x y ax by then a 2 110 1 and y 1 and b 1 0 Trace for x 1 2 2 1 0 Trace for y 1 2

Find the solution of the exponential equation as in Example 1 Enter your answers as a comma separated list 22x3 1 X

Find the solution of the exponential equation as in Example 1 Enter your answers as a comma separated list e 3x e4x 3 X

Find the exact real number value of the expression sec sin 1

Problem 7 A chain lying on the ground is 14 ft long and weighs 90 lbs How much work is required to raise one end of the chain to a height of 8ft Problem 7 6619 5010 Enter your answer symbolically as in these examples

A table of values for f g f and g is given f x g x 3 2 1 8 7 2 2 1 2 3 f x 4 57 7 a If h x f g x find h 1 h 1 b If H x g f x find H 1 H 1 g x 6 79

9x 49 441 0 a Find the standard form of the equation of the ellipse b Find the center vertices foci and eccentricity of the ellipse Order your answers from smallest to largest x then from smallest to largest y If the ellipse is degenerate find its center then enter NONE in each center vertices foci eccentricity x y x y c Select the correct graph of the ellipse Use a graphing utility to verify your answer y 10 10 10 10 10 10 10 O

Find the center vertices foci and eccentricity of the ellipse Order your answers from smallest to largest x then from smallest to large y 5 x 5 1 1 4 Center Vertices Foci Eccentricity Select the correct graph of the ellipse Use a graphing utility to verify your answer 10 4 10 6 4 fo

Find the standard form of the equation of the ellipse with the given characteristics y 1 3 3 5 2 5 4 7 3

In Silicon Valley a number of computer firms were found to be contaminating underground water supplies with toxic chemicals stored in leaking underground containers A water quality control agency ordered the companies to take immediate corrective action and contribute to a monetary pool for the testing and cleanup of the underground contamination Suppose that the required monetary pool in millions of dollars is given by the following function where x is the percentage expressed as a decimal fraction of the total contaminant removed Complete parts A through E P x 3x 1 x 0 x 1 A Where is P x increasing Decreasing Select the correct choice below and fill in the answer box es to complete your choice Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate answers as needed OA P x is increasing on OB P x is increasing on OC P x is decreasing on and decreasing on B Where is the graph of P concave upward Downward Select the correct choice below and fill in the answer box es to complete your choice Type your answer in interval notation Use integers or fractions for any numbers in the expression Use a comma to separate pooded

A car rental agency rents 220 cars per day at a rate of 40 per day For each 1 increase in rate 5 fewer cars are rented At what rate should the cars be rented to produce the maximum income What is the maximum income The rental agency will eam a maximum income of MILD when it charges per day

A company manufactures a variety of camp stoves at different locations The total cost C x in dollars of producing x camp stoves per week at plant A is shown in the figure Discuss the graph of the marginal cost function C x and interpret the graph of C x in terms of the efficiency of the production process at this plant www C x 100 000 50 000 500 1 000 Choose the correct answer below OA The graph of y C x shows that the marginal cost function is positive and decreasing Since marginal costs are decreasin the production process is becoming more efficient as production increases OB The graph of y C x shows that the marginal cost function is negative and increasing Since marginal costs are increasing the production process is becoming less efficient as production increases OC The graph of y C x shows that the marginal cost function is positive and increasing Since marginal costs are increasing the production process is becoming less efficient as production increases OD The graph of y C x shows that the marginal cost function is negative and decreasing Since marginal costs are decreasin the production process is becoming more efficient as production increases

Drag the choices above into the spaces below A B Q D AY Q U E Ay AY U C F Ay www O 57

4 Square ABCD is drawn as shown below with the diagonals intersecting at E B A D a State four pairs of equivalent vectors b State four pairs of opposite vectors c State two pairs of vectors whose magnitudes are equal but whose directions are perpendicular to each other

Part 1 of 2 A particle travels clockwise on a circular path of diameter R monitored by a sensor on the circle at point P the other endpoint of the diameter on which the sensor lies is Q Let 8 be the angle between the diameter PQ and the line from the sensor to the particle Let c be the length of the chord from the particle s position to Q Complete parts a and b below a Calculate de dc ACIER HW Score 30 43 7 of 23 points O Points 0 of 1 Sensor Particle

5 Each of the following vector expressions can be simplified and written as a single vector Write the single vector corresponding to each expression and illustrate your answer with a sketch a PORO RS b PS RQ RS PO

1 The vectors and y are drawn as shown below Draw a vector equivalent to each of the following X

Recall the following definition Let 0 be an angle in standard position The reference angle of 0 is the acute angle formed by the terminal side of 0 and the horizontal axis Below express the reference angle for 0 in the same units degrees or radians as 0 The reference angle for 254 is The reference angle for 342 is The reference angle for 220 is The reference angle for 429 is degrees degrees degrees degrees

Find an angle between 0 and 27 that is coterminal with the given angle is coterminal with 2 is coterminal with 3 47 is coterminal with 2 1 19T 5 4 19 is coterminal with 11 F A FI

A boat is towed toward a dock by a cable attached to a winch that stands 10 ft above the water level see figure Let 8 be the angle of elevation of the winch and let L be the length of the cable as the boat is towed toward the dock Complete parts a to c below b Compute de dL When L 50 ft When L 20 ft when L 50 20 and 11 ft de dL de dL 10 ft 0 0041 Round to four decimal places as needed Round to four decimal places as needed

Consider f x 2 x 1 sin a Graph f with a graphing utility b Compute and graph f c Verify that the zeros of f correspond to points at which f has a horizontal tangent line a Each of the graphs below is shown in a 1 1 0 2 by 2 8 1 viewing window Which is the graph of f OA OB OC oo x on the interval 1 1 S OD A

K ctly toward a 150 m skyscraper that stands on the edge of a harbor The angular size of the building is the angle formed by lines from the top and bottom of the building to the observer see figure Complete parts a and b below a Use a trigonometric equation to find an expression for the angle 0 in terms of x COS X 150 Type an equation 150 m

7 A trough has a trapezoidal cross section with a height of 7 m and horizontal sides of width 2 m and 7 m Assume the length of the trough is 14 m See the figure to the right Complete parts a and b below OA Sdx B 7 S 68600 7 y dy 0 7m Simplify your answer Use integers or fractions for any numbers in the expression Set up the integral that gives the work required to pump the water out of the tank Use increasing limits of integration Select the correct choice below and fill in the answer boxes to complete your choice Type exact answers of work is required to pump the water out of the trough 7m 3 5 m 14 m Litho math palette as needed Round to six decimal places as needed

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Given the following intersecting lines how can angles 4 and 6 be classified Show your work here Vertical Angles Adjacent Angles None of the above 1 6 5 2 3 Supplementary Angles Complementary Angles

Find the area of the figure The figure is not drawn to scale 14 in 6 in 16 in 7 in

Multiply the two polynomials x 2x 3 x 2x x 4 What is the resulting expression work here

Find the product and simplify 8 2a 184 2a814 1 Show your work here Hint To add an exponent a type exponent or press A