Vector Calculus Questions and Answers

A machinist is required to manufacture a circular metal disk with area 1200 cm a What radius produces such a disk Round your answer to four decimal places b If the machinist is allowed an error tolerance of 18 cm in the area of the disk how close to the ideal radius in part a must the machinist control the radius Round your answers to four decimal places cm r cm c In terms of the s definition of lim f x L what is x x a cm Oarea Otarget radius radius target area Otolerance in the area What is f x Oarea O target radius radius O target area O tolerance in the area What is a area target radius radius O target area Otolerance in the area What is L area target radius radius

Given that lim 4x 2 6 illustrate this definition by finding the largest values of that correspond to 0 5 0 1 and X 2 0 5 0 1 E 0 05 Need Help Submit Answer 8 S 8 S 8 Read It 1 DETAILS Master It 10 0 23 Points How close to 2 do we have to take x so that the following is satisfied Give the largest possible value 10 000 SCALCET9 2 4 041 MI

Values of f x y are shown in the table below 313 3 3 6 3 4 8 10 13 y 4 5 10 12 16 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 Az 0 3 and Ay 0 5 over estimate 1 under estimate

Use a table of values to estimate the value of the limit If you have a graphing device use it to confirm your result graphically tan 9x lim x 0 tan 4x Need Help Submit Answer 20 0 1 Points Read It lim DETAILS Determine the infinite limit x 6 Master It SCALCET9 2 XP 2 012

a Estimate the value of lim x x 5 x by graphing the function f x x x 5 x Round your answer to one decimal place b Use a table of values of f x to guess the value of the limit Round your answer to one decimal place c Prove that your guess is correct by evaluating lim f x X 00

Find the horizontal and vertical asymptotes of the curve You may want to use a graphing calculator or computer to check your work by graphing the curve and estimating the asymptotes Enter your answers as comma separated lists If an answer does not exist enter DNE y 4x x 2 x x 6

Evaluate the iterated integral 2x y dydx

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

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

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

Akilah flies a drone in a circular path around an object that is 240 meters north and 180 meters west of her position The drone s path takes it over a point that is 250 meters west and 190 meters south of her Find an equation for the drone s path Assume Akilah is located at the origin with the horizontal axis running east west and the vertical axis running north south The drone s path follows the equation

me function to the appropriate area below Square function Constant function Reciprocal function Square root function Absolute value function Identity function Cube root function Cube function

For each table below create a table for f x 18 x 2 4 8 12 18 f x 3 5 9 14 20

Determine whether the statement is true or false If the line 7 is a vertical asymptote of y f x then f x is not defined at 7 O False O True

In March 2014 a gas company had the rate schedule shown to the right for natural gas usage in single family residences a What is the charge for using 30 therms in a month b What is the charge for using 200 therms in a month c Develop a function that models the monthly charge C for x therms of gas d Graph the function found in part c a The charge for using 30 therms in a month is b The charge for using 200 therms in a month is c C x if 0 x if x Simplify your answers Monthly service charge Delivery charge First 40 therms Over 40 therms Natural gas cost First 40 therms Over 40 therms Simplify your answer Round to the nearest cent Simplify your answer Round to the nearest cent 18 5 0 70202 therm 0 0 3049 therm 0 5637 therm

Find the exact length of the curve of r t sint cost In cost for 0 t 7 7 points

C A e AL IlaInst 3 Above are 6 surfaces A F Match them with the following equations HINT Think about what the traces would look like 6 points a y z 0 b 2x 3y z 5 4x 2 z 1 D x 2 2 9 F d x z 1 f y x

BONUS 3 Find a vector in the direction of 4 8 8 with a length of 10 Must show work to receive credit

5 Given r t 10t 1 t for o t o find a vector orthogonal to r and r 7 points

a Find the critical point of f x y 2x y z 4y x y 6x b Find the discriminant D x y for the function D x y 0 what is the value of the discriminant at the critical point D 48 0 14 0 8 14 0 What does this tell us about the critical point The discriminant doesn t tell us anything c Sketch contours near the critical point to determine or confirm whether it is a local maximum a local minimum a saddle point or none of th The critical point is a saddle point

Adderall is a proprietary combination of amphetamine salts that is used to treat ADHD Attention Deficit Hyperactivity Disorder The patient takes one pill before 8am and the drug has completely entered the bloodstream by 8am At 8am the blood concentration of the drug is 33 8 ng ml Adderall has first order elimination kinetics 8 with 7 7 of the drug being removed from the blood each hour I a The concentration of the drug in the patient s blood t hours after 8am is measured to be Ct Write a recursive relation of C 1 in terms of Ct Assume that the patience takes no other Adderall pills What happens in the long run to the amount of Adderall in the system b Solve your recurrence equation to derive an explicit formula for Ct as a function of t When does the concentration of drug first drop below 0 1 ng ml

2 7 pts Equation of a Line Write an equation of the line passing through the points 1 2 and 4 10

Problem Set 1 In the figure to the right AB IDE and BC EF Prove that m ABC mZDEF Evidence Claims 1 ABIIDE BC IEF 2 Draw EZ label z 3 m ABC m E2C 4 m EZC M DEF m 5 mx ABC M4 DEF 1 AB II CO 2 Draw FE label w vo 1 Given 2 Construction 3 4 2 In the figure to the right AB CD Prove that mZAEC a c 3 a w Co 4 w m AEC 5 a c m AEC 5 1 Given 2 Construction 3 linear 4 Angle addition 5 substitution a B A D wo Ivo E C Co

Determine the global extreme values of the function f x y 4x 4x y 2y x y 20 x y fmin

Calculate the directional derivative of f x y z y in the direction of vi 2j at the point P 2 2 Remember to normalize the direction vector Duf 2 2 12

Calculate the directional derivative of f z y sin y in the direction of v 1 23 at the point P 0 5 0 5 Remember to normalize the direction vector D f 0 5 0 5

Find the x and y intercepts of the graph of the equation If an answer does not exist enter DNE 9x 4y 36 x intercept x intercept x y y intercept x y y intercept x y x y b x xy 5y 81 x intercept x y x intercept x y y intercept x y smaller x value larger x value smaller y value larger y value smaller x value larger x value

Find the gradient of the function f x y z xy at the point 1 1 0 Vf 1 1 0

Calculate the gradient of h x y z x 2 Vh

3 10 pts Calculate the difference quotient of f x 7x 5x 9

2 Write the equation of the following line in 4x 3y 8 a Slope intercept form b Point slope form

The point P 4 4 lies on the curve y a If Q is the point 1 3 9 mpqQ ii 3 99 mpQ iii 3 999 mpQ iv 3 9999 MpQ 1 v 4 1 MpQ vi 4 01 m mpQ vii 4 001 mpo viii 4 0001 mpo 4 3 x find the slope of the secant line PQ correct to six decimal places for the following values of x b Using the results of part a guess the value of the slope of the tangent line to the curve at P 4 4 c Using the slope from part b find an equation of the tangent line to the curve at P 4 4

A ball is thrown into the air by a baby alien on a planet in the system of Alpha Centauri with a velocity of 46 ft s Its height in feet after t seconds is given by y 46t 17t A Find the average velocity for the time period beginning when t 1 and lasting for the following number of seconds 01 sec 1 01 X Hint If the throw starts at t 1 and lasts 01 sec what will the ending time t2 be 005 sec 0 005 X 002 sec 46 4 001 sec 0 001 X NOTE For the above answers you may have to enter 6 or 7 significant digits if you are using a calculator Estimate the instanteneous velocity when t 1 12 X

Given f x 1 find the average rate of change of f x on the interval 5 5 h Your answer will x 8 be an expression involving h

8 Let n x x 4 The average rate of change of n x over the interval c 5 is equal to 3 where c is a constant Find the value of c 70 navig ont 25 no

17 1 point The two quadratic functions have the same vertex f x 3 x 2 3 f x x 4x 1 True False

Let Horizontal asymptotes y f x Find the equations of the horizontal asymptotes and the vertical asymptotes of f x If there are no asymptotes of a given type enter NONE If there is more than one asymptote of a given type give a comma seperated list i e 1 2 Vertical Asymptotes x x 3x 4 3x 11x 4

Draw the graph of the function y f x that satisfies the following properties Both the function and the slope decrease as x increases Select all graphs that satisfy both properties A X G G C B X C C X G D X

Let f x y be a differentiable function with f x y c a level curve of f Which of the following statements are true Select all that apply Du f zo yo is maximum when u is tangent to the level curve at zo yo Vf zo yo is orthogonal to the level curve at xo 30 Duf xo 30 0 when u is tangent to the level curve at zo yo Vf zo yo is tangent to the level curve at x0 90 f xo 30 c at every point 2o 3o along the level curve

A line is shown on the zy coordinate plane If 14 k is a point on this line find the value of k to a of 25

eet 2 12 pts Test the following equation for symmetry Show and label each symmetry test Make sure that you state any types of symmetry that the graph of the equation exhibits x 2y 3y4

3 e Bonus 1 35 x 10 14 mm to m

2 Convert the following numbers to decimal notation 2 points a 3 66 x 105 b 1 06 x 10 7

What is the velocity in ft s of a sandbag 2 5 s after it is released from a hot air balloon that is stationary in the The velocity is ft s

The sum of the interior angles of a polygon with n sides is given by the formula 180 n 2 To solve for n we set up the equation 180 n 2 2340 and solve for n The correct answer is option C sides

When Abdoulaye commutes to work the amount of time it takes him to arrive is normally distributed with a mean of 45 minutes and a standard deviation of 5 minutes Using the empirical rule what percentage of his commutes will be between 35 and 55 minutes

Find the surface area of the rectangular prism 2 in 4 in

e table displays the scores of students on a recent exam Find the mean of the scores to the nearest 10th Score Number of Students 60 9 65 8 70 4 75 1 80 6 85 3 90 6

Kayla is organizing textbooks on her bookshelf She has an English textbook a biology textbook and a history textbook How ma different ways can she line the textbooks up on her bookshelf

f x 40 10 9 Step 1 Select a function then move with blue dot if necessary done v V 8 x 2 7 6 5 4 3 2 1 10 9 00 10 7 6 5 4 3 2 1 1 2 6 3 4 5 y 1 for x 4 for x 5 2 3 4 5 6 7 8 9 10